Theorem vcdi 30854

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 = (1st ‘𝑊)
vciOLD.2	⊢ 𝑆 = (2nd ‘𝑊)
vciOLD.3	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
vcdi	⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))

Proof of Theorem vcdi

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vciOLD.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑊)
2		vciOLD.2	. . . . . 6 ⊢ 𝑆 = (2nd ‘𝑊)
3		vciOLD.3	. . . . . 6 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	vciOLD 30850	. . . . 5 ⊢ (𝑊 ∈ CVecOLD → (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))))
5		simpl 487	. . . . . . . . 9 ⊢ ((∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))) → ∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
6	5	ralimi 3108	. . . . . . . 8 ⊢ (∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))) → ∀𝑦 ∈ ℂ ∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
7	6	adantl 486	. . . . . . 7 ⊢ (((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))) → ∀𝑦 ∈ ℂ ∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
8	7	ralimi 3108	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℂ ∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
9	8	3ad2ant3 1151	. . . . 5 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℂ ∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
10	4, 9	syl 18	. . . 4 ⊢ (𝑊 ∈ CVecOLD → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℂ ∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
11		oveq1 7415	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥𝐺𝑧) = (𝐵𝐺𝑧))
12	11	oveq2d 7424	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑦𝑆(𝑥𝐺𝑧)) = (𝑦𝑆(𝐵𝐺𝑧)))
13		oveq2 7416	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑦𝑆𝑥) = (𝑦𝑆𝐵))
14	13	oveq1d 7423	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) = ((𝑦𝑆𝐵)𝐺(𝑦𝑆𝑧)))
15	12, 14	eqeq12d 2785	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ↔ (𝑦𝑆(𝐵𝐺𝑧)) = ((𝑦𝑆𝐵)𝐺(𝑦𝑆𝑧))))
16		oveq1 7415	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦𝑆(𝐵𝐺𝑧)) = (𝐴𝑆(𝐵𝐺𝑧)))
17		oveq1 7415	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦𝑆𝐵) = (𝐴𝑆𝐵))
18		oveq1 7415	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦𝑆𝑧) = (𝐴𝑆𝑧))
19	17, 18	oveq12d 7426	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦𝑆𝐵)𝐺(𝑦𝑆𝑧)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝑧)))
20	16, 19	eqeq12d 2785	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑦𝑆(𝐵𝐺𝑧)) = ((𝑦𝑆𝐵)𝐺(𝑦𝑆𝑧)) ↔ (𝐴𝑆(𝐵𝐺𝑧)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝑧))))
21		oveq2 7416	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝐵𝐺𝑧) = (𝐵𝐺𝐶))
22	21	oveq2d 7424	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝐴𝑆(𝐵𝐺𝑧)) = (𝐴𝑆(𝐵𝐺𝐶)))
23		oveq2 7416	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝐴𝑆𝑧) = (𝐴𝑆𝐶))
24	23	oveq2d 7424	. . . . . 6 ⊢ (𝑧 = 𝐶 → ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝑧)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))
25	22, 24	eqeq12d 2785	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝐴𝑆(𝐵𝐺𝑧)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝑧)) ↔ (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶))))
26	15, 20, 25	rspc3v 3606	. . . 4 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℂ ∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶))))
27	10, 26	syl5 35	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ 𝑋) → (𝑊 ∈ CVecOLD → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶))))
28	27	3com12 1139	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝑊 ∈ CVecOLD → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶))))
29	28	impcom 412	1 ⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085   × cxp 5657  ran crn 5660  ⟶wf 6529  ‘cfv 6533  (class class class)co 7408  1^st c1st 7980  2^nd c2nd 7981  ℂcc 11094  1c1 11097   + caddc 11099   · cmul 11101  AbelOpcablo 30833  CVec_OLDcvc 30847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7411  df-1st 7982  df-2nd 7983  df-vc 30848

This theorem is referenced by:  nvdi  30919