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Theorem hldir 29258
Description: Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
hldi.1 𝑋 = (BaseSet‘𝑈)
hldi.2 𝐺 = ( +𝑣𝑈)
hldi.4 𝑆 = ( ·𝑠OLD𝑈)
Assertion
Ref Expression
hldir ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))

Proof of Theorem hldir
StepHypRef Expression
1 hlnv 29241 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)
2 hldi.1 . . 3 𝑋 = (BaseSet‘𝑈)
3 hldi.2 . . 3 𝐺 = ( +𝑣𝑈)
4 hldi.4 . . 3 𝑆 = ( ·𝑠OLD𝑈)
52, 3, 4nvdir 28981 . 2 ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))
61, 5sylan 580 1 ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1542  wcel 2110  cfv 6431  (class class class)co 7269  cc 10862   + caddc 10867  NrmCVeccnv 28934   +𝑣 cpv 28935  BaseSetcba 28936   ·𝑠OLD cns 28937  CHilOLDchlo 29235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7580
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-ov 7272  df-oprab 7273  df-1st 7818  df-2nd 7819  df-vc 28909  df-nv 28942  df-va 28945  df-ba 28946  df-sm 28947  df-0v 28948  df-nmcv 28950  df-cbn 29213  df-hlo 29236
This theorem is referenced by:  axhvdistr2-zf  29341
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