Theorem cpmatmcllem 22836

Description: Lemma for cpmatmcl 22837. (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 2765	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	1, 2, 3, 4	cpmatelimp 22830	. . 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 22830	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦 ∈ 𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))))
7	6	adantr 485	. . . . . . 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 3293	. . . . . . . . . . . . . . . 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 3150	. . . . . . . . . . . . . . . . . . . . . . 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 3293	. . . . . . . . . . . . . . . . . . . . . . 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 280	. . . . . . . . . . . . . . . . . . . . . 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 1937	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑐(((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
13		nfra1 3289	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑐∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))
14	12, 13	nfan 1922	. . . . . . . . . . . . . . . . . . . . . . . . 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 795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑅 ∈ Ring)
16		eqid 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (Base‘𝑃) = (Base‘𝑃)
17		simplrl 788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑖 ∈ 𝑁)
18		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
19		simplrl 788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑥 ∈ (Base‘𝐶))
20	19	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 22544	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑖𝑥𝑘) ∈ (Base‘𝑃))
22		simplrr 789	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑗 ∈ 𝑁)
23		simplrr 789	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑦 ∈ (Base‘𝐶))
24	23	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 22544	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 524	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑙 = 𝑘 → (𝑖𝑥𝑙) = (𝑖𝑥𝑘))
29	28	fveq2d 6875	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑙 = 𝑘 → (coe1‘(𝑖𝑥𝑙)) = (coe1‘(𝑖𝑥𝑘)))
30	29	fveq1d 6873	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑙 = 𝑘 → ((coe1‘(𝑖𝑥𝑙))‘𝑐) = ((coe1‘(𝑖𝑥𝑘))‘𝑐))
31	30	eqeq1d 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑙 = 𝑘 → (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ↔ ((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g‘𝑅)))
32		fvoveq1 7423	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑙 = 𝑘 → (coe1‘(𝑙𝑦𝑗)) = (coe1‘(𝑘𝑦𝑗)))
33	32	fveq1d 6873	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑙 = 𝑘 → ((coe1‘(𝑙𝑦𝑗))‘𝑐) = ((coe1‘(𝑘𝑦𝑗))‘𝑐))
34	33	eqeq1d 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑙 = 𝑘 → (((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅) ↔ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g‘𝑅)))
35	31, 34	anbi12d 643	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 87	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 422	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (0g‘𝑅) = (0g‘𝑅)
45		eqid 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (.r‘𝑃) = (.r‘𝑃)
46	2, 16, 44, 45	cply1mul 22417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 66	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 664	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5198	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7416	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 20316	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
53	52	anim2i 628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Mnd))
54	53	ancomd 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin))
55	44	gsumz 18885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin) → (𝑅 Σg (𝑘 ∈ 𝑁 ↦ (0g‘𝑅))) = (0g‘𝑅))
56	54, 55	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 Σg (𝑘 ∈ 𝑁 ↦ (0g‘𝑅))) = (0g‘𝑅))
57	56	ad4antr 744	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . . . . . . . . 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 3263	. . . . . . . . . . . . . . . . . . . . . . . 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 795	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑅 ∈ Ring)
62		nnnn0 12502	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℕ → 𝑐 ∈ ℕ0)
63	62	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑐 ∈ ℕ0)
64	2	ply1ring 22367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
65	64	ad4antlr 745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑃 ∈ Ring)
66	16, 45	ringcl 20323	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑃 ∈ Ring ∧ (𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃)) → ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
67	65, 21, 25, 66	syl3anc 1394	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3157	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 22425	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . . . . . 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 87	. . . . . . . . . . . . . . . . . 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 422	. . . . . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . 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 87	. . . . . . . . . . . . 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 456	. . . . . . . . . . . 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 411	. . . . . . . . . . 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 3177	. . . . . . . . . 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 417	. . . . . . . . 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 461	. . . . . . . 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 415	. . . . . . 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 48	. . . . . 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 87	. . . . 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 417	. . . 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 415	. . 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 48	. 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 423	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 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079   ↦ cmpt 5186  ‘cfv 6525  (class class class)co 7400  Fincfn 8931  ℕcn 12224  ℕ₀cn0 12495  Basecbs 17259  ._rcmulr 17301  0_gc0g 17482   Σ_g cgsu 17483  Mndcmnd 18782  Ringcrg 20306  Poly₁cpl1 22297  coe₁cco1 22298   Mat cmat 22525   ConstPolyMat ccpmat 22821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-ofr 7665  df-om 7851  df-1st 7974  df-2nd 7975  df-supp 8145  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-er 8682  df-map 8814  df-pm 8815  df-ixp 8884  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fsupp 9310  df-sup 9390  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-9 12301  df-n0 12496  df-z 12583  df-dec 12703  df-uz 12854  df-fz 13527  df-fzo 13674  df-seq 14029  df-hash 14358  df-struct 17197  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-plusg 17313  df-mulr 17314  df-sca 17316  df-vsca 17317  df-ip 17318  df-tset 17319  df-ple 17320  df-ds 17322  df-hom 17324  df-cco 17325  df-0g 17484  df-gsum 17485  df-prds 17490  df-pws 17492  df-mre 17628  df-mrc 17629  df-acs 17631  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-mhm 18831  df-submnd 18832  df-grp 18993  df-minusg 18994  df-mulg 19125  df-subg 19180  df-ghm 19275  df-cntz 19378  df-cmn 19843  df-abl 19844  df-mgp 20208  df-rng 20222  df-ur 20255  df-ring 20308  df-subrng 20622  df-subrg 20646  df-sra 21263  df-rgmod 21264  df-dsmm 21842  df-frlm 21857  df-psr 22019  df-mpl 22021  df-opsr 22023  df-psr1 22300  df-ply1 22302  df-coe1 22303  df-mat 22526  df-cpmat 22824

This theorem is referenced by:  cpmatmcl  22837