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Theorem dmatmul 20579
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 2764 . . . . . 6 (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
31, 2matmulr 20519 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
43adantr 472 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
54eqcomd 2770 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (.r𝐴) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
65oveqd 6858 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(.r𝐴)𝑌) = (𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝑌))
7 eqid 2764 . . 3 (Base‘𝑅) = (Base‘𝑅)
8 eqid 2764 . . 3 (.r𝑅) = (.r𝑅)
9 simplr 785 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑅 ∈ Ring)
10 simpll 783 . . 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 20576 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷𝑋𝐵))
1514imp 395 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋𝐷) → 𝑋𝐵)
161, 7, 11matbas2i 20503 . . . . 5 (𝑋𝐵𝑋 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
1715, 16syl 17 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋𝐷) → 𝑋 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
1817adantrr 708 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑋 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
191, 11, 12, 13dmatmat 20576 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑌𝐷𝑌𝐵))
2019imp 395 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌𝐷) → 𝑌𝐵)
211, 7, 11matbas2i 20503 . . . . 5 (𝑌𝐵𝑌 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
2220, 21syl 17 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌𝐷) → 𝑌 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
2322adantrl 707 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑌 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
242, 7, 8, 9, 10, 10, 10, 18, 23mamuval 20467 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝑌) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))))))
25 eqid 2764 . . . . . . 7 (+g𝑅) = (+g𝑅)
26 ringcmn 18847 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
2726ad2antlr 718 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑅 ∈ CMnd)
28273ad2ant1 1163 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑅 ∈ CMnd)
2928adantl 473 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → 𝑅 ∈ CMnd)
30103ad2ant1 1163 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑁 ∈ Fin)
3130adantl 473 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → 𝑁 ∈ Fin)
32 eqid 2764 . . . . . . . 8 (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))) = (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))
33 ovexd 6875 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) ∈ V)
34 fvexd 6389 . . . . . . . 8 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (0g𝑅) ∈ V)
3532, 31, 33, 34fsuppmptdm 8492 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))) finSupp (0g𝑅))
3693ad2ant1 1163 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑅 ∈ Ring)
3736ad2antlr 718 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑅 ∈ Ring)
38 simp2 1167 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑥𝑁)
3938ad2antlr 718 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑥𝑁)
40 simpr 477 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑘𝑁)
41 eqid 2764 . . . . . . . . . . . . . 14 (Base‘𝐴) = (Base‘𝐴)
421, 41, 12, 13dmatmat 20576 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷𝑋 ∈ (Base‘𝐴)))
4342imp 395 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋𝐷) → 𝑋 ∈ (Base‘𝐴))
4443adantrr 708 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑋 ∈ (Base‘𝐴))
45443ad2ant1 1163 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑋 ∈ (Base‘𝐴))
4645ad2antlr 718 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑋 ∈ (Base‘𝐴))
471, 7matecl 20506 . . . . . . . . 9 ((𝑥𝑁𝑘𝑁𝑋 ∈ (Base‘𝐴)) → (𝑥𝑋𝑘) ∈ (Base‘𝑅))
4839, 40, 46, 47syl3anc 1490 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑥𝑋𝑘) ∈ (Base‘𝑅))
49 simplr3 1279 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑦𝑁)
501, 41, 12, 13dmatmat 20576 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑌𝐷𝑌 ∈ (Base‘𝐴)))
5150imp 395 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌𝐷) → 𝑌 ∈ (Base‘𝐴))
5251adantrl 707 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑌 ∈ (Base‘𝐴))
53523ad2ant1 1163 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑌 ∈ (Base‘𝐴))
5453ad2antlr 718 . . . . . . . . 9 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑌 ∈ (Base‘𝐴))
551, 7matecl 20506 . . . . . . . . 9 ((𝑘𝑁𝑦𝑁𝑌 ∈ (Base‘𝐴)) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
5640, 49, 54, 55syl3anc 1490 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
577, 8ringcl 18827 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑘) ∈ (Base‘𝑅) ∧ (𝑘𝑌𝑦) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) ∈ (Base‘𝑅))
5837, 48, 56, 57syl3anc 1490 . . . . . . 7 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) ∈ (Base‘𝑅))
5938adantl 473 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → 𝑥𝑁)
60 simp3 1168 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑦𝑁)
6115adantrr 708 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑋𝐵)
6261, 11syl6eleq 2853 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑋 ∈ (Base‘𝐴))
63623ad2ant1 1163 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑋 ∈ (Base‘𝐴))
641, 7matecl 20506 . . . . . . . . . 10 ((𝑥𝑁𝑦𝑁𝑋 ∈ (Base‘𝐴)) → (𝑥𝑋𝑦) ∈ (Base‘𝑅))
6538, 60, 63, 64syl3anc 1490 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑥𝑋𝑦) ∈ (Base‘𝑅))
6650a1d 25 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷 → (𝑌𝐷𝑌 ∈ (Base‘𝐴))))
6766imp32 409 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑌 ∈ (Base‘𝐴))
68673ad2ant1 1163 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑌 ∈ (Base‘𝐴))
691, 7matecl 20506 . . . . . . . . . 10 ((𝑥𝑁𝑦𝑁𝑌 ∈ (Base‘𝐴)) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
7038, 60, 68, 69syl3anc 1490 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
717, 8ringcl 18827 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑦) ∈ (Base‘𝑅) ∧ (𝑥𝑌𝑦) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
7236, 65, 70, 71syl3anc 1490 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
7372adantl 473 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
74 eqtr 2783 . . . . . . . . . . 11 ((𝑘 = 𝑥𝑥 = 𝑦) → 𝑘 = 𝑦)
7574ancoms 450 . . . . . . . . . 10 ((𝑥 = 𝑦𝑘 = 𝑥) → 𝑘 = 𝑦)
7675oveq2d 6857 . . . . . . . . 9 ((𝑥 = 𝑦𝑘 = 𝑥) → (𝑥𝑋𝑘) = (𝑥𝑋𝑦))
7776adantlr 706 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 = 𝑥) → (𝑥𝑋𝑘) = (𝑥𝑋𝑦))
78 oveq1 6848 . . . . . . . . 9 (𝑘 = 𝑥 → (𝑘𝑌𝑦) = (𝑥𝑌𝑦))
7978adantl 473 . . . . . . . 8 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 = 𝑥) → (𝑘𝑌𝑦) = (𝑥𝑌𝑦))
8077, 79oveq12d 6859 . . . . . . 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 18625 . . . . . 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 787 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑋𝐷)
8310, 9, 823jca 1158 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷))
84833ad2ant1 1163 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷))
8584ad2antlr 718 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷))
8638ad2antlr 718 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑥𝑁)
87 eldifi 3893 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑁 ∖ {𝑥}) → 𝑘𝑁)
8887adantl 473 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑘𝑁)
89 eldifsni 4475 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝑁 ∖ {𝑥}) → 𝑘𝑥)
9089necomd 2991 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑁 ∖ {𝑥}) → 𝑥𝑘)
9190adantl 473 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑥𝑘)
921, 11, 12, 13dmatelnd 20578 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷) ∧ (𝑥𝑁𝑘𝑁𝑥𝑘)) → (𝑥𝑋𝑘) = 0 )
9385, 86, 88, 91, 92syl13anc 1491 . . . . . . . . . . . 12 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → (𝑥𝑋𝑘) = 0 )
9493oveq1d 6856 . . . . . . . . . . 11 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = ( 0 (.r𝑅)(𝑘𝑌𝑦)))
9536ad2antlr 718 . . . . . . . . . . . 12 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑅 ∈ Ring)
96 simplr3 1279 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑦𝑁)
9753ad2antlr 718 . . . . . . . . . . . . 13 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → 𝑌 ∈ (Base‘𝐴))
9888, 96, 97, 55syl3anc 1490 . . . . . . . . . . . 12 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
997, 8, 12ringlz 18853 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑘𝑌𝑦) ∈ (Base‘𝑅)) → ( 0 (.r𝑅)(𝑘𝑌𝑦)) = 0 )
10095, 98, 99syl2anc 579 . . . . . . . . . . 11 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → ( 0 (.r𝑅)(𝑘𝑌𝑦)) = 0 )
10194, 100eqtrd 2798 . . . . . . . . . 10 (((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘 ∈ (𝑁 ∖ {𝑥})) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = 0 )
102101mpteq2dva 4902 . . . . . . . . 9 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))) = (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ 0 ))
103102oveq2d 6857 . . . . . . . 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 8398 . . . . . . . . . . . . 13 (𝑁 ∈ Fin → (𝑁 ∖ {𝑥}) ∈ Fin)
105 ringmnd 18822 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
106104, 105anim12ci 607 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
107106adantr 472 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
1081073ad2ant1 1163 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
109108adantl 473 . . . . . . . . 9 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 ∈ Mnd ∧ (𝑁 ∖ {𝑥}) ∈ Fin))
11012gsumz 17641 . . . . . . . . 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 2798 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘 ∈ (𝑁 ∖ {𝑥}) ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = 0 )
113112oveq1d 6856 . . . . . 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 718 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑅 ∈ Mnd)
1151143ad2ant1 1163 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → 𝑅 ∈ Mnd)
11638, 60, 53, 69syl3anc 1490 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑥𝑌𝑦) ∈ (Base‘𝑅))
11736, 65, 116, 71syl3anc 1490 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅))
118115, 117jca 507 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑅 ∈ Mnd ∧ ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅)))
119118adantl 473 . . . . . . 7 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 ∈ Mnd ∧ ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)) ∈ (Base‘𝑅)))
1207, 25, 12mndlid 17578 . . . . . . 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 2802 . . . . 5 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)))
123 iftrue 4248 . . . . . 6 (𝑥 = 𝑦 → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ) = ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)))
124123adantr 472 . . . . 5 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ) = ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)))
125122, 124eqtr4d 2801 . . . 4 ((𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))
126 simprr 789 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → 𝑌𝐷)
12710, 9, 1263jca 1158 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐷))
1281273ad2ant1 1163 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐷))
129128ad2antlr 718 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐷))
130129adantl 473 . . . . . . . . . . 11 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐷))
131 simprr 789 . . . . . . . . . . 11 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑘𝑁)
132 simplr3 1279 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑦𝑁)
133132adantl 473 . . . . . . . . . . 11 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑦𝑁)
134 df-ne 2937 . . . . . . . . . . . . . . 15 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
135 neeq1 2998 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑘 → (𝑥𝑦𝑘𝑦))
136135biimpcd 240 . . . . . . . . . . . . . . 15 (𝑥𝑦 → (𝑥 = 𝑘𝑘𝑦))
137134, 136sylbir 226 . . . . . . . . . . . . . 14 𝑥 = 𝑦 → (𝑥 = 𝑘𝑘𝑦))
138137adantr 472 . . . . . . . . . . . . 13 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑥 = 𝑘𝑘𝑦))
139138adantr 472 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑥 = 𝑘𝑘𝑦))
140139impcom 396 . . . . . . . . . . 11 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑘𝑦)
1411, 11, 12, 13dmatelnd 20578 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐷) ∧ (𝑘𝑁𝑦𝑁𝑘𝑦)) → (𝑘𝑌𝑦) = 0 )
142130, 131, 133, 140, 141syl13anc 1491 . . . . . . . . . 10 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → (𝑘𝑌𝑦) = 0 )
143142oveq2d 6857 . . . . . . . . 9 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = ((𝑥𝑋𝑘)(.r𝑅) 0 ))
14436ad2antlr 718 . . . . . . . . . . 11 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑅 ∈ Ring)
14538ad2antlr 718 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑥𝑁)
146 simpr 477 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑘𝑁)
14763ad2antlr 718 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑋 ∈ (Base‘𝐴))
148145, 146, 147, 47syl3anc 1490 . . . . . . . . . . 11 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑥𝑋𝑘) ∈ (Base‘𝑅))
1497, 8, 12ringrz 18854 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝑥𝑋𝑘) ∈ (Base‘𝑅)) → ((𝑥𝑋𝑘)(.r𝑅) 0 ) = 0 )
150144, 148, 149syl2anc 579 . . . . . . . . . 10 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → ((𝑥𝑋𝑘)(.r𝑅) 0 ) = 0 )
151150adantl 473 . . . . . . . . 9 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ((𝑥𝑋𝑘)(.r𝑅) 0 ) = 0 )
152143, 151eqtrd 2798 . . . . . . . 8 ((𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = 0 )
15384ad2antlr 718 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷))
154153adantl 473 . . . . . . . . . . 11 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷))
155145adantl 473 . . . . . . . . . . 11 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑥𝑁)
156 simprr 789 . . . . . . . . . . 11 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑘𝑁)
157 df-ne 2937 . . . . . . . . . . . . 13 (𝑥𝑘 ↔ ¬ 𝑥 = 𝑘)
158157biimpri 219 . . . . . . . . . . . 12 𝑥 = 𝑘𝑥𝑘)
159158adantr 472 . . . . . . . . . . 11 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → 𝑥𝑘)
160154, 155, 156, 159, 92syl13anc 1491 . . . . . . . . . 10 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → (𝑥𝑋𝑘) = 0 )
161160oveq1d 6856 . . . . . . . . 9 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = ( 0 (.r𝑅)(𝑘𝑌𝑦)))
16268ad2antlr 718 . . . . . . . . . . . 12 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → 𝑌 ∈ (Base‘𝐴))
163146, 132, 162, 55syl3anc 1490 . . . . . . . . . . 11 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → (𝑘𝑌𝑦) ∈ (Base‘𝑅))
164144, 163, 99syl2anc 579 . . . . . . . . . 10 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → ( 0 (.r𝑅)(𝑘𝑌𝑦)) = 0 )
165164adantl 473 . . . . . . . . 9 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ( 0 (.r𝑅)(𝑘𝑌𝑦)) = 0 )
166161, 165eqtrd 2798 . . . . . . . 8 ((¬ 𝑥 = 𝑘 ∧ ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁)) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = 0 )
167152, 166pm2.61ian 846 . . . . . . 7 (((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) ∧ 𝑘𝑁) → ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)) = 0 )
168167mpteq2dva 4902 . . . . . 6 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))) = (𝑘𝑁0 ))
169168oveq2d 6857 . . . . 5 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = (𝑅 Σg (𝑘𝑁0 )))
170105anim2i 610 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Mnd))
171170ancomd 453 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin))
17212gsumz 17641 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin) → (𝑅 Σg (𝑘𝑁0 )) = 0 )
173171, 172syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 Σg (𝑘𝑁0 )) = 0 )
174173adantr 472 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑅 Σg (𝑘𝑁0 )) = 0 )
1751743ad2ant1 1163 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑅 Σg (𝑘𝑁0 )) = 0 )
176175adantl 473 . . . . 5 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘𝑁0 )) = 0 )
177 iffalse 4251 . . . . . . 7 𝑥 = 𝑦 → if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ) = 0 )
178177eqcomd 2770 . . . . . 6 𝑥 = 𝑦0 = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))
179178adantr 472 . . . . 5 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → 0 = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))
180169, 176, 1793eqtrd 2802 . . . 4 ((¬ 𝑥 = 𝑦 ∧ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁)) → (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))
181125, 180pm2.61ian 846 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) ∧ 𝑥𝑁𝑦𝑁) → (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦)))) = if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 ))
182181mpt2eq3dva 6916 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑘𝑁 ↦ ((𝑥𝑋𝑘)(.r𝑅)(𝑘𝑌𝑦))))) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 )))
1836, 24, 1823eqtrd 2802 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 384  w3a 1107   = wceq 1652  wcel 2155  wne 2936  Vcvv 3349  cdif 3728  ifcif 4242  {csn 4333  cotp 4341  cmpt 4887   × cxp 5274  cfv 6067  (class class class)co 6841  cmpt2 6843  𝑚 cmap 8059  Fincfn 8159  Basecbs 16131  +gcplusg 16215  .rcmulr 16216  0gc0g 16367   Σg cgsu 16368  Mndcmnd 17561  CMndccmn 18458  Ringcrg 18813   maMul cmmul 20464   Mat cmat 20488   DMat cdmat 20570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-inf2 8752  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-ot 4342  df-uni 4594  df-int 4633  df-iun 4677  df-iin 4678  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-se 5236  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-isom 6076  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-of 7094  df-om 7263  df-1st 7365  df-2nd 7366  df-supp 7497  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-oadd 7767  df-er 7946  df-map 8061  df-ixp 8113  df-en 8160  df-dom 8161  df-sdom 8162  df-fin 8163  df-fsupp 8482  df-sup 8554  df-oi 8621  df-card 9015  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-nn 11274  df-2 11334  df-3 11335  df-4 11336  df-5 11337  df-6 11338  df-7 11339  df-8 11340  df-9 11341  df-n0 11538  df-z 11624  df-dec 11740  df-uz 11886  df-fz 12533  df-fzo 12673  df-seq 13008  df-hash 13321  df-struct 16133  df-ndx 16134  df-slot 16135  df-base 16137  df-sets 16138  df-ress 16139  df-plusg 16228  df-mulr 16229  df-sca 16231  df-vsca 16232  df-ip 16233  df-tset 16234  df-ple 16235  df-ds 16237  df-hom 16239  df-cco 16240  df-0g 16369  df-gsum 16370  df-prds 16375  df-pws 16377  df-mre 16513  df-mrc 16514  df-acs 16516  df-mgm 17509  df-sgrp 17551  df-mnd 17562  df-submnd 17603  df-grp 17693  df-minusg 17694  df-mulg 17809  df-cntz 18014  df-cmn 18460  df-abl 18461  df-mgp 18756  df-ur 18768  df-ring 18815  df-sra 19445  df-rgmod 19446  df-dsmm 20351  df-frlm 20366  df-mamu 20465  df-mat 20489  df-dmat 20572
This theorem is referenced by:  dmatmulcl  20582  dmatcrng  20584  scmatscmiddistr  20590  scmatcrng  20603
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