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Theorem dmatmul 22774

Description: The product of two diagonal matrices. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)

**Hypotheses**
Ref	Expression
dmatid.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
dmatid.b	⊢ 𝐵 = (Base‘𝐴)
dmatid.0	⊢ 0 = (0_g‘𝑅)
dmatid.d	⊢ 𝐷 = (𝑁 DMat 𝑅)

**Assertion**
Ref	Expression
dmatmul	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑋(._r‘𝐴)𝑌) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)), 0 )))

Distinct variable groups: 𝑥,𝐷,𝑦 𝑥,𝑁,𝑦 𝑥,𝑅,𝑦 𝑥,𝑋,𝑦 𝑥,𝑌,𝑦 Allowed substitution hints: 𝐴(𝑥, 𝑦) 𝐵(𝑥, 𝑦) 0 (𝑥, 𝑦)

Proof of Theorem dmatmul Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmatid.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		eqid 2760	. . . . . 6 ⊢ (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
3	1, 2	matmulr 22715	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (._r‘𝐴))
4	3	adantr 486	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (._r‘𝐴))
5	4	eqcomd 2766	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (._r‘𝐴) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
6	5	oveqd 7425	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑋(._r‘𝐴)𝑌) = (𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝑌))
7		eqid 2760	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2760	. . 3 ⊢ (._r‘𝑅) = (._r‘𝑅)
9		simplr 781	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑅 ∈ Ring)
10		simpll 779	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑁 ∈ Fin)
11		dmatid.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
12		dmatid.0	. . . . . . 7 ⊢ 0 = (0_g‘𝑅)
13		dmatid.d	. . . . . . 7 ⊢ 𝐷 = (𝑁 DMat 𝑅)
14	1, 11, 12, 13	dmatmat 22771	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 ∈ 𝐷 → 𝑋 ∈ 𝐵))
15	14	imp 412	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋 ∈ 𝐷) → 𝑋 ∈ 𝐵)
16	1, 7, 11	matbas2i 22699	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
17	15, 16	syl 18	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋 ∈ 𝐷) → 𝑋 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
18	17	adantrr 730	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑋 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
19	1, 11, 12, 13	dmatmat 22771	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑌 ∈ 𝐷 → 𝑌 ∈ 𝐵))
20	19	imp 412	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌 ∈ 𝐷) → 𝑌 ∈ 𝐵)
21	1, 7, 11	matbas2i 22699	. . . . 5 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
22	20, 21	syl 18	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌 ∈ 𝐷) → 𝑌 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
23	22	adantrl 729	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑌 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
24	2, 7, 8, 9, 10, 10, 10, 18, 23	mamuval 22670	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝑌) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦))))))
25		eqid 2760	. . . . . . 7 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
26		ringcmn 20473	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
27	26	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑅 ∈ CMnd)
28	27	3ad2ant1 1151	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑅 ∈ CMnd)
29	28	adantl 487	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑅 ∈ CMnd)
30	10	3ad2ant1 1151	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑁 ∈ Fin)
31	30	adantl 487	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑁 ∈ Fin)
32		eqid 2760	. . . . . . . 8 ⊢ (𝑘 ∈ 𝑁 ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦))) = (𝑘 ∈ 𝑁 ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)))
33		ovexd 7443	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)) ∈ V)
34		fvexd 6888	. . . . . . . 8 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (0_g‘𝑅) ∈ V)
35	32, 31, 33, 34	fsuppmptdm 9346	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑘 ∈ 𝑁 ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦))) finSupp (0_g‘𝑅))
36	9	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑅 ∈ Ring)
37	36	ad2antlr 740	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑅 ∈ Ring)
38		simp2 1155	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁)
39	38	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑥 ∈ 𝑁)
40		simpr 490	. . . . . . . . 9 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
41		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐴) = (Base‘𝐴)
42	1, 41, 12, 13	dmatmat 22771	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 ∈ 𝐷 → 𝑋 ∈ (Base‘𝐴)))
43	42	imp 412	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋 ∈ 𝐷) → 𝑋 ∈ (Base‘𝐴))
44	43	adantrr 730	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑋 ∈ (Base‘𝐴))
45	44	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑋 ∈ (Base‘𝐴))
46	45	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑋 ∈ (Base‘𝐴))
47	1, 7	matecl 22702	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁 ∧ 𝑋 ∈ (Base‘𝐴)) → (𝑥𝑋𝑘) ∈ (Base‘𝑅))
48	39, 40, 46, 47	syl3anc 1398	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑥𝑋𝑘) ∈ (Base‘𝑅))
49		simplr3 1236	. . . . . . . . 9 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑦 ∈ 𝑁)
50	1, 41, 12, 13	dmatmat 22771	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑌 ∈ 𝐷 → 𝑌 ∈ (Base‘𝐴)))
51	50	imp 412	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌 ∈ 𝐷) → 𝑌 ∈ (Base‘𝐴))
52	51	adantrl 729	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑌 ∈ (Base‘𝐴))
53	52	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑌 ∈ (Base‘𝐴))
54	53	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑌 ∈ (Base‘𝐴))
55	1, 7	matecl 22702	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁 ∧ 𝑌 ∈ (Base‘𝐴)) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
56	40, 49, 54, 55	syl3anc 1398	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
57	7, 8	ringcl 20439	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑘) ∈ (Base‘𝑅) ∧ (𝑘𝑌𝑦) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)) ∈ (Base‘𝑅))
58	37, 48, 56, 57	syl3anc 1398	. . . . . . 7 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)) ∈ (Base‘𝑅))
59	38	adantl 487	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
60		simp3 1156	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
61	15	adantrr 730	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑋 ∈ 𝐵)
62	61, 11	eleqtrdi 2870	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑋 ∈ (Base‘𝐴))
63	62	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑋 ∈ (Base‘𝐴))
64	1, 7	matecl 22702	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁 ∧ 𝑋 ∈ (Base‘𝐴)) → (𝑥𝑋𝑦) ∈ (Base‘𝑅))
65	38, 60, 63, 64	syl3anc 1398	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑋𝑦) ∈ (Base‘𝑅))
66	50	a1d 26	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 ∈ 𝐷 → (𝑌 ∈ 𝐷 → 𝑌 ∈ (Base‘𝐴))))
67	66	imp32 424	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑌 ∈ (Base‘𝐴))
68	67	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑌 ∈ (Base‘𝐴))
69	1, 7	matecl 22702	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁 ∧ 𝑌 ∈ (Base‘𝐴)) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
70	38, 60, 68, 69	syl3anc 1398	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
71	7, 8	ringcl 20439	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑦) ∈ (Base‘𝑅) ∧ (𝑥𝑌𝑦) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
72	36, 65, 70, 71	syl3anc 1398	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
73	72	adantl 487	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
74		eqtr 2780	. . . . . . . . . . 11 ⊢ ((𝑘 = 𝑥 ∧ 𝑥 = 𝑦) → 𝑘 = 𝑦)
75	74	ancoms 464	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑦 ∧ 𝑘 = 𝑥) → 𝑘 = 𝑦)
76	75	oveq2d 7424	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑘 = 𝑥) → (𝑥𝑋𝑘) = (𝑥𝑋𝑦))
77	76	adantlr 728	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 = 𝑥) → (𝑥𝑋𝑘) = (𝑥𝑋𝑦))
78		oveq1 7415	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (𝑘𝑌𝑦) = (𝑥𝑌𝑦))
79	78	adantl 487	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 = 𝑥) → (𝑘𝑌𝑦) = (𝑥𝑌𝑦))
80	77, 79	oveq12d 7426	. . . . . . 7 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 = 𝑥) → ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)) = ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)))
81	7, 25, 29, 31, 35, 58, 59, 73, 80	gsumdifsnd 20137	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)))) = ((𝑅 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦))))(+_g‘𝑅)((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦))))
82		simprl 783	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑋 ∈ 𝐷)
83	10, 9, 82	3jca 1146	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷))
84	83	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷))
85	84	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷))
86	38	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑥 ∈ 𝑁)
87		eldifi 4077	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑁 ∖ {𝑥}) → 𝑘 ∈ 𝑁)
88	87	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑘 ∈ 𝑁)
89		eldifsni 4752	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝑁 ∖ {𝑥}) → 𝑘 ≠ 𝑥)
90	89	necomd 3010	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑁 ∖ {𝑥}) → 𝑥 ≠ 𝑘)
91	90	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑥 ≠ 𝑘)
92	1, 11, 12, 13	dmatelnd 22773	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷) ∧ (𝑥 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁 ∧ 𝑥 ≠ 𝑘)) → (𝑥𝑋𝑘) = 0 )
93	85, 86, 88, 91, 92	syl13anc 1399	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → (𝑥𝑋𝑘) = 0 )
94	93	oveq1d 7423	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)) = ( 0 (._r‘𝑅)(𝑘𝑌𝑦)))
95	36	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑅 ∈ Ring)
96		simplr3 1236	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑦 ∈ 𝑁)
97	53	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑌 ∈ (Base‘𝐴))
98	88, 96, 97, 55	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
99	7, 8, 12	ringlz 20486	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ (𝑘𝑌𝑦) ∈ (Base‘𝑅)) → ( 0 (._r‘𝑅)(𝑘𝑌𝑦)) = 0 )
100	95, 98, 99	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → ( 0 (._r‘𝑅)(𝑘𝑌𝑦)) = 0 )
101	94, 100	eqtrd 2795	. . . . . . . . . 10 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)) = 0 )
102	101	mpteq2dva 5197	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦))) = (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ 0 ))
103	102	oveq2d 7424	. . . . . . . 8 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑅 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)))) = (𝑅 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ 0 )))
104		diffi 9168	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝑥}) ∈ Fin)
105		ringmnd 20432	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
106	104, 105	anim12ci 626	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
107	106	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
108	107	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
109	108	adantl 487	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
110	12	gsumz 18994	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin) → (𝑅 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ 0 )) = 0 )
111	109, 110	syl 18	. . . . . . . 8 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑅 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ 0 )) = 0 )
112	103, 111	eqtrd 2795	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑅 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)))) = 0 )
113	112	oveq1d 7423	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ((𝑅 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦))))(+_g‘𝑅)((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦))) = ( 0 (+_g‘𝑅)((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦))))
114	105	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑅 ∈ Mnd)
115	114	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑅 ∈ Mnd)
116	38, 60, 53, 69	syl3anc 1398	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
117	36, 65, 116, 71	syl3anc 1398	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
118	115, 117	jca 521	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑅 ∈ Mnd ∧ ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅)))
119	118	adantl 487	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑅 ∈ Mnd ∧ ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅)))
120	7, 25, 12	mndlid 18906	. . . . . . 7 ⊢ ((𝑅 ∈ Mnd ∧ ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅)) → ( 0 (+_g‘𝑅)((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦))) = ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)))
121	119, 120	syl 18	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ( 0 (+_g‘𝑅)((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦))) = ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)))
122	81, 113, 121	3eqtrd 2799	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)))) = ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)))
123		iftrue 4487	. . . . . 6 ⊢ (𝑥 = 𝑦 → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)), 0 ) = ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)))
124	123	adantr 486	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)), 0 ) = ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)))
125	122, 124	eqtr4d 2798	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)))) = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)), 0 ))
126		simprr 785	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑌 ∈ 𝐷)
127	10, 9, 126	3jca 1146	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐷))
128	127	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐷))
129	128	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐷))
130	129	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐷))
131		simprr 785	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → 𝑘 ∈ 𝑁)
132		simplr3 1236	. . . . . . . . . . . 12 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑦 ∈ 𝑁)
133	132	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → 𝑦 ∈ 𝑁)
134		df-ne 2956	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
135		neeq1 3017	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → (𝑥 ≠ 𝑦 ↔ 𝑘 ≠ 𝑦))
136	135	biimpcd 252	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≠ 𝑦 → (𝑥 = 𝑘 → 𝑘 ≠ 𝑦))
137	134, 136	sylbir 238	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑥 = 𝑦 → (𝑥 = 𝑘 → 𝑘 ≠ 𝑦))
138	137	adantr 486	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥 = 𝑘 → 𝑘 ≠ 𝑦))
139	138	adantr 486	. . . . . . . . . . . 12 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑥 = 𝑘 → 𝑘 ≠ 𝑦))
140	139	impcom 413	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → 𝑘 ≠ 𝑦)
141	1, 11, 12, 13	dmatelnd 22773	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐷) ∧ (𝑘 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁 ∧ 𝑘 ≠ 𝑦)) → (𝑘𝑌𝑦) = 0 )
142	130, 131, 133, 140, 141	syl13anc 1399	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → (𝑘𝑌𝑦) = 0 )
143	142	oveq2d 7424	. . . . . . . . 9 ⊢ ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)) = ((𝑥𝑋𝑘)(._r‘𝑅) 0 ))
144	36	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑅 ∈ Ring)
145	38	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑥 ∈ 𝑁)
146		simpr 490	. . . . . . . . . . . 12 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
147	63	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑋 ∈ (Base‘𝐴))
148	145, 146, 147, 47	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑥𝑋𝑘) ∈ (Base‘𝑅))
149	7, 8, 12	ringrz 20487	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑘) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑘)(._r‘𝑅) 0 ) = 0 )
150	144, 148, 149	syl2anc 596	. . . . . . . . . 10 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → ((𝑥𝑋𝑘)(._r‘𝑅) 0 ) = 0 )
151	150	adantl 487	. . . . . . . . 9 ⊢ ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → ((𝑥𝑋𝑘)(._r‘𝑅) 0 ) = 0 )
152	143, 151	eqtrd 2795	. . . . . . . 8 ⊢ ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)) = 0 )
153	84	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷))
154	153	adantl 487	. . . . . . . . . . 11 ⊢ ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷))
155	145	adantl 487	. . . . . . . . . . 11 ⊢ ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
156		simprr 785	. . . . . . . . . . 11 ⊢ ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → 𝑘 ∈ 𝑁)
157		neqne 2963	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 = 𝑘 → 𝑥 ≠ 𝑘)
158	157	adantr 486	. . . . . . . . . . 11 ⊢ ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → 𝑥 ≠ 𝑘)
159	154, 155, 156, 158, 92	syl13anc 1399	. . . . . . . . . 10 ⊢ ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → (𝑥𝑋𝑘) = 0 )
160	159	oveq1d 7423	. . . . . . . . 9 ⊢ ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)) = ( 0 (._r‘𝑅)(𝑘𝑌𝑦)))
161	68	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑌 ∈ (Base‘𝐴))
162	146, 132, 161, 55	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
163	144, 162, 99	syl2anc 596	. . . . . . . . . 10 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → ( 0 (._r‘𝑅)(𝑘𝑌𝑦)) = 0 )
164	163	adantl 487	. . . . . . . . 9 ⊢ ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → ( 0 (._r‘𝑅)(𝑘𝑌𝑦)) = 0 )
165	160, 164	eqtrd 2795	. . . . . . . 8 ⊢ ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)) = 0 )
166	152, 165	pm2.61ian 824	. . . . . . 7 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)) = 0 )
167	166	mpteq2dva 5197	. . . . . 6 ⊢ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑘 ∈ 𝑁 ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦))) = (𝑘 ∈ 𝑁 ↦ 0 ))
168	167	oveq2d 7424	. . . . 5 ⊢ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)))) = (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ 0 )))
169	105	anim2i 629	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Mnd))
170	169	ancomd 467	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin))
171	12	gsumz 18994	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ 0 )) = 0 )
172	170, 171	syl 18	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ 0 )) = 0 )
173	172	adantr 486	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ 0 )) = 0 )
174	173	3ad2ant1 1151	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ 0 )) = 0 )
175	174	adantl 487	. . . . 5 ⊢ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ 0 )) = 0 )
176		iffalse 4490	. . . . . . 7 ⊢ (¬ 𝑥 = 𝑦 → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)), 0 ) = 0 )
177	176	eqcomd 2766	. . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → 0 = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)), 0 ))
178	177	adantr 486	. . . . 5 ⊢ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 0 = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)), 0 ))
179	168, 175, 178	3eqtrd 2799	. . . 4 ⊢ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)))) = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)), 0 ))
180	125, 179	pm2.61ian 824	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)))) = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)), 0 ))
181	180	mpoeq3dva 7485	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦))))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)), 0 )))
182	6, 24, 181	3eqtrd 2799	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑋(._r‘𝐴)𝑌) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)), 0 )))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 Vcvv 3450 ∖ cdif 3895 ifcif 4481 {csn 4583 ⟨cotp 4591 ↦ cmpt 5185 × cxp 5645 ‘cfv 6527 (class class class)co 7408 ∈ cmpo 7410 ↑_m cmap 8825 Fincfn 8951 Basecbs 17349 +_gcplusg 17390 ._rcmulr 17391 0_gc0g 17572 Σ_g cgsu 17573 Mndcmnd 18885 CMndccmn 19956 Ringcrg 20421 maMul cmmul 22667 Mat cmat 22684 DMat cdmat 22765

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11228 ax-resscn 11229 ax-1cn 11230 ax-icn 11231 ax-addcl 11232 ax-addrcl 11233 ax-mulcl 11234 ax-mulrcl 11235 ax-mulcom 11236 ax-addass 11237 ax-mulass 11238 ax-distr 11239 ax-i2m1 11240 ax-1ne0 11241 ax-1rid 11242 ax-rnegex 11243 ax-rrecex 11244 ax-cnre 11245 ax-pre-lttri 11246 ax-pre-lttrn 11247 ax-pre-ltadd 11248 ax-pre-mulgt0 11249

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-ot 4592 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-isom 6536 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-ixp 8904 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-fsupp 9332 df-sup 9412 df-oi 9482 df-card 9992 df-pnf 11317 df-mnf 11318 df-xr 11319 df-ltxr 11320 df-le 11321 df-sub 11515 df-neg 11516 df-nn 12306 df-2 12375 df-3 12376 df-4 12377 df-5 12378 df-6 12379 df-7 12380 df-8 12381 df-9 12382 df-n0 12577 df-z 12664 df-dec 12785 df-uz 12936 df-fz 13610 df-fzo 13758 df-seq 14114 df-hash 14443 df-struct 17287 df-sets 17304 df-slot 17322 df-ndx 17334 df-base 17350 df-ress 17371 df-plusg 17403 df-mulr 17404 df-sca 17406 df-vsca 17407 df-ip 17408 df-tset 17409 df-ple 17410 df-ds 17412 df-hom 17414 df-cco 17415 df-0g 17574 df-gsum 17575 df-prds 17580 df-pws 17582 df-mre 17718 df-mrc 17719 df-acs 17721 df-mgm 18778 df-sgrp 18870 df-mnd 18886 df-submnd 18941 df-grp 19109 df-minusg 19110 df-mulg 19240 df-cntz 19493 df-cmn 19958 df-abl 19959 df-mgp 20323 df-rng 20337 df-ur 20370 df-ring 20423 df-sra 21410 df-rgmod 21411 df-dsmm 22000 df-frlm 22015 df-mamu 22668 df-mat 22685 df-dmat 22767

This theorem is used by: dmatmulcl 22777 dmatcrng 22779 scmatscmiddistr 22785 scmatcrng 22798

W3C validator