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Theorem dmatmul 21968
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 2733 . . . . . 6 (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
31, 2matmulr 21909 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
43adantr 482 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
54eqcomd 2739 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (.r𝐴) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
65oveqd 7413 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(.r𝐴)𝑌) = (𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝑌))
7 eqid 2733 . . 3 (Base‘𝑅) = (Base‘𝑅)
8 eqid 2733 . . 3 (.r𝑅) = (.r𝑅)
9 simplr 768 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑅 ∈ Ring)
10 simpll 766 . . 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 21965 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷𝑋𝐵))
1514imp 408 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋𝐷) → 𝑋𝐵)
161, 7, 11matbas2i 21893 . . . . 5 (𝑋𝐵𝑋 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
1715, 16syl 17 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋𝐷) → 𝑋 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
1817adantrr 716 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑋 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
191, 11, 12, 13dmatmat 21965 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑌𝐷𝑌𝐵))
2019imp 408 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌𝐷) → 𝑌𝐵)
211, 7, 11matbas2i 21893 . . . . 5 (𝑌𝐵𝑌 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
2220, 21syl 17 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌𝐷) → 𝑌 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
2322adantrl 715 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑌 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
242, 7, 8, 9, 10, 10, 10, 18, 23mamuval 21857 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝑌) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))))))
25 eqid 2733 . . . . . . 7 (+g𝑅) = (+g𝑅)
26 ringcmn 20080 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
2726ad2antlr 726 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑅 ∈ CMnd)
28273ad2ant1 1134 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑅 ∈ CMnd)
2928adantl 483 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → 𝑅 ∈ CMnd)
30103ad2ant1 1134 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑁 ∈ Fin)
3130adantl 483 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → 𝑁 ∈ Fin)
32 eqid 2733 . . . . . . . 8 (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))) = (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))
33 ovexd 7431 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) ∈ V)
34 fvexd 6896 . . . . . . . 8 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (0g𝑅) ∈ V)
3532, 31, 33, 34fsuppmptdm 9362 . . . . . . 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 726 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑅 ∈ Ring)
38 simp2 1138 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑥𝑁)
3938ad2antlr 726 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑥𝑁)
40 simpr 486 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑘𝑁)
41 eqid 2733 . . . . . . . . . . . . . 14 (Base‘𝐴) = (Base‘𝐴)
421, 41, 12, 13dmatmat 21965 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷𝑋 ∈ (Base‘𝐴)))
4342imp 408 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋𝐷) → 𝑋 ∈ (Base‘𝐴))
4443adantrr 716 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑋 ∈ (Base‘𝐴))
45443ad2ant1 1134 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑋 ∈ (Base‘𝐴))
4645ad2antlr 726 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑋 ∈ (Base‘𝐴))
471, 7matecl 21896 . . . . . . . . 9 ((𝑥𝑁𝑘𝑁𝑋 ∈ (Base‘𝐴)) → (𝑥𝑋𝑘) ∈ (Base‘𝑅))
4839, 40, 46, 47syl3anc 1372 . . . . . . . 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 21965 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑌𝐷𝑌 ∈ (Base‘𝐴)))
5150imp 408 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌𝐷) → 𝑌 ∈ (Base‘𝐴))
5251adantrl 715 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑌 ∈ (Base‘𝐴))
53523ad2ant1 1134 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑌 ∈ (Base‘𝐴))
5453ad2antlr 726 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑌 ∈ (Base‘𝐴))
551, 7matecl 21896 . . . . . . . . 9 ((𝑘𝑁𝑦𝑁𝑌 ∈ (Base‘𝐴)) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
5640, 49, 54, 55syl3anc 1372 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
577, 8ringcl 20055 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑘) ∈ (Base‘𝑅) ∧ (𝑘𝑌𝑦) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) ∈ (Base‘𝑅))
5837, 48, 56, 57syl3anc 1372 . . . . . . 7 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) ∈ (Base‘𝑅))
5938adantl 483 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → 𝑥𝑁)
60 simp3 1139 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑦𝑁)
6115adantrr 716 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑋𝐵)
6261, 11eleqtrdi 2844 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑋 ∈ (Base‘𝐴))
63623ad2ant1 1134 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑋 ∈ (Base‘𝐴))
641, 7matecl 21896 . . . . . . . . . 10 ((𝑥𝑁𝑦𝑁𝑋 ∈ (Base‘𝐴)) → (𝑥𝑋𝑦) ∈ (Base‘𝑅))
6538, 60, 63, 64syl3anc 1372 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑥𝑋𝑦) ∈ (Base‘𝑅))
6650a1d 25 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷 → (𝑌𝐷𝑌 ∈ (Base‘𝐴))))
6766imp32 420 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑌 ∈ (Base‘𝐴))
68673ad2ant1 1134 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑌 ∈ (Base‘𝐴))
691, 7matecl 21896 . . . . . . . . . 10 ((𝑥𝑁𝑦𝑁𝑌 ∈ (Base‘𝐴)) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
7038, 60, 68, 69syl3anc 1372 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
717, 8ringcl 20055 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑦) ∈ (Base‘𝑅) ∧ (𝑥𝑌𝑦) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
7236, 65, 70, 71syl3anc 1372 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
7372adantl 483 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
74 eqtr 2756 . . . . . . . . . . 11 ((𝑘 = 𝑥𝑥 = 𝑦) → 𝑘 = 𝑦)
7574ancoms 460 . . . . . . . . . 10 ((𝑥 = 𝑦𝑘 = 𝑥) → 𝑘 = 𝑦)
7675oveq2d 7412 . . . . . . . . 9 ((𝑥 = 𝑦𝑘 = 𝑥) → (𝑥𝑋𝑘) = (𝑥𝑋𝑦))
7776adantlr 714 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 = 𝑥) → (𝑥𝑋𝑘) = (𝑥𝑋𝑦))
78 oveq1 7403 . . . . . . . . 9 (𝑘 = 𝑥 → (𝑘𝑌𝑦) = (𝑥𝑌𝑦))
7978adantl 483 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 = 𝑥) → (𝑘𝑌𝑦) = (𝑥𝑌𝑦))
8077, 79oveq12d 7414 . . . . . . 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 19812 . . . . . 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 770 . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷))
8638ad2antlr 726 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑥𝑁)
87 eldifi 4124 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑁 ∖ {𝑥}) → 𝑘𝑁)
8887adantl 483 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑘𝑁)
89 eldifsni 4789 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝑁 ∖ {𝑥}) → 𝑘𝑥)
9089necomd 2997 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑁 ∖ {𝑥}) → 𝑥𝑘)
9190adantl 483 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑥𝑘)
921, 11, 12, 13dmatelnd 21967 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷) ∧ (𝑥𝑁𝑘𝑁𝑥𝑘)) → (𝑥𝑋𝑘) = 0 )
9385, 86, 88, 91, 92syl13anc 1373 . . . . . . . . . . . 12 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → (𝑥𝑋𝑘) = 0 )
9493oveq1d 7411 . . . . . . . . . . 11 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = ( 0 (.r𝑅)(𝑘𝑌𝑦)))
9536ad2antlr 726 . . . . . . . . . . . 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 726 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑌 ∈ (Base‘𝐴))
9888, 96, 97, 55syl3anc 1372 . . . . . . . . . . . 12 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
997, 8, 12ringlz 20088 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑘𝑌𝑦) ∈ (Base‘𝑅)) → ( 0 (.r𝑅)(𝑘𝑌𝑦)) = 0 )
10095, 98, 99syl2anc 585 . . . . . . . . . . 11 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → ( 0 (.r𝑅)(𝑘𝑌𝑦)) = 0 )
10194, 100eqtrd 2773 . . . . . . . . . 10 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = 0 )
102101mpteq2dva 5244 . . . . . . . . 9 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))) = (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ 0 ))
103102oveq2d 7412 . . . . . . . 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 9167 . . . . . . . . . . . . 13 (𝑁 ∈ Fin → (𝑁 ∖ {𝑥}) ∈ Fin)
105 ringmnd 20048 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
106104, 105anim12ci 615 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
107106adantr 482 . . . . . . . . . . 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 483 . . . . . . . . 9 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
11012gsumz 18704 . . . . . . . . 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 2773 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = 0 )
113112oveq1d 7411 . . . . . 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 726 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑅 ∈ Mnd)
1151143ad2ant1 1134 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑅 ∈ Mnd)
11638, 60, 53, 69syl3anc 1372 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
11736, 65, 116, 71syl3anc 1372 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
118115, 117jca 513 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑅 ∈ Mnd ∧ ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅)))
119118adantl 483 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 ∈ Mnd ∧ ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅)))
1207, 25, 12mndlid 18632 . . . . . . 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 2777 . . . . 5 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)))
123 iftrue 4530 . . . . . 6 (𝑥 = 𝑦 → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ) = ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)))
124123adantr 482 . . . . 5 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ) = ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)))
125122, 124eqtr4d 2776 . . . 4 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))
126 simprr 772 . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐷))
130129adantl 483 . . . . . . . . . . 11 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐷))
131 simprr 772 . . . . . . . . . . 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 483 . . . . . . . . . . 11 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑦𝑁)
134 df-ne 2942 . . . . . . . . . . . . . . 15 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
135 neeq1 3004 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑘 → (𝑥𝑦𝑘𝑦))
136135biimpcd 248 . . . . . . . . . . . . . . 15 (𝑥𝑦 → (𝑥 = 𝑘𝑘𝑦))
137134, 136sylbir 234 . . . . . . . . . . . . . 14 𝑥 = 𝑦 → (𝑥 = 𝑘𝑘𝑦))
138137adantr 482 . . . . . . . . . . . . 13 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑥 = 𝑘𝑘𝑦))
139138adantr 482 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑥 = 𝑘𝑘𝑦))
140139impcom 409 . . . . . . . . . . 11 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑘𝑦)
1411, 11, 12, 13dmatelnd 21967 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐷) ∧ (𝑘𝑁𝑦𝑁𝑘𝑦)) → (𝑘𝑌𝑦) = 0 )
142130, 131, 133, 140, 141syl13anc 1373 . . . . . . . . . 10 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → (𝑘𝑌𝑦) = 0 )
143142oveq2d 7412 . . . . . . . . 9 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = ((𝑥𝑋𝑘)(.r𝑅) 0 ))
14436ad2antlr 726 . . . . . . . . . . 11 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑅 ∈ Ring)
14538ad2antlr 726 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑥𝑁)
146 simpr 486 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑘𝑁)
14763ad2antlr 726 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑋 ∈ (Base‘𝐴))
148145, 146, 147, 47syl3anc 1372 . . . . . . . . . . 11 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑥𝑋𝑘) ∈ (Base‘𝑅))
1497, 8, 12ringrz 20089 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑘) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑘)(.r𝑅) 0 ) = 0 )
150144, 148, 149syl2anc 585 . . . . . . . . . 10 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → ((𝑥𝑋𝑘)(.r𝑅) 0 ) = 0 )
151150adantl 483 . . . . . . . . 9 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ((𝑥𝑋𝑘)(.r𝑅) 0 ) = 0 )
152143, 151eqtrd 2773 . . . . . . . 8 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = 0 )
15384ad2antlr 726 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷))
154153adantl 483 . . . . . . . . . . 11 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷))
155145adantl 483 . . . . . . . . . . 11 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑥𝑁)
156 simprr 772 . . . . . . . . . . 11 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑘𝑁)
157 neqne 2949 . . . . . . . . . . . 12 𝑥 = 𝑘𝑥𝑘)
158157adantr 482 . . . . . . . . . . 11 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑥𝑘)
159154, 155, 156, 158, 92syl13anc 1373 . . . . . . . . . 10 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → (𝑥𝑋𝑘) = 0 )
160159oveq1d 7411 . . . . . . . . 9 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = ( 0 (.r𝑅)(𝑘𝑌𝑦)))
16168ad2antlr 726 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑌 ∈ (Base‘𝐴))
162146, 132, 161, 55syl3anc 1372 . . . . . . . . . . 11 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
163144, 162, 99syl2anc 585 . . . . . . . . . 10 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → ( 0 (.r𝑅)(𝑘𝑌𝑦)) = 0 )
164163adantl 483 . . . . . . . . 9 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ( 0 (.r𝑅)(𝑘𝑌𝑦)) = 0 )
165160, 164eqtrd 2773 . . . . . . . 8 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = 0 )
166152, 165pm2.61ian 811 . . . . . . 7 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = 0 )
167166mpteq2dva 5244 . . . . . 6 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))) = (𝑘𝑁0 ))
168167oveq2d 7412 . . . . 5 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = (𝑅 Σg (𝑘𝑁0 )))
169105anim2i 618 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Mnd))
170169ancomd 463 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin))
17112gsumz 18704 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin) → (𝑅 Σg (𝑘𝑁0 )) = 0 )
172170, 171syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 Σg (𝑘𝑁0 )) = 0 )
173172adantr 482 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑅 Σg (𝑘𝑁0 )) = 0 )
1741733ad2ant1 1134 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑅 Σg (𝑘𝑁0 )) = 0 )
175174adantl 483 . . . . 5 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘𝑁0 )) = 0 )
176 iffalse 4533 . . . . . . 7 𝑥 = 𝑦 → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ) = 0 )
177176eqcomd 2739 . . . . . 6 𝑥 = 𝑦0 = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))
178177adantr 482 . . . . 5 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → 0 = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))
179168, 175, 1783eqtrd 2777 . . . 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 811 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))
181180mpoeq3dva 7473 . 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 2777 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 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  Vcvv 3475  cdif 3943  ifcif 4524  {csn 4624  cotp 4632  cmpt 5227   × cxp 5670  cfv 6535  (class class class)co 7396  cmpo 7398  m cmap 8808  Fincfn 8927  Basecbs 17131  +gcplusg 17184  .rcmulr 17185  0gc0g 17372   Σg cgsu 17373  Mndcmnd 18612  CMndccmn 19632  Ringcrg 20038   maMul cmmul 21854   Mat cmat 21876   DMat cdmat 21959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-cnex 11153  ax-resscn 11154  ax-1cn 11155  ax-icn 11156  ax-addcl 11157  ax-addrcl 11158  ax-mulcl 11159  ax-mulrcl 11160  ax-mulcom 11161  ax-addass 11162  ax-mulass 11163  ax-distr 11164  ax-i2m1 11165  ax-1ne0 11166  ax-1rid 11167  ax-rnegex 11168  ax-rrecex 11169  ax-cnre 11170  ax-pre-lttri 11171  ax-pre-lttrn 11172  ax-pre-ltadd 11173  ax-pre-mulgt0 11174
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-ot 4633  df-uni 4905  df-int 4947  df-iun 4995  df-iin 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7657  df-om 7843  df-1st 7962  df-2nd 7963  df-supp 8134  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8691  df-map 8810  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9350  df-sup 9424  df-oi 9492  df-card 9921  df-pnf 11237  df-mnf 11238  df-xr 11239  df-ltxr 11240  df-le 11241  df-sub 11433  df-neg 11434  df-nn 12200  df-2 12262  df-3 12263  df-4 12264  df-5 12265  df-6 12266  df-7 12267  df-8 12268  df-9 12269  df-n0 12460  df-z 12546  df-dec 12665  df-uz 12810  df-fz 13472  df-fzo 13615  df-seq 13954  df-hash 14278  df-struct 17067  df-sets 17084  df-slot 17102  df-ndx 17114  df-base 17132  df-ress 17161  df-plusg 17197  df-mulr 17198  df-sca 17200  df-vsca 17201  df-ip 17202  df-tset 17203  df-ple 17204  df-ds 17206  df-hom 17208  df-cco 17209  df-0g 17374  df-gsum 17375  df-prds 17380  df-pws 17382  df-mre 17517  df-mrc 17518  df-acs 17520  df-mgm 18548  df-sgrp 18597  df-mnd 18613  df-submnd 18659  df-grp 18809  df-minusg 18810  df-mulg 18936  df-cntz 19166  df-cmn 19634  df-abl 19635  df-mgp 19971  df-ur 19988  df-ring 20040  df-sra 20762  df-rgmod 20763  df-dsmm 21260  df-frlm 21275  df-mamu 21855  df-mat 21877  df-dmat 21961
This theorem is referenced by:  dmatmulcl  21971  dmatcrng  21973  scmatscmiddistr  21979  scmatcrng  21992
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