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Theorem axhvmulid-zf 31017
Description: Derive Axiom ax-hvmulid 31035 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 30998 . . . 4 ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
3 axhil.1 . . . . 5 𝑈 = ⟨⟨ + , · ⟩, norm
43fveq2i 6910 . . . 4 (BaseSet‘𝑈) = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
52, 4eqtr4i 2766 . . 3 ℋ = (BaseSet‘𝑈)
61hlnvi 30921 . . . 4 𝑈 ∈ NrmCVec
73, 6h2hsm 31004 . . 3 · = ( ·𝑠OLD𝑈)
85, 7hlmulid 30934 . 2 ((𝑈 ∈ CHilOLD𝐴 ∈ ℋ) → (1 · 𝐴) = 𝐴)
91, 8mpan 690 1 (𝐴 ∈ ℋ → (1 · 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cop 4637  cfv 6563  (class class class)co 7431  1c1 11154  BaseSetcba 30615  CHilOLDchlo 30914  chba 30948   + cva 30949   · csm 30950  normcno 30952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-1st 8013  df-2nd 8014  df-vc 30588  df-nv 30621  df-va 30624  df-ba 30625  df-sm 30626  df-0v 30627  df-nmcv 30629  df-cbn 30892  df-hlo 30915  df-hba 30998
This theorem is referenced by: (None)
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