Theorem axhvmulid-zf 31008

Description: Derive Axiom ax-hvmulid 31026 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
axhil.1	⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
axhil.2	⊢ 𝑈 ∈ CHilOLD

Assertion

Ref	Expression
axhvmulid-zf	⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴)

Proof of Theorem axhvmulid-zf

Step	Hyp	Ref	Expression
1		axhil.2	. 2 ⊢ 𝑈 ∈ CHilOLD
2		df-hba 30989	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
4	3	fveq2i 6908	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
5	2, 4	eqtr4i 2767	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 30912	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hsm 30995	. . 3 ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈)
8	5, 7	hlmulid 30925	. 2 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ ℋ) → (1 ·ℎ 𝐴) = 𝐴)
9	1, 8	mpan 690	1 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ⟨cop 4631  ‘cfv 6560  (class class class)co 7432  1c1 11157  BaseSetcba 30606  CHil_OLDchlo 30905   ℋchba 30939   +_ℎ cva 30940   ·_ℎ csm 30941  norm_ℎcno 30943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-1st 8015  df-2nd 8016  df-vc 30579  df-nv 30612  df-va 30615  df-ba 30616  df-sm 30617  df-0v 30618  df-nmcv 30620  df-cbn 30883  df-hlo 30906  df-hba 30989

This theorem is referenced by: (None)