Theorem cpmatmcllem 22693

Description: Lemma for cpmatmcl 22694. (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 2737	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	1, 2, 3, 4	cpmatelimp 22687	. . 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 22687	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦 ∈ 𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))))
7	6	adantr 480	. . . . . . 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 3266	. . . . . . . . . . . . . . . 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 3123	. . . . . . . . . . . . . . . . . . . . . . 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 3266	. . . . . . . . . . . . . . . . . . . . . . 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 277	. . . . . . . . . . . . . . . . . . . . . 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 1916	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑐(((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
13		nfra1 3262	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑐∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))
14	12, 13	nfan 1901	. . . . . . . . . . . . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑅 ∈ Ring)
16		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (Base‘𝑃) = (Base‘𝑃)
17		simplrl 777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑖 ∈ 𝑁)
18		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
19		simplrl 777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑥 ∈ (Base‘𝐶))
20	19	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 22401	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑖𝑥𝑘) ∈ (Base‘𝑃))
22		simplrr 778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑗 ∈ 𝑁)
23		simplrr 778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑦 ∈ (Base‘𝐶))
24	23	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 22401	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑙 = 𝑘 → (𝑖𝑥𝑙) = (𝑖𝑥𝑘))
29	28	fveq2d 6838	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑙 = 𝑘 → (coe1‘(𝑖𝑥𝑙)) = (coe1‘(𝑖𝑥𝑘)))
30	29	fveq1d 6836	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑙 = 𝑘 → ((coe1‘(𝑖𝑥𝑙))‘𝑐) = ((coe1‘(𝑖𝑥𝑘))‘𝑐))
31	30	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑙 = 𝑘 → (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ↔ ((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g‘𝑅)))
32		fvoveq1 7383	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑙 = 𝑘 → (coe1‘(𝑙𝑦𝑗)) = (coe1‘(𝑘𝑦𝑗)))
33	32	fveq1d 6836	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑙 = 𝑘 → ((coe1‘(𝑙𝑦𝑗))‘𝑐) = ((coe1‘(𝑘𝑦𝑗))‘𝑐))
34	33	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑙 = 𝑘 → (((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅) ↔ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g‘𝑅)))
35	31, 34	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3563	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 430	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3150	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (0g‘𝑅) = (0g‘𝑅)
45		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (.r‘𝑃) = (.r‘𝑃)
46	2, 16, 44, 45	cply1mul 22271	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3230	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 653	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5179	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 20215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
53	52	anim2i 618	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Mnd))
54	53	ancomd 461	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin))
55	44	gsumz 18795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 733	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . 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 3236	. . . . . . . . . . . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑅 ∈ Ring)
62		nnnn0 12435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℕ → 𝑐 ∈ ℕ0)
63	62	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑐 ∈ ℕ0)
64	2	ply1ring 22221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
65	64	ad4antlr 734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑃 ∈ Ring)
66	16, 45	ringcl 20222	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑃 ∈ Ring ∧ (𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃)) → ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
67	65, 21, 25, 66	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 22279	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3159	. . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . 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 3150	. . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . 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 451	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 3150	. . . . . . . . . 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 412	. . . . . . . . 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 456	. . . . . . . 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 410	. . . . . . 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 412	. . . 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 410	. . 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 418	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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ↦ cmpt 5167  ‘cfv 6492  (class class class)co 7360  Fincfn 8886  ℕcn 12165  ℕ₀cn0 12428  Basecbs 17170  ._rcmulr 17212  0_gc0g 17393   Σ_g cgsu 17394  Mndcmnd 18693  Ringcrg 20205  Poly₁cpl1 22150  coe₁cco1 22151   Mat cmat 22382   ConstPolyMat ccpmat 22678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-ofr 7625  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8104  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-er 8636  df-map 8768  df-pm 8769  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9268  df-sup 9348  df-oi 9418  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-fzo 13600  df-seq 13955  df-hash 14284  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-hom 17235  df-cco 17236  df-0g 17395  df-gsum 17396  df-prds 17401  df-pws 17403  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-submnd 18743  df-grp 18903  df-minusg 18904  df-mulg 19035  df-subg 19090  df-ghm 19179  df-cntz 19283  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-subrng 20514  df-subrg 20538  df-sra 21160  df-rgmod 21161  df-dsmm 21722  df-frlm 21737  df-psr 21899  df-mpl 21901  df-opsr 21903  df-psr1 22153  df-ply1 22155  df-coe1 22156  df-mat 22383  df-cpmat 22681

This theorem is referenced by:  cpmatmcl  22694