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Theorem vcdir 28352
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 28347 . . . . 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 486 . . . . . . . . . . 11 ((((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))) → ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
65ralimi 3155 . . . . . . . . . 10 (∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))) → ∀𝑧 ∈ ℂ ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
76adantl 485 . . . . . . . . 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 3155 . . . . . . . 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 485 . . . . . . 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 3155 . . . . . 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 1132 . . . . 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 7157 . . . . . 6 (𝑥 = 𝐶 → ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦 + 𝑧)𝑆𝐶))
14 oveq2 7157 . . . . . . 7 (𝑥 = 𝐶 → (𝑦𝑆𝑥) = (𝑦𝑆𝐶))
15 oveq2 7157 . . . . . . 7 (𝑥 = 𝐶 → (𝑧𝑆𝑥) = (𝑧𝑆𝐶))
1614, 15oveq12d 7167 . . . . . 6 (𝑥 = 𝐶 → ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) = ((𝑦𝑆𝐶)𝐺(𝑧𝑆𝐶)))
1713, 16eqeq12d 2840 . . . . 5 (𝑥 = 𝐶 → (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ↔ ((𝑦 + 𝑧)𝑆𝐶) = ((𝑦𝑆𝐶)𝐺(𝑧𝑆𝐶))))
18 oveq1 7156 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 + 𝑧) = (𝐴 + 𝑧))
1918oveq1d 7164 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 + 𝑧)𝑆𝐶) = ((𝐴 + 𝑧)𝑆𝐶))
20 oveq1 7156 . . . . . . 7 (𝑦 = 𝐴 → (𝑦𝑆𝐶) = (𝐴𝑆𝐶))
2120oveq1d 7164 . . . . . 6 (𝑦 = 𝐴 → ((𝑦𝑆𝐶)𝐺(𝑧𝑆𝐶)) = ((𝐴𝑆𝐶)𝐺(𝑧𝑆𝐶)))
2219, 21eqeq12d 2840 . . . . 5 (𝑦 = 𝐴 → (((𝑦 + 𝑧)𝑆𝐶) = ((𝑦𝑆𝐶)𝐺(𝑧𝑆𝐶)) ↔ ((𝐴 + 𝑧)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝑧𝑆𝐶))))
23 oveq2 7157 . . . . . . 7 (𝑧 = 𝐵 → (𝐴 + 𝑧) = (𝐴 + 𝐵))
2423oveq1d 7164 . . . . . 6 (𝑧 = 𝐵 → ((𝐴 + 𝑧)𝑆𝐶) = ((𝐴 + 𝐵)𝑆𝐶))
25 oveq1 7156 . . . . . . 7 (𝑧 = 𝐵 → (𝑧𝑆𝐶) = (𝐵𝑆𝐶))
2625oveq2d 7165 . . . . . 6 (𝑧 = 𝐵 → ((𝐴𝑆𝐶)𝐺(𝑧𝑆𝐶)) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))
2724, 26eqeq12d 2840 . . . . 5 (𝑧 = 𝐵 → (((𝐴 + 𝑧)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝑧𝑆𝐶)) ↔ ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶))))
2817, 22, 27rspc3v 3622 . . . 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 1124 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋) → (𝑊 ∈ CVecOLD → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶))))
3130impcom 411 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 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133   × cxp 5540  ran crn 5543  wf 6339  cfv 6343  (class class class)co 7149  1st c1st 7682  2nd c2nd 7683  cc 10533  1c1 10536   + caddc 10538   · cmul 10540  AbelOpcablo 28330  CVecOLDcvc 28344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-1st 7684  df-2nd 7685  df-vc 28345
This theorem is referenced by:  vc2OLD  28354  vc0  28360  vcm  28362  nvdir  28417
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