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Theorem nvdir 28004
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
nvdir ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))

Proof of Theorem nvdir
StepHypRef Expression
1 eqid 2798 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 27988 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 nvdi.2 . . . 4 𝐺 = ( +𝑣𝑈)
43vafval 27976 . . 3 𝐺 = (1st ‘(1st𝑈))
5 nvdi.4 . . . 4 𝑆 = ( ·𝑠OLD𝑈)
65smfval 27978 . . 3 𝑆 = (2nd ‘(1st𝑈))
7 nvdi.1 . . . 4 𝑋 = (BaseSet‘𝑈)
87, 3bafval 27977 . . 3 𝑋 = ran 𝐺
94, 6, 8vcdir 27939 . 2 (((1st𝑈) ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))
102, 9sylan 576 1 ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  cfv 6100  (class class class)co 6877  1st c1st 7398  cc 10221   + caddc 10226  CVecOLDcvc 27931  NrmCVeccnv 27957   +𝑣 cpv 27958  BaseSetcba 27959   ·𝑠OLD cns 27960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-rep 4963  ax-sep 4974  ax-nul 4982  ax-pow 5034  ax-pr 5096  ax-un 7182
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3386  df-sbc 3633  df-csb 3728  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-iun 4711  df-br 4843  df-opab 4905  df-mpt 4922  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6063  df-fun 6102  df-fn 6103  df-f 6104  df-f1 6105  df-fo 6106  df-f1o 6107  df-fv 6108  df-ov 6880  df-oprab 6881  df-1st 7400  df-2nd 7401  df-vc 27932  df-nv 27965  df-va 27968  df-ba 27969  df-sm 27970  df-0v 27971  df-nmcv 27973
This theorem is referenced by:  nvge0  28046  smcnlem  28070  ipidsq  28083  ip2i  28201  ipasslem1  28204  ipasslem11  28213  hldir  28282
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