Theorem cpmatmcllem 22636

Description: Lemma for cpmatmcl 22637. (Contributed by AV, 18-Nov-2019.)

Hypotheses

Ref	Expression
cpmatsrngpmat.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
cpmatsrngpmat.p	⊢ 𝑃 = (Poly1‘𝑅)
cpmatsrngpmat.c	⊢ 𝐶 = (𝑁 Mat 𝑃)

Assertion

Ref	Expression
cpmatmcllem	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅))

Distinct variable groups:   𝐶,𝑖,𝑗   𝑖,𝑁,𝑗   𝑅,𝑖,𝑗   𝐶,𝑐   𝑁,𝑐,𝑥,𝑦,𝑖,𝑗   𝑃,𝑐   𝑅,𝑐,𝑥,𝑦   𝑦,𝑆   𝐶,𝑘   𝑘,𝑁,𝑐,𝑖,𝑗,𝑥,𝑦   𝑃,𝑘   𝑅,𝑘

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑃(𝑥,𝑦,𝑖,𝑗)   𝑆(𝑥,𝑖,𝑗,𝑘,𝑐)

Proof of Theorem cpmatmcllem

Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cpmatsrngpmat.s	. . . 4 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
2		cpmatsrngpmat.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
3		cpmatsrngpmat.c	. . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃)
4		eqid 2725	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	1, 2, 3, 4	cpmatelimp 22630	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ 𝑆 → (𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅))))
6	1, 2, 3, 4	cpmatelimp 22630	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦 ∈ 𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))))
7	6	adantr 479	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ 𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))))
8		ralcom 3277	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅) ↔ ∀𝑗 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))
9		r19.26-2 3128	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)) ↔ (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)))
10		ralcom 3277	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)) ↔ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)))
11	9, 10	bitr3i 276	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)) ↔ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)))
12		nfv 1909	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑐(((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
13		nfra1 3272	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑐∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))
14	12, 13	nfan 1894	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑐((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)))
15		simp-4r 782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑅 ∈ Ring)
16		eqid 2725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (Base‘𝑃) = (Base‘𝑃)
17		simplrl 775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑖 ∈ 𝑁)
18		simpr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
19		simplrl 775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑥 ∈ (Base‘𝐶))
20	19	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑥 ∈ (Base‘𝐶))
21	3, 16, 4, 17, 18, 20	matecld 22344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑖𝑥𝑘) ∈ (Base‘𝑃))
22		simplrr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑗 ∈ 𝑁)
23		simplrr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑦 ∈ (Base‘𝐶))
24	23	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑦 ∈ (Base‘𝐶))
25	3, 16, 4, 18, 22, 24	matecld 22344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑦𝑗) ∈ (Base‘𝑃))
26	15, 21, 25	jca32 514	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑅 ∈ Ring ∧ ((𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃))))
27	26	adantlr 713	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))) ∧ 𝑘 ∈ 𝑁) → (𝑅 ∈ Ring ∧ ((𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃))))
28		oveq2 7423	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑙 = 𝑘 → (𝑖𝑥𝑙) = (𝑖𝑥𝑘))
29	28	fveq2d 6895	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑙 = 𝑘 → (coe1‘(𝑖𝑥𝑙)) = (coe1‘(𝑖𝑥𝑘)))
30	29	fveq1d 6893	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑙 = 𝑘 → ((coe1‘(𝑖𝑥𝑙))‘𝑐) = ((coe1‘(𝑖𝑥𝑘))‘𝑐))
31	30	eqeq1d 2727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑙 = 𝑘 → (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ↔ ((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g‘𝑅)))
32		fvoveq1 7438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑙 = 𝑘 → (coe1‘(𝑙𝑦𝑗)) = (coe1‘(𝑘𝑦𝑗)))
33	32	fveq1d 6893	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑙 = 𝑘 → ((coe1‘(𝑙𝑦𝑗))‘𝑐) = ((coe1‘(𝑘𝑦𝑗))‘𝑐))
34	33	eqeq1d 2727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑙 = 𝑘 → (((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅) ↔ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g‘𝑅)))
35	31, 34	anbi12d 630	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑙 = 𝑘 → ((((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)) ↔ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g‘𝑅))))
36	35	rspcva 3600	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑘 ∈ 𝑁 ∧ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))) → (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g‘𝑅)))
37	36	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → ((𝑘 ∈ 𝑁 ∧ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))) → (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g‘𝑅))))
38	37	exp4b 429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑐 ∈ ℕ → (𝑘 ∈ 𝑁 → (∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)) → (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g‘𝑅))))))
39	38	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑘 ∈ 𝑁 → (𝑐 ∈ ℕ → (∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)) → (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g‘𝑅))))))
40	39	imp31 416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) ∧ 𝑐 ∈ ℕ) → (∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)) → (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g‘𝑅))))
41	40	ralimdva 3157	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)) → ∀𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g‘𝑅))))
42	41	impancom 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))) → (𝑘 ∈ 𝑁 → ∀𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g‘𝑅))))
43	42	imp 405	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))) ∧ 𝑘 ∈ 𝑁) → ∀𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g‘𝑅)))
44		eqid 2725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (0g‘𝑅) = (0g‘𝑅)
45		eqid 2725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (.r‘𝑃) = (.r‘𝑃)
46	2, 16, 44, 45	cply1mul 22222	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑅 ∈ Ring ∧ ((𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃))) → (∀𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g‘𝑅)) → ∀𝑐 ∈ ℕ ((coe1‘((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))‘𝑐) = (0g‘𝑅)))
47	27, 43, 46	sylc 65	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))) ∧ 𝑘 ∈ 𝑁) → ∀𝑐 ∈ ℕ ((coe1‘((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))‘𝑐) = (0g‘𝑅))
48	47	r19.21bi 3239	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))) ∧ 𝑘 ∈ 𝑁) ∧ 𝑐 ∈ ℕ) → ((coe1‘((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))‘𝑐) = (0g‘𝑅))
49	48	an32s 650	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))) ∧ 𝑐 ∈ ℕ) ∧ 𝑘 ∈ 𝑁) → ((coe1‘((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))‘𝑐) = (0g‘𝑅))
50	49	mpteq2dva 5243	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))) ∧ 𝑐 ∈ ℕ) → (𝑘 ∈ 𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))‘𝑐)) = (𝑘 ∈ 𝑁 ↦ (0g‘𝑅)))
51	50	oveq2d 7431	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))) ∧ 𝑐 ∈ ℕ) → (𝑅 Σg (𝑘 ∈ 𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (𝑅 Σg (𝑘 ∈ 𝑁 ↦ (0g‘𝑅))))
52		ringmnd 20185	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
53	52	anim2i 615	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Mnd))
54	53	ancomd 460	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin))
55	44	gsumz 18790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin) → (𝑅 Σg (𝑘 ∈ 𝑁 ↦ (0g‘𝑅))) = (0g‘𝑅))
56	54, 55	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 Σg (𝑘 ∈ 𝑁 ↦ (0g‘𝑅))) = (0g‘𝑅))
57	56	ad4antr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))) ∧ 𝑐 ∈ ℕ) → (𝑅 Σg (𝑘 ∈ 𝑁 ↦ (0g‘𝑅))) = (0g‘𝑅))
58	51, 57	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))) ∧ 𝑐 ∈ ℕ) → (𝑅 Σg (𝑘 ∈ 𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g‘𝑅))
59	58	ex 411	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))) → (𝑐 ∈ ℕ → (𝑅 Σg (𝑘 ∈ 𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g‘𝑅)))
60	14, 59	ralrimi 3245	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))) → ∀𝑐 ∈ ℕ (𝑅 Σg (𝑘 ∈ 𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g‘𝑅))
61		simp-4r 782	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑅 ∈ Ring)
62		nnnn0 12507	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℕ → 𝑐 ∈ ℕ0)
63	62	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑐 ∈ ℕ0)
64	2	ply1ring 22173	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
65	64	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑃 ∈ Ring)
66	16, 45	ringcl 20192	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑃 ∈ Ring ∧ (𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃)) → ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
67	65, 21, 25, 66	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
68	67	ralrimiva 3136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ∀𝑘 ∈ 𝑁 ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
69	68	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → ∀𝑘 ∈ 𝑁 ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
70		simp-4l 781	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑁 ∈ Fin)
71	2, 16, 61, 63, 69, 70	coe1fzgsumd 22230	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (𝑅 Σg (𝑘 ∈ 𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))‘𝑐))))
72	71	eqeq1d 2727	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → (((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅) ↔ (𝑅 Σg (𝑘 ∈ 𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g‘𝑅)))
73	72	ralbidva 3166	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅) ↔ ∀𝑐 ∈ ℕ (𝑅 Σg (𝑘 ∈ 𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g‘𝑅)))
74	73	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))) → (∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅) ↔ ∀𝑐 ∈ ℕ (𝑅 Σg (𝑘 ∈ 𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g‘𝑅)))
75	60, 74	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅))
76	75	ex 411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅)))
77	11, 76	biimtrid 241	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅)))
78	77	expd 414	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅))))
79	78	expr 455	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖 ∈ 𝑁) → (𝑗 ∈ 𝑁 → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅)))))
80	79	com23 86	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖 ∈ 𝑁) → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) → (𝑗 ∈ 𝑁 → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅)))))
81	80	imp31 416	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖 ∈ 𝑁) ∧ ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅)) ∧ 𝑗 ∈ 𝑁) → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅)))
82	81	ralimdva 3157	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖 ∈ 𝑁) ∧ ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅)) → (∀𝑗 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅) → ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅)))
83	8, 82	biimtrid 241	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖 ∈ 𝑁) ∧ ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅)) → (∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅) → ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅)))
84	83	ex 411	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖 ∈ 𝑁) → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) → (∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅) → ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅))))
85	84	com23 86	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖 ∈ 𝑁) → (∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅) → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) → ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅))))
86	85	impancom 450	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)) → (𝑖 ∈ 𝑁 → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) → ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅))))
87	86	imp 405	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)) ∧ 𝑖 ∈ 𝑁) → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) → ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅)))
88	87	ralimdva 3157	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)) → (∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅)))
89	88	ex 411	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅) → (∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅))))
90	89	expr 455	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐶) → (∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅) → (∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅)))))
91	90	impd 409	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)) → (∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅))))
92	7, 91	syld 47	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ 𝑆 → (∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅))))
93	92	com23 86	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) → (𝑦 ∈ 𝑆 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅))))
94	93	ex 411	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ (Base‘𝐶) → (∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) → (𝑦 ∈ 𝑆 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅)))))
95	94	impd 409	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅)) → (𝑦 ∈ 𝑆 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅))))
96	5, 95	syld 47	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ 𝑆 → (𝑦 ∈ 𝑆 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅))))
97	96	imp32 417	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3051   ↦ cmpt 5226  ‘cfv 6542  (class class class)co 7415  Fincfn 8960  ℕcn 12240  ℕ₀cn0 12500  Basecbs 17177  ._rcmulr 17231  0_gc0g 17418   Σ_g cgsu 17419  Mndcmnd 18691  Ringcrg 20175  Poly₁cpl1 22102  coe₁cco1 22103   Mat cmat 22323   ConstPolyMat ccpmat 22621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-ot 4633  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-of 7681  df-ofr 7682  df-om 7868  df-1st 7989  df-2nd 7990  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-er 8721  df-map 8843  df-pm 8844  df-ixp 8913  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-fsupp 9384  df-sup 9463  df-oi 9531  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12501  df-z 12587  df-dec 12706  df-uz 12851  df-fz 13515  df-fzo 13658  df-seq 13997  df-hash 14320  df-struct 17113  df-sets 17130  df-slot 17148  df-ndx 17160  df-base 17178  df-ress 17207  df-plusg 17243  df-mulr 17244  df-sca 17246  df-vsca 17247  df-ip 17248  df-tset 17249  df-ple 17250  df-ds 17252  df-hom 17254  df-cco 17255  df-0g 17420  df-gsum 17421  df-prds 17426  df-pws 17428  df-mre 17563  df-mrc 17564  df-acs 17566  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-mhm 18737  df-submnd 18738  df-grp 18895  df-minusg 18896  df-mulg 19026  df-subg 19080  df-ghm 19170  df-cntz 19270  df-cmn 19739  df-abl 19740  df-mgp 20077  df-rng 20095  df-ur 20124  df-ring 20177  df-subrng 20485  df-subrg 20510  df-sra 21060  df-rgmod 21061  df-dsmm 21668  df-frlm 21683  df-psr 21844  df-mpl 21846  df-opsr 21848  df-psr1 22105  df-ply1 22107  df-coe1 22108  df-mat 22324  df-cpmat 22624

This theorem is referenced by:  cpmatmcl  22637