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Theorem axhvmulid-zf 29816
Description: Derive Axiom ax-hvmulid 29834 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
axhil.1 𝑈 = ⟨⟨ + , · ⟩, norm
axhil.2 𝑈 ∈ CHilOLD
Assertion
Ref Expression
axhvmulid-zf (𝐴 ∈ ℋ → (1 · 𝐴) = 𝐴)

Proof of Theorem axhvmulid-zf
StepHypRef Expression
1 axhil.2 . 2 𝑈 ∈ CHilOLD
2 df-hba 29797 . . . 4 ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
3 axhil.1 . . . . 5 𝑈 = ⟨⟨ + , · ⟩, norm
43fveq2i 6842 . . . 4 (BaseSet‘𝑈) = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
52, 4eqtr4i 2767 . . 3 ℋ = (BaseSet‘𝑈)
61hlnvi 29720 . . . 4 𝑈 ∈ NrmCVec
73, 6h2hsm 29803 . . 3 · = ( ·𝑠OLD𝑈)
85, 7hlmulid 29733 . 2 ((𝑈 ∈ CHilOLD𝐴 ∈ ℋ) → (1 · 𝐴) = 𝐴)
91, 8mpan 688 1 (𝐴 ∈ ℋ → (1 · 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cop 4590  cfv 6493  (class class class)co 7353  1c1 11048  BaseSetcba 29414  CHilOLDchlo 29713  chba 29747   + cva 29748   · csm 29749  normcno 29751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7356  df-oprab 7357  df-1st 7917  df-2nd 7918  df-vc 29387  df-nv 29420  df-va 29423  df-ba 29424  df-sm 29425  df-0v 29426  df-nmcv 29428  df-cbn 29691  df-hlo 29714  df-hba 29797
This theorem is referenced by: (None)
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