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Theorem axhvdistr2-zf 29982
Description: Derive Axiom ax-hvdistr2 30000 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
axhvdistr2-zf ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 + 𝐡) Β·β„Ž 𝐢) = ((𝐴 Β·β„Ž 𝐢) +β„Ž (𝐡 Β·β„Ž 𝐢)))

Proof of Theorem axhvdistr2-zf
StepHypRef Expression
1 axhil.2 . 2 π‘ˆ ∈ CHilOLD
2 df-hba 29960 . . . 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 6849 . . . 4 (BaseSetβ€˜π‘ˆ) = (BaseSetβ€˜βŸ¨βŸ¨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©)
52, 4eqtr4i 2764 . . 3 β„‹ = (BaseSetβ€˜π‘ˆ)
61hlnvi 29883 . . . 4 π‘ˆ ∈ NrmCVec
73, 6h2hva 29965 . . 3 +β„Ž = ( +𝑣 β€˜π‘ˆ)
83, 6h2hsm 29966 . . 3 Β·β„Ž = ( ·𝑠OLD β€˜π‘ˆ)
95, 7, 8hldir 29899 . 2 ((π‘ˆ ∈ CHilOLD ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹)) β†’ ((𝐴 + 𝐡) Β·β„Ž 𝐢) = ((𝐴 Β·β„Ž 𝐢) +β„Ž (𝐡 Β·β„Ž 𝐢)))
101, 9mpan 689 1 ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 + 𝐡) Β·β„Ž 𝐢) = ((𝐴 Β·β„Ž 𝐢) +β„Ž (𝐡 Β·β„Ž 𝐢)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βŸ¨cop 4596  β€˜cfv 6500  (class class class)co 7361  β„‚cc 11057   + caddc 11062  BaseSetcba 29577  CHilOLDchlo 29876   β„‹chba 29910   +β„Ž cva 29911   Β·β„Ž csm 29912  normβ„Žcno 29914
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 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-1st 7925  df-2nd 7926  df-vc 29550  df-nv 29583  df-va 29586  df-ba 29587  df-sm 29588  df-0v 29589  df-nmcv 29591  df-cbn 29854  df-hlo 29877  df-hba 29960
This theorem is referenced by: (None)
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