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Theorem dmatmul 22453

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 2737	. . . . . 6 ⊢ (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
3	1, 2	matmulr 22394	. . . . 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 2743	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (._r‘𝐴) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
6	5	oveqd 7385	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑋(._r‘𝐴)𝑌) = (𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝑌))
7		eqid 2737	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2737	. . 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 22450	. . . . . 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 22378	. . . . 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 718	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑋 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
19	1, 11, 12, 13	dmatmat 22450	. . . . . 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 22378	. . . . 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 717	. . 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 22349	. 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 2737	. . . . . . 7 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
26		ringcmn 20229	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
27	26	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑅 ∈ CMnd)
28	27	3ad2ant1 1134	. . . . . . . 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 1134	. . . . . . . 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 2737	. . . . . . . 8 ⊢ (𝑘 ∈ 𝑁 ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦))) = (𝑘 ∈ 𝑁 ↦ ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)))
33		ovexd 7403	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)) ∈ V)
34		fvexd 6857	. . . . . . . 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 9291	. . . . . . 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 1134	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑅 ∈ Ring)
37	36	ad2antlr 728	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑅 ∈ Ring)
38		simp2 1138	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁)
39	38	ad2antlr 728	. . . . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐴) = (Base‘𝐴)
42	1, 41, 12, 13	dmatmat 22450	. . . . . . . . . . . . 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 718	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑋 ∈ (Base‘𝐴))
45	44	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑋 ∈ (Base‘𝐴))
46	45	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑋 ∈ (Base‘𝐴))
47	1, 7	matecl 22381	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁 ∧ 𝑋 ∈ (Base‘𝐴)) → (𝑥𝑋𝑘) ∈ (Base‘𝑅))
48	39, 40, 46, 47	syl3anc 1374	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑥𝑋𝑘) ∈ (Base‘𝑅))
49		simplr3 1219	. . . . . . . . 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 22450	. . . . . . . . . . . . 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 717	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑌 ∈ (Base‘𝐴))
53	52	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑌 ∈ (Base‘𝐴))
54	53	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑌 ∈ (Base‘𝐴))
55	1, 7	matecl 22381	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁 ∧ 𝑌 ∈ (Base‘𝐴)) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
56	40, 49, 54, 55	syl3anc 1374	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
57	7, 8	ringcl 20197	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑘) ∈ (Base‘𝑅) ∧ (𝑘𝑌𝑦) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑘)(._r‘𝑅)(𝑘𝑌𝑦)) ∈ (Base‘𝑅))
58	37, 48, 56, 57	syl3anc 1374	. . . . . . 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 1139	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
61	15	adantrr 718	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑋 ∈ 𝐵)
62	61, 11	eleqtrdi 2847	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑋 ∈ (Base‘𝐴))
63	62	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑋 ∈ (Base‘𝐴))
64	1, 7	matecl 22381	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁 ∧ 𝑋 ∈ (Base‘𝐴)) → (𝑥𝑋𝑦) ∈ (Base‘𝑅))
65	38, 60, 63, 64	syl3anc 1374	. . . . . . . . 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 1134	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑌 ∈ (Base‘𝐴))
69	1, 7	matecl 22381	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁 ∧ 𝑌 ∈ (Base‘𝐴)) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
70	38, 60, 68, 69	syl3anc 1374	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
71	7, 8	ringcl 20197	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑦) ∈ (Base‘𝑅) ∧ (𝑥𝑌𝑦) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
72	36, 65, 70, 71	syl3anc 1374	. . . . . . . 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 2757	. . . . . . . . . . 11 ⊢ ((𝑘 = 𝑥 ∧ 𝑥 = 𝑦) → 𝑘 = 𝑦)
75	74	ancoms 458	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑦 ∧ 𝑘 = 𝑥) → 𝑘 = 𝑦)
76	75	oveq2d 7384	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑘 = 𝑥) → (𝑥𝑋𝑘) = (𝑥𝑋𝑦))
77	76	adantlr 716	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 = 𝑥) → (𝑥𝑋𝑘) = (𝑥𝑋𝑦))
78		oveq1 7375	. . . . . . . . 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 7386	. . . . . . 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 19902	. . . . . 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 1129	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷))
84	83	3ad2ant1 1134	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷))
85	84	ad2antlr 728	. . . . . . . . . . . . 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 728	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑥 ∈ 𝑁)
87		eldifi 4085	. . . . . . . . . . . . . 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 4748	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝑁 ∖ {𝑥}) → 𝑘 ≠ 𝑥)
90	89	necomd 2988	. . . . . . . . . . . . . 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 22452	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷) ∧ (𝑥 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁 ∧ 𝑥 ≠ 𝑘)) → (𝑥𝑋𝑘) = 0 )
93	85, 86, 88, 91, 92	syl13anc 1375	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → (𝑥𝑋𝑘) = 0 )
94	93	oveq1d 7383	. . . . . . . . . . 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 728	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑅 ∈ Ring)
96		simplr3 1219	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑦 ∈ 𝑁)
97	53	ad2antlr 728	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 20240	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ (𝑘𝑌𝑦) ∈ (Base‘𝑅)) → ( 0 (._r‘𝑅)(𝑘𝑌𝑦)) = 0 )
100	95, 98, 99	syl2anc 585	. . . . . . . . . . 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 2772	. . . . . . . . . 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 5193	. . . . . . . . 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 7384	. . . . . . . 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 9111	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝑥}) ∈ Fin)
105		ringmnd 20190	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
106	104, 105	anim12ci 615	. . . . . . . . . . . 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 1134	. . . . . . . . . 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 18773	. . . . . . . . 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 2772	. . . . . . 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 7383	. . . . . 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 728	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → 𝑅 ∈ Mnd)
115	114	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑅 ∈ Mnd)
116	38, 60, 53, 69	syl3anc 1374	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
117	36, 65, 116, 71	syl3anc 1374	. . . . . . . . 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 18691	. . . . . . 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 2776	. . . . 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 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 2775	. . . 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 1129	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐷))
128	127	3ad2ant1 1134	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐷))
129	128	ad2antlr 728	. . . . . . . . . . . 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 1219	. . . . . . . . . . . 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 2934	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
135		neeq1 2995	. . . . . . . . . . . . . . . 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 22452	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐷) ∧ (𝑘 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁 ∧ 𝑘 ≠ 𝑦)) → (𝑘𝑌𝑦) = 0 )
142	130, 131, 133, 140, 141	syl13anc 1375	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → (𝑘𝑌𝑦) = 0 )
143	142	oveq2d 7384	. . . . . . . . 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 728	. . . . . . . . . . 11 ⊢ (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑅 ∈ Ring)
145	38	ad2antlr 728	. . . . . . . . . . . 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 728	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 20241	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑘) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑘)(._r‘𝑅) 0 ) = 0 )
150	144, 148, 149	syl2anc 585	. . . . . . . . . 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 2772	. . . . . . . 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 728	. . . . . . . . . . . 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 2941	. . . . . . . . . . . 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 1375	. . . . . . . . . 10 ⊢ ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁)) → (𝑥𝑋𝑘) = 0 )
160	159	oveq1d 7383	. . . . . . . . 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 728	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 585	. . . . . . . . . 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 2772	. . . . . . . 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 5193	. . . . . 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 7384	. . . . 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 618	. . . . . . . . . 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 18773	. . . . . . . . 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 1134	. . . . . 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 4490	. . . . . . 7 ⊢ (¬ 𝑥 = 𝑦 → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(._r‘𝑅)(𝑥𝑌𝑦)), 0 ) = 0 )
177	176	eqcomd 2743	. . . . . 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 2776	. . . 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 7445	. 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 2776	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 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3442 ∖ cdif 3900 ifcif 4481 {csn 4582 ⟨cotp 4590 ↦ cmpt 5181 × cxp 5630 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 ↑_m cmap 8775 Fincfn 8895 Basecbs 17148 +_gcplusg 17189 ._rcmulr 17190 0_gc0g 17371 Σ_g cgsu 17372 Mndcmnd 18671 CMndccmn 19721 Ringcrg 20180 maMul cmmul 22346 Mat cmat 22363 DMat cdmat 22444

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-ot 4591 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-fzo 13583 df-seq 13937 df-hash 14266 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-hom 17213 df-cco 17214 df-0g 17373 df-gsum 17374 df-prds 17379 df-pws 17381 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18878 df-minusg 18879 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-sra 21137 df-rgmod 21138 df-dsmm 21699 df-frlm 21714 df-mamu 22347 df-mat 22364 df-dmat 22446

This theorem is referenced by: dmatmulcl 22456 dmatcrng 22458 scmatscmiddistr 22464 scmatcrng 22477

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