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Theorem cpmatmcllem 22211

Description: Lemma for cpmatmcl 22212. (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 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘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 2732	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	1, 2, 3, 4	cpmatelimp 22205	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ 𝑆 → (𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅))))
6	1, 2, 3, 4	cpmatelimp 22205	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦 ∈ 𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))))
7	6	adantr 481	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ 𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))))
8		ralcom 3286	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅) ↔ ∀𝑗 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))
9		r19.26-2 3138	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)) ↔ (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)))
10		ralcom 3286	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)) ↔ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)))
11	9, 10	bitr3i 276	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)) ↔ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)))
12		nfv 1917	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑐(((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
13		nfra1 3281	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑐∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))
14	12, 13	nfan 1902	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑐((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘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 2732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 21919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 21919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 516	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))) ∧ 𝑘 ∈ 𝑁) → (𝑅 ∈ Ring ∧ ((𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃))))
28		oveq2 7413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑙 = 𝑘 → (𝑖𝑥𝑙) = (𝑖𝑥𝑘))
29	28	fveq2d 6892	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑙 = 𝑘 → (coe₁‘(𝑖𝑥𝑙)) = (coe₁‘(𝑖𝑥𝑘)))
30	29	fveq1d 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑙 = 𝑘 → ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = ((coe₁‘(𝑖𝑥𝑘))‘𝑐))
31	30	eqeq1d 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑙 = 𝑘 → (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ↔ ((coe₁‘(𝑖𝑥𝑘))‘𝑐) = (0_g‘𝑅)))
32		fvoveq1 7428	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑙 = 𝑘 → (coe₁‘(𝑙𝑦𝑗)) = (coe₁‘(𝑘𝑦𝑗)))
33	32	fveq1d 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑙 = 𝑘 → ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = ((coe₁‘(𝑘𝑦𝑗))‘𝑐))
34	33	eqeq1d 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑙 = 𝑘 → (((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅) ↔ ((coe₁‘(𝑘𝑦𝑗))‘𝑐) = (0_g‘𝑅)))
35	31, 34	anbi12d 631	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑙 = 𝑘 → ((((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)) ↔ (((coe₁‘(𝑖𝑥𝑘))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑘𝑦𝑗))‘𝑐) = (0_g‘𝑅))))
36	35	rspcva 3610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑘 ∈ 𝑁 ∧ ∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))) → (((coe₁‘(𝑖𝑥𝑘))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑘𝑦𝑗))‘𝑐) = (0_g‘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) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))) → (((coe₁‘(𝑖𝑥𝑘))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑘𝑦𝑗))‘𝑐) = (0_g‘𝑅))))
38	37	exp4b 431	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑐 ∈ ℕ → (𝑘 ∈ 𝑁 → (∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)) → (((coe₁‘(𝑖𝑥𝑘))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑘𝑦𝑗))‘𝑐) = (0_g‘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) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)) → (((coe₁‘(𝑖𝑥𝑘))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑘𝑦𝑗))‘𝑐) = (0_g‘𝑅))))))
40	39	imp31 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) ∧ 𝑐 ∈ ℕ) → (∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)) → (((coe₁‘(𝑖𝑥𝑘))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑘𝑦𝑗))‘𝑐) = (0_g‘𝑅))))
41	40	ralimdva 3167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)) → ∀𝑐 ∈ ℕ (((coe₁‘(𝑖𝑥𝑘))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑘𝑦𝑗))‘𝑐) = (0_g‘𝑅))))
42	41	impancom 452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))) → (𝑘 ∈ 𝑁 → ∀𝑐 ∈ ℕ (((coe₁‘(𝑖𝑥𝑘))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑘𝑦𝑗))‘𝑐) = (0_g‘𝑅))))
43	42	imp 407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))) ∧ 𝑘 ∈ 𝑁) → ∀𝑐 ∈ ℕ (((coe₁‘(𝑖𝑥𝑘))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑘𝑦𝑗))‘𝑐) = (0_g‘𝑅)))
44		eqid 2732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
45		eqid 2732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (._r‘𝑃) = (._r‘𝑃)
46	2, 16, 44, 45	cply1mul 21809	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑅 ∈ Ring ∧ ((𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃))) → (∀𝑐 ∈ ℕ (((coe₁‘(𝑖𝑥𝑘))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑘𝑦𝑗))‘𝑐) = (0_g‘𝑅)) → ∀𝑐 ∈ ℕ ((coe₁‘((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))‘𝑐) = (0_g‘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) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))) ∧ 𝑘 ∈ 𝑁) → ∀𝑐 ∈ ℕ ((coe₁‘((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))‘𝑐) = (0_g‘𝑅))
48	47	r19.21bi 3248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))) ∧ 𝑘 ∈ 𝑁) ∧ 𝑐 ∈ ℕ) → ((coe₁‘((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))‘𝑐) = (0_g‘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) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))) ∧ 𝑐 ∈ ℕ) ∧ 𝑘 ∈ 𝑁) → ((coe₁‘((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))‘𝑐) = (0_g‘𝑅))
50	49	mpteq2dva 5247	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))) ∧ 𝑐 ∈ ℕ) → (𝑘 ∈ 𝑁 ↦ ((coe₁‘((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))‘𝑐)) = (𝑘 ∈ 𝑁 ↦ (0_g‘𝑅)))
51	50	oveq2d 7421	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))) ∧ 𝑐 ∈ ℕ) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((coe₁‘((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ (0_g‘𝑅))))
52		ringmnd 20059	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
53	52	anim2i 617	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Mnd))
54	53	ancomd 462	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin))
55	44	gsumz 18713	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ (0_g‘𝑅))) = (0_g‘𝑅))
56	54, 55	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ (0_g‘𝑅))) = (0_g‘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) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))) ∧ 𝑐 ∈ ℕ) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ (0_g‘𝑅))) = (0_g‘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) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))) ∧ 𝑐 ∈ ℕ) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((coe₁‘((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0_g‘𝑅))
59	58	ex 413	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))) → (𝑐 ∈ ℕ → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((coe₁‘((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0_g‘𝑅)))
60	14, 59	ralrimi 3254	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))) → ∀𝑐 ∈ ℕ (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((coe₁‘((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0_g‘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 12475	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℕ → 𝑐 ∈ ℕ₀)
63	62	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑐 ∈ ℕ₀)
64	2	ply1ring 21761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20066	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑃 ∈ Ring ∧ (𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃)) → ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
67	65, 21, 25, 66	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3146	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 21817	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((coe₁‘((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))‘𝑐))))
72	71	eqeq1d 2734	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑐 ∈ ℕ) → (((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅) ↔ (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((coe₁‘((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0_g‘𝑅)))
73	72	ralbidva 3175	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅) ↔ ∀𝑐 ∈ ℕ (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((coe₁‘((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0_g‘𝑅)))
74	73	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))) → (∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅) ↔ ∀𝑐 ∈ ℕ (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((coe₁‘((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0_g‘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) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅))) → ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅))
76	75	ex 413	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (∀𝑐 ∈ ℕ ∀𝑙 ∈ 𝑁 (((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) ∧ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)) → ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘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) = (0_g‘𝑅) ∧ ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)) → ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅)))
78	77	expd 416	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅) → ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅))))
79	78	expr 457	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖 ∈ 𝑁) → (𝑗 ∈ 𝑁 → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅) → ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘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) = (0_g‘𝑅) → (𝑗 ∈ 𝑁 → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅) → ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅)))))
81	80	imp31 418	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖 ∈ 𝑁) ∧ ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅)) ∧ 𝑗 ∈ 𝑁) → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅) → ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅)))
82	81	ralimdva 3167	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖 ∈ 𝑁) ∧ ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅)) → (∀𝑗 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅) → ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘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) = (0_g‘𝑅)) → (∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅) → ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅)))
84	83	ex 413	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖 ∈ 𝑁) → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) → (∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅) → ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘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) = (0_g‘𝑅) → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) → ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅))))
86	85	impancom 452	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)) → (𝑖 ∈ 𝑁 → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) → ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅))))
87	86	imp 407	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)) ∧ 𝑖 ∈ 𝑁) → (∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) → ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅)))
88	87	ralimdva 3167	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)) → (∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅)))
89	88	ex 413	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅) → (∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅))))
90	89	expr 457	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐶) → (∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅) → (∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅)))))
91	90	impd 411	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑙𝑦𝑗))‘𝑐) = (0_g‘𝑅)) → (∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘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) = (0_g‘𝑅) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅))))
93	92	com23 86	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) → (𝑦 ∈ 𝑆 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅))))
94	93	ex 413	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ (Base‘𝐶) → (∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅) → (𝑦 ∈ 𝑆 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅)))))
95	94	impd 411	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖 ∈ 𝑁 ∀𝑙 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑖𝑥𝑙))‘𝑐) = (0_g‘𝑅)) → (𝑦 ∈ 𝑆 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅))))
96	5, 95	syld 47	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ 𝑆 → (𝑦 ∈ 𝑆 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅))))
97	96	imp32 419	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(._r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0_g‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ↦ cmpt 5230 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 ℕcn 12208 ℕ₀cn0 12468 Basecbs 17140 ._rcmulr 17194 0_gc0g 17381 Σ_g cgsu 17382 Mndcmnd 18621 Ringcrg 20049 Poly₁cpl1 21692 coe₁cco1 21693 Mat cmat 21898 ConstPolyMat ccpmat 22196

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-mulg 18945 df-subg 18997 df-ghm 19084 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-subrg 20353 df-sra 20777 df-rgmod 20778 df-dsmm 21278 df-frlm 21293 df-psr 21453 df-mpl 21455 df-opsr 21457 df-psr1 21695 df-ply1 21697 df-coe1 21698 df-mat 21899 df-cpmat 22199

This theorem is referenced by: cpmatmcl 22212

W3C validator