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Theorem nvdi 27469
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 2621 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 27454 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 nvdi.2 . . . 4 𝐺 = ( +𝑣𝑈)
43vafval 27442 . . 3 𝐺 = (1st ‘(1st𝑈))
5 nvdi.4 . . . 4 𝑆 = ( ·𝑠OLD𝑈)
65smfval 27444 . . 3 𝑆 = (2nd ‘(1st𝑈))
7 nvdi.1 . . . 4 𝑋 = (BaseSet‘𝑈)
87, 3bafval 27443 . . 3 𝑋 = ran 𝐺
94, 6, 8vcdi 27404 . 2 (((1st𝑈) ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))
102, 9sylan 488 1 ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037   = wceq 1482  wcel 1989  cfv 5886  (class class class)co 6647  1st c1st 7163  cc 9931  CVecOLDcvc 27397  NrmCVeccnv 27423   +𝑣 cpv 27424  BaseSetcba 27425   ·𝑠OLD cns 27426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-1st 7165  df-2nd 7166  df-vc 27398  df-nv 27431  df-va 27434  df-ba 27435  df-sm 27436  df-0v 27437  df-nmcv 27439
This theorem is referenced by:  nvmdi  27487  nvaddsub4  27496  nvdif  27505  nvpi  27506  ipdirilem  27668  hldi  27747
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