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Theorem nvsass 28386
Description: Associative law for the scalar product of a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvscl.1 𝑋 = (BaseSet‘𝑈)
nvscl.4 𝑆 = ( ·𝑠OLD𝑈)
Assertion
Ref Expression
nvsass ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))

Proof of Theorem nvsass
StepHypRef Expression
1 eqid 2820 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 28373 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 eqid 2820 . . . 4 ( +𝑣𝑈) = ( +𝑣𝑈)
43vafval 28361 . . 3 ( +𝑣𝑈) = (1st ‘(1st𝑈))
5 nvscl.4 . . . 4 𝑆 = ( ·𝑠OLD𝑈)
65smfval 28363 . . 3 𝑆 = (2nd ‘(1st𝑈))
7 nvscl.1 . . . 4 𝑋 = (BaseSet‘𝑈)
87, 3bafval 28362 . . 3 𝑋 = ran ( +𝑣𝑈)
94, 6, 8vcass 28325 . 2 (((1st𝑈) ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))
102, 9sylan 582 1 ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1537  wcel 2114  cfv 6327  (class class class)co 7129  1st c1st 7661  cc 10509   · cmul 10516  CVecOLDcvc 28316  NrmCVeccnv 28342   +𝑣 cpv 28343  BaseSetcba 28344   ·𝑠OLD cns 28345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5162  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3472  df-sbc 3749  df-csb 3857  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-iun 4893  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7132  df-oprab 7133  df-1st 7663  df-2nd 7664  df-vc 28317  df-nv 28350  df-va 28353  df-ba 28354  df-sm 28355  df-0v 28356  df-nmcv 28358
This theorem is referenced by:  nvscom  28387  nvmul0or  28408  nvpi  28425  smcnlem  28455  ipval3  28467  ipdirilem  28587  ipasslem2  28590  ipasslem4  28592  ipasslem5  28593  ipasslem10  28597  ipasslem11  28598  minvecolem2  28633  hlmulass  28664
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