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Theorem vcdir 27993
Description: Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
vciOLD.1 𝐺 = (1st𝑊)
vciOLD.2 𝑆 = (2nd𝑊)
vciOLD.3 𝑋 = ran 𝐺
Assertion
Ref Expression
vcdir ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))

Proof of Theorem vcdir
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vciOLD.1 . . . . . 6 𝐺 = (1st𝑊)
2 vciOLD.2 . . . . . 6 𝑆 = (2nd𝑊)
3 vciOLD.3 . . . . . 6 𝑋 = ran 𝐺
41, 2, 3vciOLD 27988 . . . . 5 (𝑊 ∈ CVecOLD → (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))))
5 simpl 476 . . . . . . . . . . 11 ((((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))) → ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
65ralimi 3133 . . . . . . . . . 10 (∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))) → ∀𝑧 ∈ ℂ ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
76adantl 475 . . . . . . . . 9 ((∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))) → ∀𝑧 ∈ ℂ ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
87ralimi 3133 . . . . . . . 8 (∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))) → ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
98adantl 475 . . . . . . 7 (((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))) → ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
109ralimi 3133 . . . . . 6 (∀𝑥𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))) → ∀𝑥𝑋𝑦 ∈ ℂ ∀𝑧 ∈ ℂ ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
11103ad2ant3 1126 . . . . 5 ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))) → ∀𝑥𝑋𝑦 ∈ ℂ ∀𝑧 ∈ ℂ ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
124, 11syl 17 . . . 4 (𝑊 ∈ CVecOLD → ∀𝑥𝑋𝑦 ∈ ℂ ∀𝑧 ∈ ℂ ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
13 oveq2 6930 . . . . . 6 (𝑥 = 𝐶 → ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦 + 𝑧)𝑆𝐶))
14 oveq2 6930 . . . . . . 7 (𝑥 = 𝐶 → (𝑦𝑆𝑥) = (𝑦𝑆𝐶))
15 oveq2 6930 . . . . . . 7 (𝑥 = 𝐶 → (𝑧𝑆𝑥) = (𝑧𝑆𝐶))
1614, 15oveq12d 6940 . . . . . 6 (𝑥 = 𝐶 → ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) = ((𝑦𝑆𝐶)𝐺(𝑧𝑆𝐶)))
1713, 16eqeq12d 2792 . . . . 5 (𝑥 = 𝐶 → (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ↔ ((𝑦 + 𝑧)𝑆𝐶) = ((𝑦𝑆𝐶)𝐺(𝑧𝑆𝐶))))
18 oveq1 6929 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 + 𝑧) = (𝐴 + 𝑧))
1918oveq1d 6937 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 + 𝑧)𝑆𝐶) = ((𝐴 + 𝑧)𝑆𝐶))
20 oveq1 6929 . . . . . . 7 (𝑦 = 𝐴 → (𝑦𝑆𝐶) = (𝐴𝑆𝐶))
2120oveq1d 6937 . . . . . 6 (𝑦 = 𝐴 → ((𝑦𝑆𝐶)𝐺(𝑧𝑆𝐶)) = ((𝐴𝑆𝐶)𝐺(𝑧𝑆𝐶)))
2219, 21eqeq12d 2792 . . . . 5 (𝑦 = 𝐴 → (((𝑦 + 𝑧)𝑆𝐶) = ((𝑦𝑆𝐶)𝐺(𝑧𝑆𝐶)) ↔ ((𝐴 + 𝑧)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝑧𝑆𝐶))))
23 oveq2 6930 . . . . . . 7 (𝑧 = 𝐵 → (𝐴 + 𝑧) = (𝐴 + 𝐵))
2423oveq1d 6937 . . . . . 6 (𝑧 = 𝐵 → ((𝐴 + 𝑧)𝑆𝐶) = ((𝐴 + 𝐵)𝑆𝐶))
25 oveq1 6929 . . . . . . 7 (𝑧 = 𝐵 → (𝑧𝑆𝐶) = (𝐵𝑆𝐶))
2625oveq2d 6938 . . . . . 6 (𝑧 = 𝐵 → ((𝐴𝑆𝐶)𝐺(𝑧𝑆𝐶)) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))
2724, 26eqeq12d 2792 . . . . 5 (𝑧 = 𝐵 → (((𝐴 + 𝑧)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝑧𝑆𝐶)) ↔ ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶))))
2817, 22, 27rspc3v 3526 . . . 4 ((𝐶𝑋𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∀𝑥𝑋𝑦 ∈ ℂ ∀𝑧 ∈ ℂ ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶))))
2912, 28syl5 34 . . 3 ((𝐶𝑋𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑊 ∈ CVecOLD → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶))))
30293coml 1118 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋) → (𝑊 ∈ CVecOLD → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶))))
3130impcom 398 1 ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2106  wral 3089   × cxp 5353  ran crn 5356  wf 6131  cfv 6135  (class class class)co 6922  1st c1st 7443  2nd c2nd 7444  cc 10270  1c1 10273   + caddc 10275   · cmul 10277  AbelOpcablo 27971  CVecOLDcvc 27985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-1st 7445  df-2nd 7446  df-vc 27986
This theorem is referenced by:  vc2OLD  27995  vc0  28001  vcm  28003  nvdir  28058
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