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Theorem axhvdistr1-zf 30243
Description: Derive Axiom ax-hvdistr1 30261 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
axhvdistr1-zf ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 Β·β„Ž (𝐡 +β„Ž 𝐢)) = ((𝐴 Β·β„Ž 𝐡) +β„Ž (𝐴 Β·β„Ž 𝐢)))
Proof of Theorem axhvdistr1-zf
StepHypRef Expression
1 axhil.2 . 2 π‘ˆ ∈ CHilOLD
2 df-hba 30222 . . . 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 2764 . . 3 β„‹ = (BaseSetβ€˜π‘ˆ)
61hlnvi 30145 . . . 4 π‘ˆ ∈ NrmCVec
73, 6h2hva 30227 . . 3 +β„Ž = ( +𝑣 β€˜π‘ˆ)
83, 6h2hsm 30228 . . 3 Β·β„Ž = ( ·𝑠OLD β€˜π‘ˆ)
95, 7, 8hldi 30160 . 2 ((π‘ˆ ∈ CHilOLD ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹)) β†’ (𝐴 Β·β„Ž (𝐡 +β„Ž 𝐢)) = ((𝐴 Β·β„Ž 𝐡) +β„Ž (𝐴 Β·β„Ž 𝐢)))
101, 9mpan 689 1 ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 Β·β„Ž (𝐡 +β„Ž 𝐢)) = ((𝐴 Β·β„Ž 𝐡) +β„Ž (𝐴 Β·β„Ž 𝐢)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βŸ¨cop 4635  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108  BaseSetcba 29839  CHilOLDchlo 30138   β„‹chba 30172   +β„Ž cva 30173   Β·β„Ž csm 30174  normβ„Žcno 30176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  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 7412  df-oprab 7413  df-1st 7975  df-2nd 7976  df-vc 29812  df-nv 29845  df-va 29848  df-ba 29849  df-sm 29850  df-0v 29851  df-nmcv 29853  df-cbn 30116  df-hlo 30139  df-hba 30222
This theorem is referenced by: (None)
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