Theorem axhvmulid-zf 29329

Description: Derive Axiom ax-hvmulid 29347 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 29310	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
4	3	fveq2i 6771	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
5	2, 4	eqtr4i 2770	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 29233	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hsm 29316	. . 3 ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈)
8	5, 7	hlmulid 29246	. 2 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ ℋ) → (1 ·ℎ 𝐴) = 𝐴)
9	1, 8	mpan 686	1 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2109  ⟨cop 4572  ‘cfv 6430  (class class class)co 7268  1c1 10856  BaseSetcba 28927  CHil_OLDchlo 29226   ℋchba 29260   +_ℎ cva 29261   ·_ℎ csm 29262  norm_ℎcno 29264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-1st 7817  df-2nd 7818  df-vc 28900  df-nv 28933  df-va 28936  df-ba 28937  df-sm 28938  df-0v 28939  df-nmcv 28941  df-cbn 29204  df-hlo 29227  df-hba 29310

This theorem is referenced by: (None)