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Theorem axhvmulass-zf 30506
Description: Derive Axiom ax-hvmulass 30524 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
axhvmulass-zf ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 Β· 𝐡) Β·β„Ž 𝐢) = (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢)))
Proof of Theorem axhvmulass-zf
StepHypRef Expression
1 axhil.2 . 2 π‘ˆ ∈ CHilOLD
2 df-hba 30486 . . . 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 6895 . . . 4 (BaseSetβ€˜π‘ˆ) = (BaseSetβ€˜βŸ¨βŸ¨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©)
52, 4eqtr4i 2762 . . 3 β„‹ = (BaseSetβ€˜π‘ˆ)
61hlnvi 30409 . . . 4 π‘ˆ ∈ NrmCVec
73, 6h2hsm 30492 . . 3 Β·β„Ž = ( ·𝑠OLD β€˜π‘ˆ)
85, 7hlmulass 30423 . 2 ((π‘ˆ ∈ CHilOLD ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹)) β†’ ((𝐴 Β· 𝐡) Β·β„Ž 𝐢) = (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢)))
91, 8mpan 687 1 ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 Β· 𝐡) Β·β„Ž 𝐢) = (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βŸ¨cop 4635  β€˜cfv 6544  (class class class)co 7412  β„‚cc 11111   Β· cmul 11118  BaseSetcba 30103  CHilOLDchlo 30402   β„‹chba 30436   +β„Ž cva 30437   Β·β„Ž csm 30438  normβ„Žcno 30440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-1st 7978  df-2nd 7979  df-vc 30076  df-nv 30109  df-va 30112  df-ba 30113  df-sm 30114  df-0v 30115  df-nmcv 30117  df-cbn 30380  df-hlo 30403  df-hba 30486
This theorem is referenced by: (None)
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