Theorem axhvmulid-zf 28767

Description: Derive axiom ax-hvmulid 28785 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 28748	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
4	3	fveq2i 6675	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
5	2, 4	eqtr4i 2849	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 28671	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hsm 28754	. . 3 ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈)
8	5, 7	hlmulid 28684	. 2 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ ℋ) → (1 ·ℎ 𝐴) = 𝐴)
9	1, 8	mpan 688	1 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ⟨cop 4575  ‘cfv 6357  (class class class)co 7158  1c1 10540  BaseSetcba 28365  CHil_OLDchlo 28664   ℋchba 28698   +_ℎ cva 28699   ·_ℎ csm 28700  norm_ℎcno 28702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-1st 7691  df-2nd 7692  df-vc 28338  df-nv 28371  df-va 28374  df-ba 28375  df-sm 28376  df-0v 28377  df-nmcv 28379  df-cbn 28642  df-hlo 28665  df-hba 28748

This theorem is referenced by: (None)