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Theorem dmatmul 22503
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 = (0g𝑅)
dmatid.d 𝐷 = (𝑁 DMat 𝑅)
Assertion
Ref Expression
dmatmul (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(.r𝐴)𝑌) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(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.
StepHypRef Expression
1 dmatid.a . . . . . 6 𝐴 = (𝑁 Mat 𝑅)
2 eqid 2737 . . . . . 6 (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
31, 2matmulr 22444 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
43adantr 480 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
54eqcomd 2743 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (.r𝐴) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
65oveqd 7448 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(.r𝐴)𝑌) = (𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝑌))
7 eqid 2737 . . 3 (Base‘𝑅) = (Base‘𝑅)
8 eqid 2737 . . 3 (.r𝑅) = (.r𝑅)
9 simplr 769 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑅 ∈ Ring)
10 simpll 767 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑁 ∈ Fin)
11 dmatid.b . . . . . . 7 𝐵 = (Base‘𝐴)
12 dmatid.0 . . . . . . 7 0 = (0g𝑅)
13 dmatid.d . . . . . . 7 𝐷 = (𝑁 DMat 𝑅)
141, 11, 12, 13dmatmat 22500 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷𝑋𝐵))
1514imp 406 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋𝐷) → 𝑋𝐵)
161, 7, 11matbas2i 22428 . . . . 5 (𝑋𝐵𝑋 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
1715, 16syl 17 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋𝐷) → 𝑋 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
1817adantrr 717 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑋 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
191, 11, 12, 13dmatmat 22500 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑌𝐷𝑌𝐵))
2019imp 406 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌𝐷) → 𝑌𝐵)
211, 7, 11matbas2i 22428 . . . . 5 (𝑌𝐵𝑌 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
2220, 21syl 17 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌𝐷) → 𝑌 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
2322adantrl 716 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑌 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
242, 7, 8, 9, 10, 10, 10, 18, 23mamuval 22397 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝑌) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))))))
25 eqid 2737 . . . . . . 7 (+g𝑅) = (+g𝑅)
26 ringcmn 20279 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
2726ad2antlr 727 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑅 ∈ CMnd)
28273ad2ant1 1134 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑅 ∈ CMnd)
2928adantl 481 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → 𝑅 ∈ CMnd)
30103ad2ant1 1134 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑁 ∈ Fin)
3130adantl 481 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → 𝑁 ∈ Fin)
32 eqid 2737 . . . . . . . 8 (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))) = (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))
33 ovexd 7466 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) ∈ V)
34 fvexd 6921 . . . . . . . 8 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (0g𝑅) ∈ V)
3532, 31, 33, 34fsuppmptdm 9416 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))) finSupp (0g𝑅))
3693ad2ant1 1134 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑅 ∈ Ring)
3736ad2antlr 727 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑅 ∈ Ring)
38 simp2 1138 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑥𝑁)
3938ad2antlr 727 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑥𝑁)
40 simpr 484 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑘𝑁)
41 eqid 2737 . . . . . . . . . . . . . 14 (Base‘𝐴) = (Base‘𝐴)
421, 41, 12, 13dmatmat 22500 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷𝑋 ∈ (Base‘𝐴)))
4342imp 406 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋𝐷) → 𝑋 ∈ (Base‘𝐴))
4443adantrr 717 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑋 ∈ (Base‘𝐴))
45443ad2ant1 1134 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑋 ∈ (Base‘𝐴))
4645ad2antlr 727 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑋 ∈ (Base‘𝐴))
471, 7matecl 22431 . . . . . . . . 9 ((𝑥𝑁𝑘𝑁𝑋 ∈ (Base‘𝐴)) → (𝑥𝑋𝑘) ∈ (Base‘𝑅))
4839, 40, 46, 47syl3anc 1373 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑥𝑋𝑘) ∈ (Base‘𝑅))
49 simplr3 1218 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑦𝑁)
501, 41, 12, 13dmatmat 22500 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑌𝐷𝑌 ∈ (Base‘𝐴)))
5150imp 406 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌𝐷) → 𝑌 ∈ (Base‘𝐴))
5251adantrl 716 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑌 ∈ (Base‘𝐴))
53523ad2ant1 1134 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑌 ∈ (Base‘𝐴))
5453ad2antlr 727 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑌 ∈ (Base‘𝐴))
551, 7matecl 22431 . . . . . . . . 9 ((𝑘𝑁𝑦𝑁𝑌 ∈ (Base‘𝐴)) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
5640, 49, 54, 55syl3anc 1373 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
577, 8ringcl 20247 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑘) ∈ (Base‘𝑅) ∧ (𝑘𝑌𝑦) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) ∈ (Base‘𝑅))
5837, 48, 56, 57syl3anc 1373 . . . . . . 7 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) ∈ (Base‘𝑅))
5938adantl 481 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → 𝑥𝑁)
60 simp3 1139 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑦𝑁)
6115adantrr 717 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑋𝐵)
6261, 11eleqtrdi 2851 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑋 ∈ (Base‘𝐴))
63623ad2ant1 1134 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑋 ∈ (Base‘𝐴))
641, 7matecl 22431 . . . . . . . . . 10 ((𝑥𝑁𝑦𝑁𝑋 ∈ (Base‘𝐴)) → (𝑥𝑋𝑦) ∈ (Base‘𝑅))
6538, 60, 63, 64syl3anc 1373 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑥𝑋𝑦) ∈ (Base‘𝑅))
6650a1d 25 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷 → (𝑌𝐷𝑌 ∈ (Base‘𝐴))))
6766imp32 418 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑌 ∈ (Base‘𝐴))
68673ad2ant1 1134 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑌 ∈ (Base‘𝐴))
691, 7matecl 22431 . . . . . . . . . 10 ((𝑥𝑁𝑦𝑁𝑌 ∈ (Base‘𝐴)) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
7038, 60, 68, 69syl3anc 1373 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
717, 8ringcl 20247 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑦) ∈ (Base‘𝑅) ∧ (𝑥𝑌𝑦) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
7236, 65, 70, 71syl3anc 1373 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
7372adantl 481 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
74 eqtr 2760 . . . . . . . . . . 11 ((𝑘 = 𝑥𝑥 = 𝑦) → 𝑘 = 𝑦)
7574ancoms 458 . . . . . . . . . 10 ((𝑥 = 𝑦𝑘 = 𝑥) → 𝑘 = 𝑦)
7675oveq2d 7447 . . . . . . . . 9 ((𝑥 = 𝑦𝑘 = 𝑥) → (𝑥𝑋𝑘) = (𝑥𝑋𝑦))
7776adantlr 715 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 = 𝑥) → (𝑥𝑋𝑘) = (𝑥𝑋𝑦))
78 oveq1 7438 . . . . . . . . 9 (𝑘 = 𝑥 → (𝑘𝑌𝑦) = (𝑥𝑌𝑦))
7978adantl 481 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 = 𝑥) → (𝑘𝑌𝑦) = (𝑥𝑌𝑦))
8077, 79oveq12d 7449 . . . . . . 7 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 = 𝑥) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)))
817, 25, 29, 31, 35, 58, 59, 73, 80gsumdifsnd 19979 . . . . . 6 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = ((𝑅 Σg (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))))(+g𝑅)((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦))))
82 simprl 771 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑋𝐷)
8310, 9, 823jca 1129 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷))
84833ad2ant1 1134 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷))
8584ad2antlr 727 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷))
8638ad2antlr 727 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑥𝑁)
87 eldifi 4131 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑁 ∖ {𝑥}) → 𝑘𝑁)
8887adantl 481 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑘𝑁)
89 eldifsni 4790 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝑁 ∖ {𝑥}) → 𝑘𝑥)
9089necomd 2996 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑁 ∖ {𝑥}) → 𝑥𝑘)
9190adantl 481 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑥𝑘)
921, 11, 12, 13dmatelnd 22502 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷) ∧ (𝑥𝑁𝑘𝑁𝑥𝑘)) → (𝑥𝑋𝑘) = 0 )
9385, 86, 88, 91, 92syl13anc 1374 . . . . . . . . . . . 12 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → (𝑥𝑋𝑘) = 0 )
9493oveq1d 7446 . . . . . . . . . . 11 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = ( 0 (.r𝑅)(𝑘𝑌𝑦)))
9536ad2antlr 727 . . . . . . . . . . . 12 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑅 ∈ Ring)
96 simplr3 1218 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑦𝑁)
9753ad2antlr 727 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑌 ∈ (Base‘𝐴))
9888, 96, 97, 55syl3anc 1373 . . . . . . . . . . . 12 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
997, 8, 12ringlz 20290 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑘𝑌𝑦) ∈ (Base‘𝑅)) → ( 0 (.r𝑅)(𝑘𝑌𝑦)) = 0 )
10095, 98, 99syl2anc 584 . . . . . . . . . . 11 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → ( 0 (.r𝑅)(𝑘𝑌𝑦)) = 0 )
10194, 100eqtrd 2777 . . . . . . . . . 10 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = 0 )
102101mpteq2dva 5242 . . . . . . . . 9 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))) = (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ 0 ))
103102oveq2d 7447 . . . . . . . 8 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = (𝑅 Σg (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ 0 )))
104 diffi 9215 . . . . . . . . . . . . 13 (𝑁 ∈ Fin → (𝑁 ∖ {𝑥}) ∈ Fin)
105 ringmnd 20240 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
106104, 105anim12ci 614 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
107106adantr 480 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
1081073ad2ant1 1134 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
109108adantl 481 . . . . . . . . 9 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
11012gsumz 18849 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin) → (𝑅 Σg (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ 0 )) = 0 )
111109, 110syl 17 . . . . . . . 8 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ 0 )) = 0 )
112103, 111eqtrd 2777 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = 0 )
113112oveq1d 7446 . . . . . 6 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → ((𝑅 Σg (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))))(+g𝑅)((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦))) = ( 0 (+g𝑅)((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦))))
114105ad2antlr 727 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑅 ∈ Mnd)
1151143ad2ant1 1134 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑅 ∈ Mnd)
11638, 60, 53, 69syl3anc 1373 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
11736, 65, 116, 71syl3anc 1373 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
118115, 117jca 511 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑅 ∈ Mnd ∧ ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅)))
119118adantl 481 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 ∈ Mnd ∧ ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅)))
1207, 25, 12mndlid 18767 . . . . . . 7 ((𝑅 ∈ Mnd ∧ ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅)) → ( 0 (+g𝑅)((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦))) = ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)))
121119, 120syl 17 . . . . . 6 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → ( 0 (+g𝑅)((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦))) = ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)))
12281, 113, 1213eqtrd 2781 . . . . 5 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)))
123 iftrue 4531 . . . . . 6 (𝑥 = 𝑦 → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ) = ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)))
124123adantr 480 . . . . 5 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ) = ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)))
125122, 124eqtr4d 2780 . . . 4 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))
126 simprr 773 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑌𝐷)
12710, 9, 1263jca 1129 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐷))
1281273ad2ant1 1134 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐷))
129128ad2antlr 727 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐷))
130129adantl 481 . . . . . . . . . . 11 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐷))
131 simprr 773 . . . . . . . . . . 11 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑘𝑁)
132 simplr3 1218 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑦𝑁)
133132adantl 481 . . . . . . . . . . 11 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑦𝑁)
134 df-ne 2941 . . . . . . . . . . . . . . 15 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
135 neeq1 3003 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑘 → (𝑥𝑦𝑘𝑦))
136135biimpcd 249 . . . . . . . . . . . . . . 15 (𝑥𝑦 → (𝑥 = 𝑘𝑘𝑦))
137134, 136sylbir 235 . . . . . . . . . . . . . 14 𝑥 = 𝑦 → (𝑥 = 𝑘𝑘𝑦))
138137adantr 480 . . . . . . . . . . . . 13 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑥 = 𝑘𝑘𝑦))
139138adantr 480 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑥 = 𝑘𝑘𝑦))
140139impcom 407 . . . . . . . . . . 11 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑘𝑦)
1411, 11, 12, 13dmatelnd 22502 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐷) ∧ (𝑘𝑁𝑦𝑁𝑘𝑦)) → (𝑘𝑌𝑦) = 0 )
142130, 131, 133, 140, 141syl13anc 1374 . . . . . . . . . 10 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → (𝑘𝑌𝑦) = 0 )
143142oveq2d 7447 . . . . . . . . 9 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = ((𝑥𝑋𝑘)(.r𝑅) 0 ))
14436ad2antlr 727 . . . . . . . . . . 11 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑅 ∈ Ring)
14538ad2antlr 727 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑥𝑁)
146 simpr 484 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑘𝑁)
14763ad2antlr 727 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑋 ∈ (Base‘𝐴))
148145, 146, 147, 47syl3anc 1373 . . . . . . . . . . 11 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑥𝑋𝑘) ∈ (Base‘𝑅))
1497, 8, 12ringrz 20291 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑘) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑘)(.r𝑅) 0 ) = 0 )
150144, 148, 149syl2anc 584 . . . . . . . . . 10 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → ((𝑥𝑋𝑘)(.r𝑅) 0 ) = 0 )
151150adantl 481 . . . . . . . . 9 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ((𝑥𝑋𝑘)(.r𝑅) 0 ) = 0 )
152143, 151eqtrd 2777 . . . . . . . 8 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = 0 )
15384ad2antlr 727 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷))
154153adantl 481 . . . . . . . . . . 11 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷))
155145adantl 481 . . . . . . . . . . 11 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑥𝑁)
156 simprr 773 . . . . . . . . . . 11 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑘𝑁)
157 neqne 2948 . . . . . . . . . . . 12 𝑥 = 𝑘𝑥𝑘)
158157adantr 480 . . . . . . . . . . 11 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑥𝑘)
159154, 155, 156, 158, 92syl13anc 1374 . . . . . . . . . 10 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → (𝑥𝑋𝑘) = 0 )
160159oveq1d 7446 . . . . . . . . 9 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = ( 0 (.r𝑅)(𝑘𝑌𝑦)))
16168ad2antlr 727 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑌 ∈ (Base‘𝐴))
162146, 132, 161, 55syl3anc 1373 . . . . . . . . . . 11 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
163144, 162, 99syl2anc 584 . . . . . . . . . 10 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → ( 0 (.r𝑅)(𝑘𝑌𝑦)) = 0 )
164163adantl 481 . . . . . . . . 9 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ( 0 (.r𝑅)(𝑘𝑌𝑦)) = 0 )
165160, 164eqtrd 2777 . . . . . . . 8 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = 0 )
166152, 165pm2.61ian 812 . . . . . . 7 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = 0 )
167166mpteq2dva 5242 . . . . . 6 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))) = (𝑘𝑁0 ))
168167oveq2d 7447 . . . . 5 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = (𝑅 Σg (𝑘𝑁0 )))
169105anim2i 617 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Mnd))
170169ancomd 461 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin))
17112gsumz 18849 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin) → (𝑅 Σg (𝑘𝑁0 )) = 0 )
172170, 171syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 Σg (𝑘𝑁0 )) = 0 )
173172adantr 480 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑅 Σg (𝑘𝑁0 )) = 0 )
1741733ad2ant1 1134 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑅 Σg (𝑘𝑁0 )) = 0 )
175174adantl 481 . . . . 5 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘𝑁0 )) = 0 )
176 iffalse 4534 . . . . . . 7 𝑥 = 𝑦 → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ) = 0 )
177176eqcomd 2743 . . . . . 6 𝑥 = 𝑦0 = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))
178177adantr 480 . . . . 5 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → 0 = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))
179168, 175, 1783eqtrd 2781 . . . 4 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))
180125, 179pm2.61ian 812 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))
181180mpoeq3dva 7510 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))))) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 )))
1826, 24, 1813eqtrd 2781 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(.r𝐴)𝑌) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  cdif 3948  ifcif 4525  {csn 4626  cotp 4634  cmpt 5225   × cxp 5683  cfv 6561  (class class class)co 7431  cmpo 7433  m cmap 8866  Fincfn 8985  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  0gc0g 17484   Σg cgsu 17485  Mndcmnd 18747  CMndccmn 19798  Ringcrg 20230   maMul cmmul 22394   Mat cmat 22411   DMat cdmat 22494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-sra 21172  df-rgmod 21173  df-dsmm 21752  df-frlm 21767  df-mamu 22395  df-mat 22412  df-dmat 22496
This theorem is referenced by:  dmatmulcl  22506  dmatcrng  22508  scmatscmiddistr  22514  scmatcrng  22527
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