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Theorem mdetuni0 20914
Description: Lemma for mdetuni 20915. (Contributed by SO, 15-Jul-2018.)
Hypotheses
Ref Expression
mdetuni.a 𝐴 = (𝑁 Mat 𝑅)
mdetuni.b 𝐵 = (Base‘𝐴)
mdetuni.k 𝐾 = (Base‘𝑅)
mdetuni.0g 0 = (0g𝑅)
mdetuni.1r 1 = (1r𝑅)
mdetuni.pg + = (+g𝑅)
mdetuni.tg · = (.r𝑅)
mdetuni.n (𝜑𝑁 ∈ Fin)
mdetuni.r (𝜑𝑅 ∈ Ring)
mdetuni.ff (𝜑𝐷:𝐵𝐾)
mdetuni.al (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))
mdetuni.li (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))
mdetuni.sc (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))
mdetuni.e 𝐸 = (𝑁 maDet 𝑅)
mdetuni.cr (𝜑𝑅 ∈ CRing)
mdetuni.f (𝜑𝐹𝐵)
Assertion
Ref Expression
mdetuni0 (𝜑 → (𝐷𝐹) = ((𝐷‘(1r𝐴)) · (𝐸𝐹)))
Distinct variable groups:   𝜑,𝑥,𝑦,𝑧,𝑤   𝑥,𝐵,𝑦,𝑧,𝑤   𝑥,𝐾,𝑦,𝑧,𝑤   𝑥,𝑁,𝑦,𝑧,𝑤   𝑥,𝐷,𝑦,𝑧,𝑤   𝑥, · ,𝑦,𝑧,𝑤   𝑥, + ,𝑦,𝑧,𝑤   𝑥, 0 ,𝑦,𝑧,𝑤   𝑥, 1 ,𝑦,𝑧,𝑤   𝑥,𝑅,𝑦,𝑧,𝑤   𝑥,𝐴,𝑦,𝑧,𝑤   𝑥,𝐸,𝑦,𝑧,𝑤   𝑥,𝐹,𝑦,𝑧,𝑤

Proof of Theorem mdetuni0
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mdetuni.a . . . . 5 𝐴 = (𝑁 Mat 𝑅)
2 mdetuni.b . . . . 5 𝐵 = (Base‘𝐴)
3 mdetuni.k . . . . 5 𝐾 = (Base‘𝑅)
4 mdetuni.0g . . . . 5 0 = (0g𝑅)
5 mdetuni.1r . . . . 5 1 = (1r𝑅)
6 mdetuni.pg . . . . 5 + = (+g𝑅)
7 mdetuni.tg . . . . 5 · = (.r𝑅)
8 mdetuni.n . . . . 5 (𝜑𝑁 ∈ Fin)
9 mdetuni.r . . . . 5 (𝜑𝑅 ∈ Ring)
10 ringgrp 18992 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
119, 10syl 17 . . . . . . . 8 (𝜑𝑅 ∈ Grp)
1211adantr 481 . . . . . . 7 ((𝜑𝑎𝐵) → 𝑅 ∈ Grp)
13 mdetuni.ff . . . . . . . 8 (𝜑𝐷:𝐵𝐾)
1413ffvelrnda 6716 . . . . . . 7 ((𝜑𝑎𝐵) → (𝐷𝑎) ∈ 𝐾)
159adantr 481 . . . . . . . 8 ((𝜑𝑎𝐵) → 𝑅 ∈ Ring)
168, 9jca 512 . . . . . . . . . . 11 (𝜑 → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
171matring 20736 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
18 eqid 2795 . . . . . . . . . . . 12 (1r𝐴) = (1r𝐴)
192, 18ringidcl 19008 . . . . . . . . . . 11 (𝐴 ∈ Ring → (1r𝐴) ∈ 𝐵)
2016, 17, 193syl 18 . . . . . . . . . 10 (𝜑 → (1r𝐴) ∈ 𝐵)
2113, 20ffvelrnd 6717 . . . . . . . . 9 (𝜑 → (𝐷‘(1r𝐴)) ∈ 𝐾)
2221adantr 481 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝐷‘(1r𝐴)) ∈ 𝐾)
23 mdetuni.cr . . . . . . . . . 10 (𝜑𝑅 ∈ CRing)
24 mdetuni.e . . . . . . . . . . 11 𝐸 = (𝑁 maDet 𝑅)
2524, 1, 2, 3mdetf 20888 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝐸:𝐵𝐾)
2623, 25syl 17 . . . . . . . . 9 (𝜑𝐸:𝐵𝐾)
2726ffvelrnda 6716 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝐸𝑎) ∈ 𝐾)
283, 7ringcl 19001 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑎) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) ∈ 𝐾)
2915, 22, 27, 28syl3anc 1364 . . . . . . 7 ((𝜑𝑎𝐵) → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) ∈ 𝐾)
30 eqid 2795 . . . . . . . 8 (-g𝑅) = (-g𝑅)
313, 30grpsubcl 17936 . . . . . . 7 ((𝑅 ∈ Grp ∧ (𝐷𝑎) ∈ 𝐾 ∧ ((𝐷‘(1r𝐴)) · (𝐸𝑎)) ∈ 𝐾) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) ∈ 𝐾)
3212, 14, 29, 31syl3anc 1364 . . . . . 6 ((𝜑𝑎𝐵) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) ∈ 𝐾)
3332fmpttd 6742 . . . . 5 (𝜑 → (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))):𝐵𝐾)
34 simpr1 1187 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → 𝑏𝐵)
35 fveq2 6538 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝐷𝑎) = (𝐷𝑏))
36 fveq2 6538 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (𝐸𝑎) = (𝐸𝑏))
3736oveq2d 7032 . . . . . . . . . . . 12 (𝑎 = 𝑏 → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸𝑏)))
3835, 37oveq12d 7034 . . . . . . . . . . 11 (𝑎 = 𝑏 → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
39 eqid 2795 . . . . . . . . . . 11 (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))) = (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))
40 ovex 7048 . . . . . . . . . . 11 ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))) ∈ V
4138, 39, 40fvmpt 6635 . . . . . . . . . 10 (𝑏𝐵 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
4234, 41syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
43423adant3 1125 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
44 simp1 1129 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝜑)
45 simp21 1199 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑏𝐵)
46 simp3r 1195 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))
47 oveq2 7024 . . . . . . . . . . . . 13 (𝑒 = 𝑤 → (𝑐𝑏𝑒) = (𝑐𝑏𝑤))
48 oveq2 7024 . . . . . . . . . . . . 13 (𝑒 = 𝑤 → (𝑑𝑏𝑒) = (𝑑𝑏𝑤))
4947, 48eqeq12d 2810 . . . . . . . . . . . 12 (𝑒 = 𝑤 → ((𝑐𝑏𝑒) = (𝑑𝑏𝑒) ↔ (𝑐𝑏𝑤) = (𝑑𝑏𝑤)))
5049cbvralv 3403 . . . . . . . . . . 11 (∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒) ↔ ∀𝑤𝑁 (𝑐𝑏𝑤) = (𝑑𝑏𝑤))
5146, 50sylib 219 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ∀𝑤𝑁 (𝑐𝑏𝑤) = (𝑑𝑏𝑤))
52 simp22 1200 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑐𝑁)
53 simp23 1201 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑑𝑁)
54 simp3l 1194 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑐𝑑)
55 mdetuni.al . . . . . . . . . . 11 (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))
56 mdetuni.li . . . . . . . . . . 11 (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))
57 mdetuni.sc . . . . . . . . . . 11 (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))
581, 2, 3, 4, 5, 6, 7, 8, 9, 13, 55, 56, 57mdetunilem1 20905 . . . . . . . . . 10 (((𝜑𝑏𝐵 ∧ ∀𝑤𝑁 (𝑐𝑏𝑤) = (𝑑𝑏𝑤)) ∧ (𝑐𝑁𝑑𝑁𝑐𝑑)) → (𝐷𝑏) = 0 )
5944, 45, 51, 52, 53, 54, 58syl33anc 1378 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → (𝐷𝑏) = 0 )
60233ad2ant1 1126 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑅 ∈ CRing)
6124, 1, 2, 4, 60, 45, 52, 53, 54, 46mdetralt 20901 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → (𝐸𝑏) = 0 )
6261oveq2d 7032 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝐷‘(1r𝐴)) · (𝐸𝑏)) = ((𝐷‘(1r𝐴)) · 0 ))
6359, 62oveq12d 7034 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))) = ( 0 (-g𝑅)((𝐷‘(1r𝐴)) · 0 )))
643, 7, 4ringrz 19028 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · 0 ) = 0 )
659, 21, 64syl2anc 584 . . . . . . . . . . 11 (𝜑 → ((𝐷‘(1r𝐴)) · 0 ) = 0 )
6665oveq2d 7032 . . . . . . . . . 10 (𝜑 → ( 0 (-g𝑅)((𝐷‘(1r𝐴)) · 0 )) = ( 0 (-g𝑅) 0 ))
673, 4grpidcl 17889 . . . . . . . . . . 11 (𝑅 ∈ Grp → 0𝐾)
683, 4, 30grpsubid 17940 . . . . . . . . . . 11 ((𝑅 ∈ Grp ∧ 0𝐾) → ( 0 (-g𝑅) 0 ) = 0 )
6911, 67, 68syl2anc2 585 . . . . . . . . . 10 (𝜑 → ( 0 (-g𝑅) 0 ) = 0 )
7066, 69eqtrd 2831 . . . . . . . . 9 (𝜑 → ( 0 (-g𝑅)((𝐷‘(1r𝐴)) · 0 )) = 0 )
71703ad2ant1 1126 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ( 0 (-g𝑅)((𝐷‘(1r𝐴)) · 0 )) = 0 )
7243, 63, 713eqtrd 2835 . . . . . . 7 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = 0 )
73723expia 1114 . . . . . 6 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → ((𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒)) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = 0 ))
7473ralrimivvva 3159 . . . . 5 (𝜑 → ∀𝑏𝐵𝑐𝑁𝑑𝑁 ((𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒)) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = 0 ))
75 simp1 1129 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝜑)
76 simp2ll 1233 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑏𝐵)
77 simp2lr 1234 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑐𝐵)
78 simp2rl 1235 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑑𝐵)
79 simp2rr 1236 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑒𝑁)
80 simp31 1202 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))))
81 simp32 1203 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))
82 simp33 1204 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))
831, 2, 3, 4, 5, 6, 7, 8, 9, 13, 55, 56, 57mdetunilem3 20907 . . . . . . . . . . . 12 (((𝜑𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁 ∧ (𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁)))) ∧ ((𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝐷𝑏) = ((𝐷𝑐) + (𝐷𝑑)))
8475, 76, 77, 78, 79, 80, 81, 82, 83syl332anc 1394 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝐷𝑏) = ((𝐷𝑐) + (𝐷𝑑)))
85233ad2ant1 1126 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑅 ∈ CRing)
8624, 1, 2, 6, 85, 76, 77, 78, 79, 80, 81, 82mdetrlin 20895 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝐸𝑏) = ((𝐸𝑐) + (𝐸𝑑)))
8786oveq2d 7032 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝐷‘(1r𝐴)) · (𝐸𝑏)) = ((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑))))
8884, 87oveq12d 7034 . . . . . . . . . 10 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))) = (((𝐷𝑐) + (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑)))))
89 simprll 775 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑏𝐵)
9089, 41syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
91903adant3 1125 . . . . . . . . . 10 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
92 simprlr 776 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑐𝐵)
93 fveq2 6538 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑐 → (𝐷𝑎) = (𝐷𝑐))
94 fveq2 6538 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑐 → (𝐸𝑎) = (𝐸𝑐))
9594oveq2d 7032 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑐 → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸𝑐)))
9693, 95oveq12d 7034 . . . . . . . . . . . . . . 15 (𝑎 = 𝑐 → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝑐)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑐))))
97 ovex 7048 . . . . . . . . . . . . . . 15 ((𝐷𝑐)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑐))) ∈ V
9896, 39, 97fvmpt 6635 . . . . . . . . . . . . . 14 (𝑐𝐵 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑐) = ((𝐷𝑐)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑐))))
9992, 98syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑐) = ((𝐷𝑐)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑐))))
100 simprrl 777 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑑𝐵)
101 fveq2 6538 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑑 → (𝐷𝑎) = (𝐷𝑑))
102 fveq2 6538 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑑 → (𝐸𝑎) = (𝐸𝑑))
103102oveq2d 7032 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑑 → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸𝑑)))
104101, 103oveq12d 7034 . . . . . . . . . . . . . . 15 (𝑎 = 𝑑 → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑))))
105 ovex 7048 . . . . . . . . . . . . . . 15 ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑))) ∈ V
106104, 39, 105fvmpt 6635 . . . . . . . . . . . . . 14 (𝑑𝐵 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑) = ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑))))
107100, 106syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑) = ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑))))
10899, 107oveq12d 7034 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑐) + ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑)) = (((𝐷𝑐)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑐))) + ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑)))))
109 ringabl 19020 . . . . . . . . . . . . . . 15 (𝑅 ∈ Ring → 𝑅 ∈ Abel)
1109, 109syl 17 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ Abel)
111110adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ Abel)
11213adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝐷:𝐵𝐾)
113112, 92ffvelrnd 6717 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷𝑐) ∈ 𝐾)
114112, 100ffvelrnd 6717 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷𝑑) ∈ 𝐾)
1159adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ Ring)
11621adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷‘(1r𝐴)) ∈ 𝐾)
11726adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝐸:𝐵𝐾)
118117, 92ffvelrnd 6717 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐸𝑐) ∈ 𝐾)
1193, 7ringcl 19001 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑐) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · (𝐸𝑐)) ∈ 𝐾)
120115, 116, 118, 119syl3anc 1364 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝐷‘(1r𝐴)) · (𝐸𝑐)) ∈ 𝐾)
121117, 100ffvelrnd 6717 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐸𝑑) ∈ 𝐾)
1223, 7ringcl 19001 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑑) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · (𝐸𝑑)) ∈ 𝐾)
123115, 116, 121, 122syl3anc 1364 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝐷‘(1r𝐴)) · (𝐸𝑑)) ∈ 𝐾)
1243, 6, 30ablsub4 18658 . . . . . . . . . . . . 13 ((𝑅 ∈ Abel ∧ ((𝐷𝑐) ∈ 𝐾 ∧ (𝐷𝑑) ∈ 𝐾) ∧ (((𝐷‘(1r𝐴)) · (𝐸𝑐)) ∈ 𝐾 ∧ ((𝐷‘(1r𝐴)) · (𝐸𝑑)) ∈ 𝐾)) → (((𝐷𝑐) + (𝐷𝑑))(-g𝑅)(((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑)))) = (((𝐷𝑐)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑐))) + ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑)))))
125111, 113, 114, 120, 123, 124syl122anc 1372 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝐷𝑐) + (𝐷𝑑))(-g𝑅)(((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑)))) = (((𝐷𝑐)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑐))) + ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑)))))
1263, 6, 7ringdi 19006 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ ((𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑐) ∈ 𝐾 ∧ (𝐸𝑑) ∈ 𝐾)) → ((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑))) = (((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑))))
127115, 116, 118, 121, 126syl13anc 1365 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑))) = (((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑))))
128127eqcomd 2801 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑))) = ((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑))))
129128oveq2d 7032 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝐷𝑐) + (𝐷𝑑))(-g𝑅)(((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑)))) = (((𝐷𝑐) + (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑)))))
130108, 125, 1293eqtr2d 2837 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑐) + ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑)) = (((𝐷𝑐) + (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑)))))
1311303adant3 1125 . . . . . . . . . 10 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑐) + ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑)) = (((𝐷𝑐) + (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑)))))
13288, 91, 1313eqtr4d 2841 . . . . . . . . 9 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = (((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑐) + ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑)))
1331323expia 1114 . . . . . . . 8 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = (((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑐) + ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑))))
134133anassrs 468 . . . . . . 7 (((𝜑 ∧ (𝑏𝐵𝑐𝐵)) ∧ (𝑑𝐵𝑒𝑁)) → (((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = (((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑐) + ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑))))
135134ralrimivva 3158 . . . . . 6 ((𝜑 ∧ (𝑏𝐵𝑐𝐵)) → ∀𝑑𝐵𝑒𝑁 (((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = (((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑐) + ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑))))
136135ralrimivva 3158 . . . . 5 (𝜑 → ∀𝑏𝐵𝑐𝐵𝑑𝐵𝑒𝑁 (((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘𝑓 + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = (((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑐) + ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑))))
137 simp1 1129 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝜑)
138 simp2ll 1233 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑏𝐵)
139 simp2lr 1234 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑐𝐾)
140 simp2rl 1235 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑑𝐵)
141 simp2rr 1236 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑒𝑁)
142 simp3l 1194 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))))
143 simp3r 1195 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))
1441, 2, 3, 4, 5, 6, 7, 8, 9, 13, 55, 56, 57mdetunilem4 20908 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏𝐵𝑐𝐾𝑑𝐵) ∧ (𝑒𝑁 ∧ (𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝐷𝑏) = (𝑐 · (𝐷𝑑)))
145137, 138, 139, 140, 141, 142, 143, 144syl133anc 1386 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝐷𝑏) = (𝑐 · (𝐷𝑑)))
146233ad2ant1 1126 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑅 ∈ CRing)
14724, 1, 2, 3, 7, 146, 138, 139, 140, 141, 142, 143mdetrsca 20896 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝐸𝑏) = (𝑐 · (𝐸𝑑)))
148147oveq2d 7032 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝐷‘(1r𝐴)) · (𝐸𝑏)) = ((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑))))
149145, 148oveq12d 7034 . . . . . . . . . 10 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))) = ((𝑐 · (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑)))))
150 simprll 775 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝑏𝐵)
151150, 41syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
1521513adant3 1125 . . . . . . . . . 10 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
153 simprrl 777 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝑑𝐵)
154153, 106syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑) = ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑))))
155154oveq2d 7032 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 · ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑)) = (𝑐 · ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑)))))
1569adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ Ring)
157 simprlr 776 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝑐𝐾)
15813adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝐷:𝐵𝐾)
159158, 153ffvelrnd 6717 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷𝑑) ∈ 𝐾)
16021adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷‘(1r𝐴)) ∈ 𝐾)
16126adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝐸:𝐵𝐾)
162161, 153ffvelrnd 6717 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝐸𝑑) ∈ 𝐾)
163156, 160, 162, 122syl3anc 1364 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → ((𝐷‘(1r𝐴)) · (𝐸𝑑)) ∈ 𝐾)
1643, 7, 30, 156, 157, 159, 163ringsubdi 19039 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 · ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑)))) = ((𝑐 · (𝐷𝑑))(-g𝑅)(𝑐 · ((𝐷‘(1r𝐴)) · (𝐸𝑑)))))
165 eqid 2795 . . . . . . . . . . . . . . . . 17 (mulGrp‘𝑅) = (mulGrp‘𝑅)
166165crngmgp 18995 . . . . . . . . . . . . . . . 16 (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
16723, 166syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
168167adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (mulGrp‘𝑅) ∈ CMnd)
169165, 3mgpbas 18935 . . . . . . . . . . . . . . 15 𝐾 = (Base‘(mulGrp‘𝑅))
170165, 7mgpplusg 18933 . . . . . . . . . . . . . . 15 · = (+g‘(mulGrp‘𝑅))
171169, 170cmn12 18653 . . . . . . . . . . . . . 14 (((mulGrp‘𝑅) ∈ CMnd ∧ (𝑐𝐾 ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑑) ∈ 𝐾)) → (𝑐 · ((𝐷‘(1r𝐴)) · (𝐸𝑑))) = ((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑))))
172168, 157, 160, 162, 171syl13anc 1365 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 · ((𝐷‘(1r𝐴)) · (𝐸𝑑))) = ((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑))))
173172oveq2d 7032 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑐 · (𝐷𝑑))(-g𝑅)(𝑐 · ((𝐷‘(1r𝐴)) · (𝐸𝑑)))) = ((𝑐 · (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑)))))
174155, 164, 1733eqtrd 2835 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 · ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑)) = ((𝑐 · (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑)))))
1751743adant3 1125 . . . . . . . . . 10 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑐 · ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑)) = ((𝑐 · (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑)))))
176149, 152, 1753eqtr4d 2841 . . . . . . . . 9 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = (𝑐 · ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑)))
1771763expia 1114 . . . . . . . 8 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = (𝑐 · ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑))))
178177anassrs 468 . . . . . . 7 (((𝜑 ∧ (𝑏𝐵𝑐𝐾)) ∧ (𝑑𝐵𝑒𝑁)) → (((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = (𝑐 · ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑))))
179178ralrimivva 3158 . . . . . 6 ((𝜑 ∧ (𝑏𝐵𝑐𝐾)) → ∀𝑑𝐵𝑒𝑁 (((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = (𝑐 · ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑))))
180179ralrimivva 3158 . . . . 5 (𝜑 → ∀𝑏𝐵𝑐𝐾𝑑𝐵𝑒𝑁 (((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = (𝑐 · ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑))))
181 eqidd 2796 . . . . . 6 (𝜑 → (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))) = (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))))
182 fveq2 6538 . . . . . . . 8 (𝑎 = (1r𝐴) → (𝐷𝑎) = (𝐷‘(1r𝐴)))
183 fveq2 6538 . . . . . . . . 9 (𝑎 = (1r𝐴) → (𝐸𝑎) = (𝐸‘(1r𝐴)))
184183oveq2d 7032 . . . . . . . 8 (𝑎 = (1r𝐴) → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴))))
185182, 184oveq12d 7034 . . . . . . 7 (𝑎 = (1r𝐴) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷‘(1r𝐴))(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴)))))
18624, 1, 18, 5mdet1 20894 . . . . . . . . . . . 12 ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐸‘(1r𝐴)) = 1 )
18723, 8, 186syl2anc 584 . . . . . . . . . . 11 (𝜑 → (𝐸‘(1r𝐴)) = 1 )
188187oveq2d 7032 . . . . . . . . . 10 (𝜑 → ((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴))) = ((𝐷‘(1r𝐴)) · 1 ))
1893, 7, 5ringridm 19012 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · 1 ) = (𝐷‘(1r𝐴)))
1909, 21, 189syl2anc 584 . . . . . . . . . 10 (𝜑 → ((𝐷‘(1r𝐴)) · 1 ) = (𝐷‘(1r𝐴)))
191188, 190eqtrd 2831 . . . . . . . . 9 (𝜑 → ((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴))) = (𝐷‘(1r𝐴)))
192191oveq2d 7032 . . . . . . . 8 (𝜑 → ((𝐷‘(1r𝐴))(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴)))) = ((𝐷‘(1r𝐴))(-g𝑅)(𝐷‘(1r𝐴))))
1933, 4, 30grpsubid 17940 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ (𝐷‘(1r𝐴)) ∈ 𝐾) → ((𝐷‘(1r𝐴))(-g𝑅)(𝐷‘(1r𝐴))) = 0 )
19411, 21, 193syl2anc 584 . . . . . . . 8 (𝜑 → ((𝐷‘(1r𝐴))(-g𝑅)(𝐷‘(1r𝐴))) = 0 )
195192, 194eqtrd 2831 . . . . . . 7 (𝜑 → ((𝐷‘(1r𝐴))(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴)))) = 0 )
196185, 195sylan9eqr 2853 . . . . . 6 ((𝜑𝑎 = (1r𝐴)) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = 0 )
1974fvexi 6552 . . . . . . 7 0 ∈ V
198197a1i 11 . . . . . 6 (𝜑0 ∈ V)
199181, 196, 20, 198fvmptd 6641 . . . . 5 (𝜑 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘(1r𝐴)) = 0 )
200 eqid 2795 . . . . 5 {𝑏 ∣ ∀𝑐𝐵𝑑 ∈ (𝑁𝑚 𝑁)(∀𝑒𝑏 (𝑐𝑒) = if(𝑒𝑑, 1 , 0 ) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑐) = 0 )} = {𝑏 ∣ ∀𝑐𝐵𝑑 ∈ (𝑁𝑚 𝑁)(∀𝑒𝑏 (𝑐𝑒) = if(𝑒𝑑, 1 , 0 ) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑐) = 0 )}
2011, 2, 3, 4, 5, 6, 7, 8, 9, 33, 74, 136, 180, 199, 200mdetunilem9 20913 . . . 4 (𝜑 → (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))) = (𝐵 × { 0 }))
202201fveq1d 6540 . . 3 (𝜑 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝐹) = ((𝐵 × { 0 })‘𝐹))
203 fveq2 6538 . . . . . 6 (𝑎 = 𝐹 → (𝐷𝑎) = (𝐷𝐹))
204 fveq2 6538 . . . . . . 7 (𝑎 = 𝐹 → (𝐸𝑎) = (𝐸𝐹))
205204oveq2d 7032 . . . . . 6 (𝑎 = 𝐹 → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸𝐹)))
206203, 205oveq12d 7034 . . . . 5 (𝑎 = 𝐹 → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))))
207206adantl 482 . . . 4 ((𝜑𝑎 = 𝐹) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))))
208 mdetuni.f . . . 4 (𝜑𝐹𝐵)
209 ovexd 7050 . . . 4 (𝜑 → ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))) ∈ V)
210181, 207, 208, 209fvmptd 6641 . . 3 (𝜑 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝐹) = ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))))
211197fvconst2 6833 . . . 4 (𝐹𝐵 → ((𝐵 × { 0 })‘𝐹) = 0 )
212208, 211syl 17 . . 3 (𝜑 → ((𝐵 × { 0 })‘𝐹) = 0 )
213202, 210, 2123eqtr3d 2839 . 2 (𝜑 → ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))) = 0 )
21413, 208ffvelrnd 6717 . . 3 (𝜑 → (𝐷𝐹) ∈ 𝐾)
21526, 208ffvelrnd 6717 . . . 4 (𝜑 → (𝐸𝐹) ∈ 𝐾)
2163, 7ringcl 19001 . . . 4 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝐹) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · (𝐸𝐹)) ∈ 𝐾)
2179, 21, 215, 216syl3anc 1364 . . 3 (𝜑 → ((𝐷‘(1r𝐴)) · (𝐸𝐹)) ∈ 𝐾)
2183, 4, 30grpsubeq0 17942 . . 3 ((𝑅 ∈ Grp ∧ (𝐷𝐹) ∈ 𝐾 ∧ ((𝐷‘(1r𝐴)) · (𝐸𝐹)) ∈ 𝐾) → (((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))) = 0 ↔ (𝐷𝐹) = ((𝐷‘(1r𝐴)) · (𝐸𝐹))))
21911, 214, 217, 218syl3anc 1364 . 2 (𝜑 → (((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))) = 0 ↔ (𝐷𝐹) = ((𝐷‘(1r𝐴)) · (𝐸𝐹))))
220213, 219mpbid 233 1 (𝜑 → (𝐷𝐹) = ((𝐷‘(1r𝐴)) · (𝐸𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  {cab 2775  wne 2984  wral 3105  Vcvv 3437  cdif 3856  ifcif 4381  {csn 4472  cmpt 5041   × cxp 5441  cres 5445  wf 6221  cfv 6225  (class class class)co 7016  𝑓 cof 7265  𝑚 cmap 8256  Fincfn 8357  Basecbs 16312  +gcplusg 16394  .rcmulr 16395  0gc0g 16542  Grpcgrp 17861  -gcsg 17863  CMndccmn 18633  Abelcabl 18634  mulGrpcmgp 18929  1rcur 18941  Ringcrg 18987  CRingccrg 18988   Mat cmat 20700   maDet cmdat 20877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-addf 10462  ax-mulf 10463
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-xor 1497  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-ot 4481  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267  df-om 7437  df-1st 7545  df-2nd 7546  df-supp 7682  df-tpos 7743  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-er 8139  df-map 8258  df-pm 8259  df-ixp 8311  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fsupp 8680  df-sup 8752  df-oi 8820  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-xnn0 11816  df-z 11830  df-dec 11948  df-uz 12094  df-rp 12240  df-fz 12743  df-fzo 12884  df-seq 13220  df-exp 13280  df-hash 13541  df-word 13708  df-lsw 13761  df-concat 13769  df-s1 13794  df-substr 13839  df-pfx 13869  df-splice 13948  df-reverse 13957  df-s2 14046  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-mulr 16408  df-starv 16409  df-sca 16410  df-vsca 16411  df-ip 16412  df-tset 16413  df-ple 16414  df-ds 16416  df-unif 16417  df-hom 16418  df-cco 16419  df-0g 16544  df-gsum 16545  df-prds 16550  df-pws 16552  df-mre 16686  df-mrc 16687  df-acs 16689  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-mhm 17774  df-submnd 17775  df-grp 17864  df-minusg 17865  df-sbg 17866  df-mulg 17982  df-subg 18030  df-ghm 18097  df-gim 18140  df-cntz 18188  df-oppg 18215  df-symg 18237  df-pmtr 18301  df-psgn 18350  df-evpm 18351  df-cmn 18635  df-abl 18636  df-mgp 18930  df-ur 18942  df-srg 18946  df-ring 18989  df-cring 18990  df-oppr 19063  df-dvdsr 19081  df-unit 19082  df-invr 19112  df-dvr 19123  df-rnghom 19157  df-drng 19194  df-subrg 19223  df-lmod 19326  df-lss 19394  df-sra 19634  df-rgmod 19635  df-cnfld 20228  df-zring 20300  df-zrh 20333  df-dsmm 20558  df-frlm 20573  df-mamu 20677  df-mat 20701  df-mdet 20878
This theorem is referenced by:  mdetuni  20915  mdetmul  20916
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