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Theorem nvdi 27825
 Description: Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvdi.1 𝑋 = (BaseSet‘𝑈)
nvdi.2 𝐺 = ( +𝑣𝑈)
nvdi.4 𝑆 = ( ·𝑠OLD𝑈)
Assertion
Ref Expression
nvdi ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))

Proof of Theorem nvdi
StepHypRef Expression
1 eqid 2771 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 27810 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 nvdi.2 . . . 4 𝐺 = ( +𝑣𝑈)
43vafval 27798 . . 3 𝐺 = (1st ‘(1st𝑈))
5 nvdi.4 . . . 4 𝑆 = ( ·𝑠OLD𝑈)
65smfval 27800 . . 3 𝑆 = (2nd ‘(1st𝑈))
7 nvdi.1 . . . 4 𝑋 = (BaseSet‘𝑈)
87, 3bafval 27799 . . 3 𝑋 = ran 𝐺
94, 6, 8vcdi 27760 . 2 (((1st𝑈) ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))
102, 9sylan 569 1 ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ‘cfv 6031  (class class class)co 6793  1st c1st 7313  ℂcc 10136  CVecOLDcvc 27753  NrmCVeccnv 27779   +𝑣 cpv 27780  BaseSetcba 27781   ·𝑠OLD cns 27782 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-1st 7315  df-2nd 7316  df-vc 27754  df-nv 27787  df-va 27790  df-ba 27791  df-sm 27792  df-0v 27793  df-nmcv 27795 This theorem is referenced by:  nvmdi  27843  nvaddsub4  27852  nvdif  27861  nvpi  27862  ipdirilem  28024  hldi  28103
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