Theorem vciOLD 30562

Description: Obsolete version of cvsi 25077. The properties of a complex vector space, which is an Abelian group (i.e. the vectors, with the operation of vector addition) accompanied by a scalar multiplication operation on the field of complex numbers. The variable 𝑊 was chosen because V is already used for the universal class. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
vciOLD	⊢ (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)))))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐺   𝑥,𝑆,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Allowed substitution hints:   𝑊(𝑥,𝑦,𝑧)

Proof of Theorem vciOLD

Dummy variables 𝑔 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vciOLD.1	. . . . 5 ⊢ 𝐺 = (1st ‘𝑊)
2	1	eqeq2i 2746	. . . 4 ⊢ (𝑔 = 𝐺 ↔ 𝑔 = (1st ‘𝑊))
3		eleq1 2821	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ AbelOp ↔ 𝐺 ∈ AbelOp))
4		rneq 5882	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
5		vciOLD.3	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
6	4, 5	eqtr4di 2786	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
7		xpeq2 5642	. . . . . . . 8 ⊢ (ran 𝑔 = 𝑋 → (ℂ × ran 𝑔) = (ℂ × 𝑋))
8	7	feq2d 6643	. . . . . . 7 ⊢ (ran 𝑔 = 𝑋 → (𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ↔ 𝑠:(ℂ × 𝑋)⟶ran 𝑔))
9		feq3 6639	. . . . . . 7 ⊢ (ran 𝑔 = 𝑋 → (𝑠:(ℂ × 𝑋)⟶ran 𝑔 ↔ 𝑠:(ℂ × 𝑋)⟶𝑋))
10	8, 9	bitrd 279	. . . . . 6 ⊢ (ran 𝑔 = 𝑋 → (𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ↔ 𝑠:(ℂ × 𝑋)⟶𝑋))
11	6, 10	syl 17	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ↔ 𝑠:(ℂ × 𝑋)⟶𝑋))
12		oveq 7361	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑧) = (𝑥𝐺𝑧))
13	12	oveq2d 7371	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (𝑦𝑠(𝑥𝑔𝑧)) = (𝑦𝑠(𝑥𝐺𝑧)))
14		oveq 7361	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)))
15	13, 14	eqeq12d 2749	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ↔ (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧))))
16	6, 15	raleqbidv 3313	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ↔ ∀𝑧 ∈ 𝑋 (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧))))
17		oveq 7361	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)))
18	17	eqeq2d 2744	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ↔ ((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥))))
19	18	anbi1d 631	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))) ↔ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)))))
20	19	ralbidv 3156	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))) ↔ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)))))
21	16, 20	anbi12d 632	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((∀𝑧 ∈ ran 𝑔(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)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))
22	21	ralbidv 3156	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(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)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))
23	22	anbi2d 630	. . . . . 6 ⊢ (𝑔 = 𝐺 → (((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))) ↔ ((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)))))))
24	6, 23	raleqbidv 3313	. . . . 5 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))) ↔ ∀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)))))))
25	3, 11, 24	3anbi123d 1438	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)))))) ↔ (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))))))))
26	2, 25	sylbir 235	. . 3 ⊢ (𝑔 = (1st ‘𝑊) → ((𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)))))) ↔ (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))))))))
27		vciOLD.2	. . . . 5 ⊢ 𝑆 = (2nd ‘𝑊)
28	27	eqeq2i 2746	. . . 4 ⊢ (𝑠 = 𝑆 ↔ 𝑠 = (2nd ‘𝑊))
29		feq1 6637	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑠:(ℂ × 𝑋)⟶𝑋 ↔ 𝑆:(ℂ × 𝑋)⟶𝑋))
30		oveq 7361	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1𝑠𝑥) = (1𝑆𝑥))
31	30	eqeq1d 2735	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((1𝑠𝑥) = 𝑥 ↔ (1𝑆𝑥) = 𝑥))
32		oveq 7361	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑦𝑠(𝑥𝐺𝑧)) = (𝑦𝑆(𝑥𝐺𝑧)))
33		oveq 7361	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (𝑦𝑠𝑥) = (𝑦𝑆𝑥))
34		oveq 7361	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (𝑦𝑠𝑧) = (𝑦𝑆𝑧))
35	33, 34	oveq12d 7373	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
36	32, 35	eqeq12d 2749	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → ((𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) ↔ (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧))))
37	36	ralbidv 3156	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ 𝑋 (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) ↔ ∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧))))
38		oveq 7361	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → ((𝑦 + 𝑧)𝑠𝑥) = ((𝑦 + 𝑧)𝑆𝑥))
39		oveq 7361	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑆 → (𝑧𝑠𝑥) = (𝑧𝑆𝑥))
40	33, 39	oveq12d 7373	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
41	38, 40	eqeq12d 2749	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ↔ ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥))))
42		oveq 7361	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → ((𝑦 · 𝑧)𝑠𝑥) = ((𝑦 · 𝑧)𝑆𝑥))
43	39	oveq2d 7371	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑆 → (𝑦𝑠(𝑧𝑠𝑥)) = (𝑦𝑠(𝑧𝑆𝑥)))
44		oveq 7361	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑆 → (𝑦𝑠(𝑧𝑆𝑥)) = (𝑦𝑆(𝑧𝑆𝑥)))
45	43, 44	eqtrd 2768	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (𝑦𝑠(𝑧𝑠𝑥)) = (𝑦𝑆(𝑧𝑆𝑥)))
46	42, 45	eqeq12d 2749	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)) ↔ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))
47	41, 46	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → ((((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))) ↔ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))
48	47	ralbidv 3156	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))) ↔ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))
49	37, 48	anbi12d 632	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → ((∀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)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))))
50	49	ralbidv 3156	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀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)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))))
51	31, 50	anbi12d 632	. . . . . 6 ⊢ (𝑠 = 𝑆 → (((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))) ↔ ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))))
52	51	ralbidv 3156	. . . . 5 ⊢ (𝑠 = 𝑆 → (∀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 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))))
53	29, 52	3anbi23d 1441	. . . 4 ⊢ (𝑠 = 𝑆 → ((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)))))) ↔ (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))))))))
54	28, 53	sylbir 235	. . 3 ⊢ (𝑠 = (2nd ‘𝑊) → ((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)))))) ↔ (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))))))))
55	26, 54	elopabi 8003	. 2 ⊢ (𝑊 ∈ {⟨𝑔, 𝑠⟩ ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))} → (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)))))))
56		df-vc 30560	. 2 ⊢ CVecOLD = {⟨𝑔, 𝑠⟩ ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))}
57	55, 56	eleq2s 2851	1 ⊢ (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)))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048  {copab 5157   × cxp 5619  ran crn 5622  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355  1^st c1st 7928  2^nd c2nd 7929  ℂcc 11015  1c1 11018   + caddc 11020   · cmul 11022  AbelOpcablo 30545  CVec_OLDcvc 30559

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-ov 7358  df-1st 7930  df-2nd 7931  df-vc 30560

This theorem is referenced by:  vcsm  30563  vcidOLD  30565  vcdi  30566  vcdir  30567  vcass  30568  vcablo  30570