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Theorem dmatmul 22519

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 2735	. . . . . 6 ⊢ (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
3	1, 2	matmulr 22460	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (._r‘𝐴))
4	3	adantr 480	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (._r‘𝐴))
5	4	eqcomd 2741	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (._r‘𝐴) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
6	5	oveqd 7448	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑋(._r‘𝐴)𝑌) = (𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝑌))
7		eqid 2735	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2735	. . 3 ⊢ (._r‘𝑅) = (._r‘𝑅)
9		simplr 769	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑅 ∈ Ring)
10		simpll 767	. . 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 22516	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 ∈ 𝐷 → 𝑋 ∈ 𝐵))
15	14	imp 406	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋 ∈ 𝐷) → 𝑋 ∈ 𝐵)
16	1, 7, 11	matbas2i 22444	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
17	15, 16	syl 17	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋 ∈ 𝐷) → 𝑋 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
18	17	adantrr 717	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑋 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
19	1, 11, 12, 13	dmatmat 22516	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑌 ∈ 𝐷 → 𝑌 ∈ 𝐵))
20	19	imp 406	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌 ∈ 𝐷) → 𝑌 ∈ 𝐵)
21	1, 7, 11	matbas2i 22444	. . . . 5 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
22	20, 21	syl 17	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌 ∈ 𝐷) → 𝑌 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
23	22	adantrl 716	. . 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 22413	. 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 2735	. . . . . . 7 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
26		ringcmn 20296	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
27	26	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑅 ∈ CMnd)
28	27	3ad2ant1 1132	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑅 ∈ CMnd)
29	28	adantl 481	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑅 ∈ CMnd)
30	10	3ad2ant1 1132	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑁 ∈ Fin)
31	30	adantl 481	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑁 ∈ Fin)
32		eqid 2735	. . . . . . . 8 ⊢ (𝑘 ∈ 𝑁 ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦))) = (𝑘 ∈ 𝑁 ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)))
33		ovexd 7466	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)) ∈ V)
34		fvexd 6922	. . . . . . . 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 9414	. . . . . . 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 1132	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑅 ∈ Ring)
37	36	ad2antlr 727	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑅 ∈ Ring)
38		simp2 1136	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁)
39	38	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑥 ∈ 𝑁)
40		simpr 484	. . . . . . . . 9 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
41		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐴) = (Base‘𝐴)
42	1, 41, 12, 13	dmatmat 22516	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 ∈ 𝐷 → 𝑋 ∈ (Base‘𝐴)))
43	42	imp 406	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋 ∈ 𝐷) → 𝑋 ∈ (Base‘𝐴))
44	43	adantrr 717	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑋 ∈ (Base‘𝐴))
45	44	3ad2ant1 1132	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑋 ∈ (Base‘𝐴))
46	45	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑋 ∈ (Base‘𝐴))
47	1, 7	matecl 22447	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁 ∧ 𝑋 ∈ (Base‘𝐴)) → (𝑥𝑋𝑘) ∈ (Base‘𝑅))
48	39, 40, 46, 47	syl3anc 1370	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑥𝑋𝑘) ∈ (Base‘𝑅))
49		simplr3 1216	. . . . . . . . 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 22516	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑌 ∈ 𝐷 → 𝑌 ∈ (Base‘𝐴)))
51	50	imp 406	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌 ∈ 𝐷) → 𝑌 ∈ (Base‘𝐴))
52	51	adantrl 716	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑌 ∈ (Base‘𝐴))
53	52	3ad2ant1 1132	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑌 ∈ (Base‘𝐴))
54	53	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑌 ∈ (Base‘𝐴))
55	1, 7	matecl 22447	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁 ∧ 𝑌 ∈ (Base‘𝐴)) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
56	40, 49, 54, 55	syl3anc 1370	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
57	7, 8	ringcl 20268	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑘) ∈ (Base‘𝑅) ∧ (𝑘𝑌𝑦) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)) ∈ (Base‘𝑅))
58	37, 48, 56, 57	syl3anc 1370	. . . . . . 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 481	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
60		simp3 1137	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
61	15	adantrr 717	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑋 ∈ 𝐵)
62	61, 11	eleqtrdi 2849	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑋 ∈ (Base‘𝐴))
63	62	3ad2ant1 1132	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑋 ∈ (Base‘𝐴))
64	1, 7	matecl 22447	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁 ∧ 𝑋 ∈ (Base‘𝐴)) → (𝑥𝑋𝑦) ∈ (Base‘𝑅))
65	38, 60, 63, 64	syl3anc 1370	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑋𝑦) ∈ (Base‘𝑅))
66	50	a1d 25	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 ∈ 𝐷 → (𝑌 ∈ 𝐷 → 𝑌 ∈ (Base‘𝐴))))
67	66	imp32 418	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑌 ∈ (Base‘𝐴))
68	67	3ad2ant1 1132	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑌 ∈ (Base‘𝐴))
69	1, 7	matecl 22447	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁 ∧ 𝑌 ∈ (Base‘𝐴)) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
70	38, 60, 68, 69	syl3anc 1370	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
71	7, 8	ringcl 20268	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑦) ∈ (Base‘𝑅) ∧ (𝑥𝑌𝑦) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
72	36, 65, 70, 71	syl3anc 1370	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
73	72	adantl 481	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
74		eqtr 2758	. . . . . . . . . . 11 ⊢ ((𝑘 = 𝑥 ∧ 𝑥 = 𝑦) → 𝑘 = 𝑦)
75	74	ancoms 458	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑦 ∧ 𝑘 = 𝑥) → 𝑘 = 𝑦)
76	75	oveq2d 7447	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑘 = 𝑥) → (𝑥𝑋𝑘) = (𝑥𝑋𝑦))
77	76	adantlr 715	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 = 𝑥) → (𝑥𝑋𝑘) = (𝑥𝑋𝑦))
78		oveq1 7438	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (𝑘𝑌𝑦) = (𝑥𝑌𝑦))
79	78	adantl 481	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 = 𝑥) → (𝑘𝑌𝑦) = (𝑥𝑌𝑦))
80	77, 79	oveq12d 7449	. . . . . . 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 19994	. . . . . 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 771	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑋 ∈ 𝐷)
83	10, 9, 82	3jca 1127	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷))
84	83	3ad2ant1 1132	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷))
85	84	ad2antlr 727	. . . . . . . . . . . . 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 727	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑥 ∈ 𝑁)
87		eldifi 4141	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑁 ∖ {𝑥}) → 𝑘 ∈ 𝑁)
88	87	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑘 ∈ 𝑁)
89		eldifsni 4795	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝑁 ∖ {𝑥}) → 𝑘 ≠ 𝑥)
90	89	necomd 2994	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑁 ∖ {𝑥}) → 𝑥 ≠ 𝑘)
91	90	adantl 481	. . . . . . . . . . . . 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 22518	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷) ∧ (𝑥 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁 ∧ 𝑥 ≠ 𝑘)) → (𝑥𝑋𝑘) = 0 )
93	85, 86, 88, 91, 92	syl13anc 1371	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → (𝑥𝑋𝑘) = 0 )
94	93	oveq1d 7446	. . . . . . . . . . 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 727	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑅 ∈ Ring)
96		simplr3 1216	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑦 ∈ 𝑁)
97	53	ad2antlr 727	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 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 20307	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ (𝑘𝑌𝑦) ∈ (Base‘𝑅)) → ( 0 (._r‘𝑅)(𝑘𝑌𝑦)) = 0 )
100	95, 98, 99	syl2anc 584	. . . . . . . . . . 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 2775	. . . . . . . . . 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 5248	. . . . . . . . 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 7447	. . . . . . . 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 9214	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝑥}) ∈ Fin)
105		ringmnd 20261	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
106	104, 105	anim12ci 614	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
107	106	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
108	107	3ad2ant1 1132	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
109	108	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
110	12	gsumz 18862	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin) → (𝑅 Σ_g (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ 0 )) = 0 )
111	109, 110	syl 17	. . . . . . . 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 2775	. . . . . . 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 7446	. . . . . 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 727	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑅 ∈ Mnd)
115	114	3ad2ant1 1132	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑅 ∈ Mnd)
116	38, 60, 53, 69	syl3anc 1370	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
117	36, 65, 116, 71	syl3anc 1370	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
118	115, 117	jca 511	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑅 ∈ Mnd ∧ ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅)))
119	118	adantl 481	. . . . . . 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 18780	. . . . . . 7 ⊢ ((𝑅 ∈ Mnd ∧ ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅)) → ( 0 (+_g‘𝑅)((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦))) = ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)))
121	119, 120	syl 17	. . . . . 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 2779	. . . . 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 4537	. . . . . 6 ⊢ (𝑥 = 𝑦 → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)), 0 ) = ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)))
124	123	adantr 480	. . . . 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 2778	. . . 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 773	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑌 ∈ 𝐷)
127	10, 9, 126	3jca 1127	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐷))
128	127	3ad2ant1 1132	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐷))
129	128	ad2antlr 727	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 773	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → 𝑘 ∈ 𝑁)
132		simplr3 1216	. . . . . . . . . . . 12 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑦 ∈ 𝑁)
133	132	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → 𝑦 ∈ 𝑁)
134		df-ne 2939	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
135		neeq1 3001	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → (𝑥 ≠ 𝑦 ↔ 𝑘 ≠ 𝑦))
136	135	biimpcd 249	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≠ 𝑦 → (𝑥 = 𝑘 → 𝑘 ≠ 𝑦))
137	134, 136	sylbir 235	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑥 = 𝑦 → (𝑥 = 𝑘 → 𝑘 ≠ 𝑦))
138	137	adantr 480	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥 = 𝑘 → 𝑘 ≠ 𝑦))
139	138	adantr 480	. . . . . . . . . . . 12 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑥 = 𝑘 → 𝑘 ≠ 𝑦))
140	139	impcom 407	. . . . . . . . . . 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 22518	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐷) ∧ (𝑘 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁 ∧ 𝑘 ≠ 𝑦)) → (𝑘𝑌𝑦) = 0 )
142	130, 131, 133, 140, 141	syl13anc 1371	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → (𝑘𝑌𝑦) = 0 )
143	142	oveq2d 7447	. . . . . . . . 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 727	. . . . . . . . . . 11 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑅 ∈ Ring)
145	38	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑥 ∈ 𝑁)
146		simpr 484	. . . . . . . . . . . 12 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
147	63	ad2antlr 727	. . . . . . . . . . . 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 1370	. . . . . . . . . . 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 20308	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑘) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑘)(._r‘𝑅) 0 ) = 0 )
150	144, 148, 149	syl2anc 584	. . . . . . . . . 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 481	. . . . . . . . 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 2775	. . . . . . . 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 727	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
156		simprr 773	. . . . . . . . . . 11 ⊢ ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → 𝑘 ∈ 𝑁)
157		neqne 2946	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 = 𝑘 → 𝑥 ≠ 𝑘)
158	157	adantr 480	. . . . . . . . . . 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 1371	. . . . . . . . . 10 ⊢ ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → (𝑥𝑋𝑘) = 0 )
160	159	oveq1d 7446	. . . . . . . . 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 727	. . . . . . . . . . . 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 1370	. . . . . . . . . . 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 584	. . . . . . . . . 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 481	. . . . . . . . 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 2775	. . . . . . . 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 812	. . . . . . 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 5248	. . . . . 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 7447	. . . . 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 617	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Mnd))
170	169	ancomd 461	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin))
171	12	gsumz 18862	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ 0 )) = 0 )
172	170, 171	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ 0 )) = 0 )
173	172	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ 0 )) = 0 )
174	173	3ad2ant1 1132	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ 0 )) = 0 )
175	174	adantl 481	. . . . 5 ⊢ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ 0 )) = 0 )
176		iffalse 4540	. . . . . . 7 ⊢ (¬ 𝑥 = 𝑦 → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)), 0 ) = 0 )
177	176	eqcomd 2741	. . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → 0 = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)), 0 ))
178	177	adantr 480	. . . . 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 2779	. . . 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 812	. . 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 7510	. 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 2779	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

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∖ cdif 3960 ifcif 4531 {csn 4631 ⟨cotp 4639 ↦ cmpt 5231 × cxp 5687 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ↑_m cmap 8865 Fincfn 8984 Basecbs 17245 +_gcplusg 17298 ._rcmulr 17299 0_gc0g 17486 Σ_g cgsu 17487 Mndcmnd 18760 CMndccmn 19813 Ringcrg 20251 maMul cmmul 22410 Mat cmat 22427 DMat cdmat 22510

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-sra 21190 df-rgmod 21191 df-dsmm 21770 df-frlm 21785 df-mamu 22411 df-mat 22428 df-dmat 22512

This theorem is referenced by: dmatmulcl 22522 dmatcrng 22524 scmatscmiddistr 22530 scmatcrng 22543

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