Theorem axhvcom-zf

Description: Derive Axiom ax-hvcom from Hilbert space under ZF set theory. (Contributed by NM, 27-May-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	axhil.1	⊢ 𝑈 = ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩
	axhil.2	⊢ 𝑈 ∈ CHilOLD
Assertion	axhvcom-zf	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 +ℎ 𝐵 ) = ( 𝐵 +ℎ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		axhil.1	⊢ 𝑈 = ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩
2		axhil.2	⊢ 𝑈 ∈ CHilOLD
3		df-hba	⊢ ℋ = ( BaseSet ‘ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
4	1	fveq2i	⊢ ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
5	3 4	eqtr4i	⊢ ℋ = ( BaseSet ‘ 𝑈 )
6	2	hlnvi	⊢ 𝑈 ∈ NrmCVec
7	1 6	h2hva	⊢ +ℎ = ( +𝑣 ‘ 𝑈 )
8	5 7	hlcom	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 +ℎ 𝐵 ) = ( 𝐵 +ℎ 𝐴 ) )
9	2 8	mp3an1	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 +ℎ 𝐵 ) = ( 𝐵 +ℎ 𝐴 ) )