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Theorem axhvmulid-zf 30772
Description: Derive Axiom ax-hvmulid 30790 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
axhil.1 π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©
axhil.2 π‘ˆ ∈ CHilOLD
Assertion
Ref Expression
axhvmulid-zf (𝐴 ∈ β„‹ β†’ (1 Β·β„Ž 𝐴) = 𝐴)
Proof of Theorem axhvmulid-zf
StepHypRef Expression
1 axhil.2 . 2 π‘ˆ ∈ CHilOLD
2 df-hba 30753 . . . 4 β„‹ = (BaseSetβ€˜βŸ¨βŸ¨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©)
3 axhil.1 . . . . 5 π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©
43fveq2i 6894 . . . 4 (BaseSetβ€˜π‘ˆ) = (BaseSetβ€˜βŸ¨βŸ¨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©)
52, 4eqtr4i 2758 . . 3 β„‹ = (BaseSetβ€˜π‘ˆ)
61hlnvi 30676 . . . 4 π‘ˆ ∈ NrmCVec
73, 6h2hsm 30759 . . 3 Β·β„Ž = ( ·𝑠OLD β€˜π‘ˆ)
85, 7hlmulid 30689 . 2 ((π‘ˆ ∈ CHilOLD ∧ 𝐴 ∈ β„‹) β†’ (1 Β·β„Ž 𝐴) = 𝐴)
91, 8mpan 689 1 (𝐴 ∈ β„‹ β†’ (1 Β·β„Ž 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  βŸ¨cop 4630  β€˜cfv 6542  (class class class)co 7414  1c1 11125  BaseSetcba 30370  CHilOLDchlo 30669   β„‹chba 30703   +β„Ž cva 30704   Β·β„Ž csm 30705  normβ„Žcno 30707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-1st 7985  df-2nd 7986  df-vc 30343  df-nv 30376  df-va 30379  df-ba 30380  df-sm 30381  df-0v 30382  df-nmcv 30384  df-cbn 30647  df-hlo 30670  df-hba 30753
This theorem is referenced by: (None)
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