Theorem vcdi 30537

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 30533	. . . . 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 482	. . . . . . . . 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 3069	. . . . . . . 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 481	. . . . . . 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 3069	. . . . . 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 1135	. . . . 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 17	. . . 4 ⊢ (𝑊 ∈ CVecOLD → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℂ ∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
11		oveq1 7348	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥𝐺𝑧) = (𝐵𝐺𝑧))
12	11	oveq2d 7357	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑦𝑆(𝑥𝐺𝑧)) = (𝑦𝑆(𝐵𝐺𝑧)))
13		oveq2 7349	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑦𝑆𝑥) = (𝑦𝑆𝐵))
14	13	oveq1d 7356	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) = ((𝑦𝑆𝐵)𝐺(𝑦𝑆𝑧)))
15	12, 14	eqeq12d 2747	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ↔ (𝑦𝑆(𝐵𝐺𝑧)) = ((𝑦𝑆𝐵)𝐺(𝑦𝑆𝑧))))
16		oveq1 7348	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦𝑆(𝐵𝐺𝑧)) = (𝐴𝑆(𝐵𝐺𝑧)))
17		oveq1 7348	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦𝑆𝐵) = (𝐴𝑆𝐵))
18		oveq1 7348	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦𝑆𝑧) = (𝐴𝑆𝑧))
19	17, 18	oveq12d 7359	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦𝑆𝐵)𝐺(𝑦𝑆𝑧)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝑧)))
20	16, 19	eqeq12d 2747	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑦𝑆(𝐵𝐺𝑧)) = ((𝑦𝑆𝐵)𝐺(𝑦𝑆𝑧)) ↔ (𝐴𝑆(𝐵𝐺𝑧)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝑧))))
21		oveq2 7349	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝐵𝐺𝑧) = (𝐵𝐺𝐶))
22	21	oveq2d 7357	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝐴𝑆(𝐵𝐺𝑧)) = (𝐴𝑆(𝐵𝐺𝐶)))
23		oveq2 7349	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝐴𝑆𝑧) = (𝐴𝑆𝐶))
24	23	oveq2d 7357	. . . . . 6 ⊢ (𝑧 = 𝐶 → ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝑧)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))
25	22, 24	eqeq12d 2747	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝐴𝑆(𝐵𝐺𝑧)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝑧)) ↔ (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶))))
26	15, 20, 25	rspc3v 3588	. . . 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 34	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ 𝑋) → (𝑊 ∈ CVecOLD → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶))))
28	27	3com12 1123	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝑊 ∈ CVecOLD → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶))))
29	28	impcom 407	1 ⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047   × cxp 5609  ran crn 5612  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341  1^st c1st 7914  2^nd c2nd 7915  ℂcc 10999  1c1 11002   + caddc 11004   · cmul 11006  AbelOpcablo 30516  CVec_OLDcvc 30530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-ov 7344  df-1st 7916  df-2nd 7917  df-vc 30531

This theorem is referenced by:  nvdi  30602