Theorem cpmatmcllem 20342

Description: Lemma for cpmatmcl 20343. (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 2610	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	1, 2, 3, 4	cpmatelimp 20336	. . 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 20336	. . . . . . . 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 3079	. . . . . . . . . . . . . . . 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 3047	. . . . . . . . . . . . . . . . . . . . . . 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 3079	. . . . . . . . . . . . . . . . . . . . . . 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 265	. . . . . . . . . . . . . . . . . . . . . 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 1830	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑐(((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
13		nfra1 2925	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑐∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅))
14	12, 13	nfan 1816	. . . . . . . . . . . . . . . . . . . . . . . . 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 803	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑅 ∈ Ring)
16		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (Base‘𝑃) = (Base‘𝑃)
17		simplrl 796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑖 ∈ 𝑁)
18		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
19		simplrl 796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20051	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑖𝑥𝑘) ∈ (Base‘𝑃))
22		simplrr 797	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑗 ∈ 𝑁)
23		simplrr 797	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20051	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑙 = 𝑘 → (𝑖𝑥𝑙) = (𝑖𝑥𝑘))
29	28	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑙 = 𝑘 → (coe1‘(𝑖𝑥𝑙)) = (coe1‘(𝑖𝑥𝑘)))
30	29	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑙 = 𝑘 → ((coe1‘(𝑖𝑥𝑙))‘𝑐) = ((coe1‘(𝑖𝑥𝑘))‘𝑐))
31	30	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑙 = 𝑘 → (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ↔ ((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g‘𝑅)))
32		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑙 = 𝑘 → (𝑙𝑦𝑗) = (𝑘𝑦𝑗))
33	32	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑙 = 𝑘 → (coe1‘(𝑙𝑦𝑗)) = (coe1‘(𝑘𝑦𝑗)))
34	33	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑙 = 𝑘 → ((coe1‘(𝑙𝑦𝑗))‘𝑐) = ((coe1‘(𝑘𝑦𝑗))‘𝑐))
35	34	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑙 = 𝑘 → (((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅) ↔ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g‘𝑅)))
36	31, 35	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑙 = 𝑘 → ((((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g‘𝑅)) ↔ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g‘𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g‘𝑅))))
37	36	rspcva 3280	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
38	37	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))))
39	38	exp4b 630	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))
40	39	com23 84	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))
41	40	imp31 447	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
42	41	ralimdva 2945	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
43	42	impancom 455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
44	43	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
45		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (0g‘𝑅) = (0g‘𝑅)
46		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (.r‘𝑃) = (.r‘𝑃)
47	2, 16, 45, 46	cply1mul 19485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
48	27, 44, 47	sylc 63	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
49	48	r19.21bi 2916	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
50	49	an32s 842	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
51	50	mpteq2dva 4672	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
52	51	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
53		ringmnd 18379	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
54	53	anim2i 591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Mnd))
55	54	ancomd 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin))
56	45	gsumz 17197	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin) → (𝑅 Σg (𝑘 ∈ 𝑁 ↦ (0g‘𝑅))) = (0g‘𝑅))
57	55, 56	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 Σg (𝑘 ∈ 𝑁 ↦ (0g‘𝑅))) = (0g‘𝑅))
58	57	ad4antr 764	. . . . . . . . . . . . . . . . . . . . . . . . . . 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))
59	52, 58	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . 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))
60	59	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . 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)))
61	14, 60	ralrimi 2940	. . . . . . . . . . . . . . . . . . . . . . . 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))
62		simp-4r 803	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑅 ∈ Ring)
63		nnnn0 11176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℕ → 𝑐 ∈ ℕ0)
64	63	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑐 ∈ ℕ0)
65	2	ply1ring 19439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
66	65	ad4antlr 765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑃 ∈ Ring)
67	16, 46	ringcl 18384	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑃 ∈ Ring ∧ (𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃)) → ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
68	66, 21, 25, 67	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
69	68	ralrimiva 2949	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ∀𝑘 ∈ 𝑁 ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
70	69	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))
71		simp-4l 802	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑁 ∈ Fin)
72	2, 16, 62, 64, 70, 71	coe1fzgsumd 19493	. . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
73	72	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . 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)))
74	73	ralbidva 2968	. . . . . . . . . . . . . . . . . . . . . . . . 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)))
75	74	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)))
76	61, 75	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . 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))
77	76	ex 449	. . . . . . . . . . . . . . . . . . . . . 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)))
78	11, 77	syl5bi 231	. . . . . . . . . . . . . . . . . . . . 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)))
79	78	expd 451	. . . . . . . . . . . . . . . . . . . 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))))
80	79	expr 641	. . . . . . . . . . . . . . . . . . 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)))))
81	80	com23 84	. . . . . . . . . . . . . . . . . 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)))))
82	81	imp31 447	. . . . . . . . . . . . . . . . 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)))
83	82	ralimdva 2945	. . . . . . . . . . . . . . . 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)))
84	8, 83	syl5bi 231	. . . . . . . . . . . . . . 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)))
85	84	ex 449	. . . . . . . . . . . . . 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))))
86	85	com23 84	. . . . . . . . . . . . 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))))
87	86	impancom 455	. . . . . . . . . . . 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))))
88	87	imp 444	. . . . . . . . . . 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)))
89	88	ralimdva 2945	. . . . . . . . . 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)))
90	89	ex 449	. . . . . . . . 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))))
91	90	expr 641	. . . . . . . 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)))))
92	91	impd 446	. . . . . . 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))))
93	7, 92	syld 46	. . . . . 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))))
94	93	com23 84	. . . . 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))))
95	94	ex 449	. . . 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)))))
96	95	impd 446	. . 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))))
97	5, 96	syld 46	. 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))))
98	97	imp32 448	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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  ℕcn 10897  ℕ₀cn0 11169  Basecbs 15695  ._rcmulr 15769  0_gc0g 15923   Σ_g cgsu 15924  Mndcmnd 17117  Ringcrg 18370  Poly₁cpl1 19368  coe₁cco1 19369   Mat cmat 20032   ConstPolyMat ccpmat 20327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-pws 15933  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-subrg 18601  df-sra 18993  df-rgmod 18994  df-psr 19177  df-mpl 19179  df-opsr 19181  df-psr1 19371  df-ply1 19373  df-coe1 19374  df-dsmm 19895  df-frlm 19910  df-mat 20033  df-cpmat 20330

This theorem is referenced by:  cpmatmcl  20343