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Theorem vcdir 26805
 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 26800 . . . . 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 472 . . . . . . . . . . 11 ((((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))) → ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
65ralimi 2936 . . . . . . . . . 10 (∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))) → ∀𝑧 ∈ ℂ ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
76adantl 481 . . . . . . . . 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 2936 . . . . . . . 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 481 . . . . . . 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 2936 . . . . . 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 1077 . . . . 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 6557 . . . . . 6 (𝑥 = 𝐶 → ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦 + 𝑧)𝑆𝐶))
14 oveq2 6557 . . . . . . 7 (𝑥 = 𝐶 → (𝑦𝑆𝑥) = (𝑦𝑆𝐶))
15 oveq2 6557 . . . . . . 7 (𝑥 = 𝐶 → (𝑧𝑆𝑥) = (𝑧𝑆𝐶))
1614, 15oveq12d 6567 . . . . . 6 (𝑥 = 𝐶 → ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) = ((𝑦𝑆𝐶)𝐺(𝑧𝑆𝐶)))
1713, 16eqeq12d 2625 . . . . 5 (𝑥 = 𝐶 → (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ↔ ((𝑦 + 𝑧)𝑆𝐶) = ((𝑦𝑆𝐶)𝐺(𝑧𝑆𝐶))))
18 oveq1 6556 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 + 𝑧) = (𝐴 + 𝑧))
1918oveq1d 6564 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 + 𝑧)𝑆𝐶) = ((𝐴 + 𝑧)𝑆𝐶))
20 oveq1 6556 . . . . . . 7 (𝑦 = 𝐴 → (𝑦𝑆𝐶) = (𝐴𝑆𝐶))
2120oveq1d 6564 . . . . . 6 (𝑦 = 𝐴 → ((𝑦𝑆𝐶)𝐺(𝑧𝑆𝐶)) = ((𝐴𝑆𝐶)𝐺(𝑧𝑆𝐶)))
2219, 21eqeq12d 2625 . . . . 5 (𝑦 = 𝐴 → (((𝑦 + 𝑧)𝑆𝐶) = ((𝑦𝑆𝐶)𝐺(𝑧𝑆𝐶)) ↔ ((𝐴 + 𝑧)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝑧𝑆𝐶))))
23 oveq2 6557 . . . . . . 7 (𝑧 = 𝐵 → (𝐴 + 𝑧) = (𝐴 + 𝐵))
2423oveq1d 6564 . . . . . 6 (𝑧 = 𝐵 → ((𝐴 + 𝑧)𝑆𝐶) = ((𝐴 + 𝐵)𝑆𝐶))
25 oveq1 6556 . . . . . . 7 (𝑧 = 𝐵 → (𝑧𝑆𝐶) = (𝐵𝑆𝐶))
2625oveq2d 6565 . . . . . 6 (𝑧 = 𝐵 → ((𝐴𝑆𝐶)𝐺(𝑧𝑆𝐶)) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))
2724, 26eqeq12d 2625 . . . . 5 (𝑧 = 𝐵 → (((𝐴 + 𝑧)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝑧𝑆𝐶)) ↔ ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶))))
2817, 22, 27rspc3v 3296 . . . 4 ((𝐶𝑋𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∀𝑥𝑋𝑦 ∈ ℂ ∀𝑧 ∈ ℂ ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶))))
2912, 28syl5 33 . . 3 ((𝐶𝑋𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑊 ∈ CVecOLD → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶))))
30293coml 1264 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋) → (𝑊 ∈ CVecOLD → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶))))
3130impcom 445 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   × cxp 5036  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  ℂcc 9813  1c1 9816   + caddc 9818   · cmul 9820  AbelOpcablo 26782  CVecOLDcvc 26797 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-1st 7059  df-2nd 7060  df-vc 26798 This theorem is referenced by:  vc2OLD  26807  vc0  26813  vcm  26815  nvdir  26870
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