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Theorem axhvmulid-zf 30840
Description: Derive Axiom ax-hvmulid 30858 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 30821 . . . 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 2756 . . 3 β„‹ = (BaseSetβ€˜π‘ˆ)
61hlnvi 30744 . . . 4 π‘ˆ ∈ NrmCVec
73, 6h2hsm 30827 . . 3 Β·β„Ž = ( ·𝑠OLD β€˜π‘ˆ)
85, 7hlmulid 30757 . 2 ((π‘ˆ ∈ CHilOLD ∧ 𝐴 ∈ β„‹) β†’ (1 Β·β„Ž 𝐴) = 𝐴)
91, 8mpan 688 1 (𝐴 ∈ β„‹ β†’ (1 Β·β„Ž 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4630  β€˜cfv 6542  (class class class)co 7415  1c1 11137  BaseSetcba 30438  CHilOLDchlo 30737   β„‹chba 30771   +β„Ž cva 30772   Β·β„Ž csm 30773  normβ„Žcno 30775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  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 7418  df-oprab 7419  df-1st 7989  df-2nd 7990  df-vc 30411  df-nv 30444  df-va 30447  df-ba 30448  df-sm 30449  df-0v 30450  df-nmcv 30452  df-cbn 30715  df-hlo 30738  df-hba 30821
This theorem is referenced by: (None)
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