Understanding Proofs in Spivak's Calculus: Properties P1-P3 Explained

[chemistry1]

I'm studying Spivak's calculus and I have a really simple question :

I'm only in the first chapter on "The basic properties of numbers"

So far, we have the following propostion

P1 : (a+b)+c=a+(b+c)

P3 : a+(-a)=(-a)+a=0

P2 : a+0=0+a=a

Now, he tries to prove P2 (He doesn't do it for P3, so it's granted) He also says :

"The proof of this assertion involves nothing more than subtracting a from both sides of the equation, in other word, adding -a to both sides." Now, that I understand

"as the following detailled proof shows, all three properties P1-P3 must be used to justify this operation." That I don't understand. First, how can you use a proof of something you haven't proven ? Second, when he says all three properties to justify this operation, he means to substract "a" from both sides, right ? If so, I don't understand how they (properties) can be used ...

He starts with this :

If a+x=a

then (-a)+(a+x)=(-a)+a=0

hence ((-a)+a)+x=0

hence 0+x=0

hence x=0

My comments : For the first line, he starts with the assertion that an equation a+x=a exists. Now, he substract "a" from borth sides and with property 3 the right hand sides equals 0. With property 1 we regroup and cancel with property 3.Now we have 0+x=0 and we subtract zero from both sides to have x=0. Where is property 2 used ? How is subtracting "a" from both sides proven with all three properties ?

Thank you

 

[HallsofIvy]

I think you are mistaken when you say this is a proof of P2. He starts with "a+ x= a" and concludes "x= 0". That is NOT "P2". It is a separate theorem completely. His proof uses P1, P2, and P3.

 

[chemistry1]

Yeah, I didn't understand it in the correct way. Thank you !