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Theorem mdetuni0 22643
Description: Lemma for mdetuni 22644. (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 20256 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
119, 10syl 17 . . . . . . . 8 (𝜑𝑅 ∈ Grp)
1211adantr 480 . . . . . . 7 ((𝜑𝑎𝐵) → 𝑅 ∈ Grp)
13 mdetuni.ff . . . . . . . 8 (𝜑𝐷:𝐵𝐾)
1413ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑎𝐵) → (𝐷𝑎) ∈ 𝐾)
159adantr 480 . . . . . . . 8 ((𝜑𝑎𝐵) → 𝑅 ∈ Ring)
168, 9jca 511 . . . . . . . . . . 11 (𝜑 → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
171matring 22465 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
18 eqid 2735 . . . . . . . . . . . 12 (1r𝐴) = (1r𝐴)
192, 18ringidcl 20280 . . . . . . . . . . 11 (𝐴 ∈ Ring → (1r𝐴) ∈ 𝐵)
2016, 17, 193syl 18 . . . . . . . . . 10 (𝜑 → (1r𝐴) ∈ 𝐵)
2113, 20ffvelcdmd 7105 . . . . . . . . 9 (𝜑 → (𝐷‘(1r𝐴)) ∈ 𝐾)
2221adantr 480 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝐷‘(1r𝐴)) ∈ 𝐾)
23 mdetuni.cr . . . . . . . . . 10 (𝜑𝑅 ∈ CRing)
24 mdetuni.e . . . . . . . . . . 11 𝐸 = (𝑁 maDet 𝑅)
2524, 1, 2, 3mdetf 22617 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝐸:𝐵𝐾)
2623, 25syl 17 . . . . . . . . 9 (𝜑𝐸:𝐵𝐾)
2726ffvelcdmda 7104 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝐸𝑎) ∈ 𝐾)
283, 7ringcl 20268 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑎) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) ∈ 𝐾)
2915, 22, 27, 28syl3anc 1370 . . . . . . 7 ((𝜑𝑎𝐵) → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) ∈ 𝐾)
30 eqid 2735 . . . . . . . 8 (-g𝑅) = (-g𝑅)
313, 30grpsubcl 19051 . . . . . . 7 ((𝑅 ∈ Grp ∧ (𝐷𝑎) ∈ 𝐾 ∧ ((𝐷‘(1r𝐴)) · (𝐸𝑎)) ∈ 𝐾) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) ∈ 𝐾)
3212, 14, 29, 31syl3anc 1370 . . . . . 6 ((𝜑𝑎𝐵) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) ∈ 𝐾)
3332fmpttd 7135 . . . . 5 (𝜑 → (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))):𝐵𝐾)
34 simpr1 1193 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → 𝑏𝐵)
35 fveq2 6907 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝐷𝑎) = (𝐷𝑏))
36 fveq2 6907 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (𝐸𝑎) = (𝐸𝑏))
3736oveq2d 7447 . . . . . . . . . . . 12 (𝑎 = 𝑏 → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸𝑏)))
3835, 37oveq12d 7449 . . . . . . . . . . 11 (𝑎 = 𝑏 → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
39 eqid 2735 . . . . . . . . . . 11 (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))) = (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))
40 ovex 7464 . . . . . . . . . . 11 ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))) ∈ V
4138, 39, 40fvmpt 7016 . . . . . . . . . 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 1131 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
44 simp1 1135 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝜑)
45 simp21 1205 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑏𝐵)
46 simp3r 1201 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))
47 oveq2 7439 . . . . . . . . . . . . 13 (𝑒 = 𝑤 → (𝑐𝑏𝑒) = (𝑐𝑏𝑤))
48 oveq2 7439 . . . . . . . . . . . . 13 (𝑒 = 𝑤 → (𝑑𝑏𝑒) = (𝑑𝑏𝑤))
4947, 48eqeq12d 2751 . . . . . . . . . . . 12 (𝑒 = 𝑤 → ((𝑐𝑏𝑒) = (𝑑𝑏𝑒) ↔ (𝑐𝑏𝑤) = (𝑑𝑏𝑤)))
5049cbvralvw 3235 . . . . . . . . . . 11 (∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒) ↔ ∀𝑤𝑁 (𝑐𝑏𝑤) = (𝑑𝑏𝑤))
5146, 50sylib 218 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ∀𝑤𝑁 (𝑐𝑏𝑤) = (𝑑𝑏𝑤))
52 simp22 1206 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑐𝑁)
53 simp23 1207 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑑𝑁)
54 simp3l 1200 . . . . . . . . . 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 22634 . . . . . . . . . 10 (((𝜑𝑏𝐵 ∧ ∀𝑤𝑁 (𝑐𝑏𝑤) = (𝑑𝑏𝑤)) ∧ (𝑐𝑁𝑑𝑁𝑐𝑑)) → (𝐷𝑏) = 0 )
5944, 45, 51, 52, 53, 54, 58syl33anc 1384 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → (𝐷𝑏) = 0 )
60233ad2ant1 1132 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑅 ∈ CRing)
6124, 1, 2, 4, 60, 45, 52, 53, 54, 46mdetralt 22630 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → (𝐸𝑏) = 0 )
6261oveq2d 7447 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝐷‘(1r𝐴)) · (𝐸𝑏)) = ((𝐷‘(1r𝐴)) · 0 ))
6359, 62oveq12d 7449 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))) = ( 0 (-g𝑅)((𝐷‘(1r𝐴)) · 0 )))
643, 7, 4ringrz 20308 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · 0 ) = 0 )
659, 21, 64syl2anc 584 . . . . . . . . . . 11 (𝜑 → ((𝐷‘(1r𝐴)) · 0 ) = 0 )
6665oveq2d 7447 . . . . . . . . . 10 (𝜑 → ( 0 (-g𝑅)((𝐷‘(1r𝐴)) · 0 )) = ( 0 (-g𝑅) 0 ))
673, 4grpidcl 18996 . . . . . . . . . . 11 (𝑅 ∈ Grp → 0𝐾)
683, 4, 30grpsubid 19055 . . . . . . . . . . 11 ((𝑅 ∈ Grp ∧ 0𝐾) → ( 0 (-g𝑅) 0 ) = 0 )
6911, 67, 68syl2anc2 585 . . . . . . . . . 10 (𝜑 → ( 0 (-g𝑅) 0 ) = 0 )
7066, 69eqtrd 2775 . . . . . . . . 9 (𝜑 → ( 0 (-g𝑅)((𝐷‘(1r𝐴)) · 0 )) = 0 )
71703ad2ant1 1132 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ( 0 (-g𝑅)((𝐷‘(1r𝐴)) · 0 )) = 0 )
7243, 63, 713eqtrd 2779 . . . . . . 7 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = 0 )
73723expia 1120 . . . . . 6 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → ((𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒)) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = 0 ))
7473ralrimivvva 3203 . . . . 5 (𝜑 → ∀𝑏𝐵𝑐𝑁𝑑𝑁 ((𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒)) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = 0 ))
75 simp1 1135 . . . . . . . . . . . 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 1239 . . . . . . . . . . . 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 1240 . . . . . . . . . . . 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 1241 . . . . . . . . . . . 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 1242 . . . . . . . . . . . 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 1208 . . . . . . . . . . . 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 1209 . . . . . . . . . . . 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 1210 . . . . . . . . . . . 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 22636 . . . . . . . . . . . 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 1400 . . . . . . . . . . 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 1132 . . . . . . . . . . . . 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 22624 . . . . . . . . . . . 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 7447 . . . . . . . . . . 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 7449 . . . . . . . . . 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 779 . . . . . . . . . . . 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 1131 . . . . . . . . . 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 780 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑐𝐵)
93 fveq2 6907 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑐 → (𝐷𝑎) = (𝐷𝑐))
94 fveq2 6907 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑐 → (𝐸𝑎) = (𝐸𝑐))
9594oveq2d 7447 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑐 → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸𝑐)))
9693, 95oveq12d 7449 . . . . . . . . . . . . . . 15 (𝑎 = 𝑐 → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝑐)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑐))))
97 ovex 7464 . . . . . . . . . . . . . . 15 ((𝐷𝑐)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑐))) ∈ V
9896, 39, 97fvmpt 7016 . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑑𝐵)
101 fveq2 6907 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑑 → (𝐷𝑎) = (𝐷𝑑))
102 fveq2 6907 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑑 → (𝐸𝑎) = (𝐸𝑑))
103102oveq2d 7447 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑑 → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸𝑑)))
104101, 103oveq12d 7449 . . . . . . . . . . . . . . 15 (𝑎 = 𝑑 → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑))))
105 ovex 7464 . . . . . . . . . . . . . . 15 ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑))) ∈ V
106104, 39, 105fvmpt 7016 . . . . . . . . . . . . . 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 7449 . . . . . . . . . . . 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 20295 . . . . . . . . . . . . . . 15 (𝑅 ∈ Ring → 𝑅 ∈ Abel)
1109, 109syl 17 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ Abel)
111110adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ Abel)
11213adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝐷:𝐵𝐾)
113112, 92ffvelcdmd 7105 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷𝑐) ∈ 𝐾)
114112, 100ffvelcdmd 7105 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷𝑑) ∈ 𝐾)
1159adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ Ring)
11621adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷‘(1r𝐴)) ∈ 𝐾)
11726adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝐸:𝐵𝐾)
118117, 92ffvelcdmd 7105 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐸𝑐) ∈ 𝐾)
1193, 7ringcl 20268 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑐) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · (𝐸𝑐)) ∈ 𝐾)
120115, 116, 118, 119syl3anc 1370 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝐷‘(1r𝐴)) · (𝐸𝑐)) ∈ 𝐾)
121117, 100ffvelcdmd 7105 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐸𝑑) ∈ 𝐾)
1223, 7ringcl 20268 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑑) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · (𝐸𝑑)) ∈ 𝐾)
123115, 116, 121, 122syl3anc 1370 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝐷‘(1r𝐴)) · (𝐸𝑑)) ∈ 𝐾)
1243, 6, 30ablsub4 19843 . . . . . . . . . . . . 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 1378 . . . . . . . . . . . 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 20278 . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑))) = (((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑))))
128127eqcomd 2741 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑))) = ((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑))))
129128oveq2d 7447 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝐷𝑐) + (𝐷𝑑))(-g𝑅)(((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑)))) = (((𝐷𝑐) + (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑)))))
130108, 125, 1293eqtr2d 2781 . . . . . . . . . . 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 1131 . . . . . . . . . 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 2785 . . . . . . . . 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 1120 . . . . . . . 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 467 . . . . . . 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 3200 . . . . . 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 3200 . . . . 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 1135 . . . . . . . . . . . 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 1239 . . . . . . . . . . . 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 1240 . . . . . . . . . . . 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 1241 . . . . . . . . . . . 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 1242 . . . . . . . . . . . 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 1200 . . . . . . . . . . . 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 1201 . . . . . . . . . . . 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 22637 . . . . . . . . . . . 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 1392 . . . . . . . . . . 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 1132 . . . . . . . . . . . . 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 22625 . . . . . . . . . . . 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 7447 . . . . . . . . . . 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 7449 . . . . . . . . . 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 779 . . . . . . . . . . . 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 1131 . . . . . . . . . 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 781 . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 · ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑)) = (𝑐 · ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑)))))
1569adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ Ring)
157 simprlr 780 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝑐𝐾)
15813adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝐷:𝐵𝐾)
159158, 153ffvelcdmd 7105 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷𝑑) ∈ 𝐾)
16021adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷‘(1r𝐴)) ∈ 𝐾)
16126adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝐸:𝐵𝐾)
162161, 153ffvelcdmd 7105 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝐸𝑑) ∈ 𝐾)
163156, 160, 162, 122syl3anc 1370 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → ((𝐷‘(1r𝐴)) · (𝐸𝑑)) ∈ 𝐾)
1643, 7, 30, 156, 157, 159, 163ringsubdi 20321 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 · ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑)))) = ((𝑐 · (𝐷𝑑))(-g𝑅)(𝑐 · ((𝐷‘(1r𝐴)) · (𝐸𝑑)))))
165 eqid 2735 . . . . . . . . . . . . . . . . 17 (mulGrp‘𝑅) = (mulGrp‘𝑅)
166165crngmgp 20259 . . . . . . . . . . . . . . . 16 (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
16723, 166syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
168167adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (mulGrp‘𝑅) ∈ CMnd)
169165, 3mgpbas 20158 . . . . . . . . . . . . . . 15 𝐾 = (Base‘(mulGrp‘𝑅))
170165, 7mgpplusg 20156 . . . . . . . . . . . . . . 15 · = (+g‘(mulGrp‘𝑅))
171169, 170cmn12 19835 . . . . . . . . . . . . . 14 (((mulGrp‘𝑅) ∈ CMnd ∧ (𝑐𝐾 ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑑) ∈ 𝐾)) → (𝑐 · ((𝐷‘(1r𝐴)) · (𝐸𝑑))) = ((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑))))
172168, 157, 160, 162, 171syl13anc 1371 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 · ((𝐷‘(1r𝐴)) · (𝐸𝑑))) = ((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑))))
173172oveq2d 7447 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑐 · (𝐷𝑑))(-g𝑅)(𝑐 · ((𝐷‘(1r𝐴)) · (𝐸𝑑)))) = ((𝑐 · (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑)))))
174155, 164, 1733eqtrd 2779 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 · ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑)) = ((𝑐 · (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑)))))
1751743adant3 1131 . . . . . . . . . 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 2785 . . . . . . . . 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 1120 . . . . . . . 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 467 . . . . . . 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 3200 . . . . . 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 3200 . . . . 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 2736 . . . . . 6 (𝜑 → (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))) = (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))))
182 fveq2 6907 . . . . . . . 8 (𝑎 = (1r𝐴) → (𝐷𝑎) = (𝐷‘(1r𝐴)))
183 fveq2 6907 . . . . . . . . 9 (𝑎 = (1r𝐴) → (𝐸𝑎) = (𝐸‘(1r𝐴)))
184183oveq2d 7447 . . . . . . . 8 (𝑎 = (1r𝐴) → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴))))
185182, 184oveq12d 7449 . . . . . . 7 (𝑎 = (1r𝐴) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷‘(1r𝐴))(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴)))))
18624, 1, 18, 5mdet1 22623 . . . . . . . . . . . 12 ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐸‘(1r𝐴)) = 1 )
18723, 8, 186syl2anc 584 . . . . . . . . . . 11 (𝜑 → (𝐸‘(1r𝐴)) = 1 )
188187oveq2d 7447 . . . . . . . . . 10 (𝜑 → ((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴))) = ((𝐷‘(1r𝐴)) · 1 ))
1893, 7, 5ringridm 20284 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · 1 ) = (𝐷‘(1r𝐴)))
1909, 21, 189syl2anc 584 . . . . . . . . . 10 (𝜑 → ((𝐷‘(1r𝐴)) · 1 ) = (𝐷‘(1r𝐴)))
191188, 190eqtrd 2775 . . . . . . . . 9 (𝜑 → ((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴))) = (𝐷‘(1r𝐴)))
192191oveq2d 7447 . . . . . . . 8 (𝜑 → ((𝐷‘(1r𝐴))(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴)))) = ((𝐷‘(1r𝐴))(-g𝑅)(𝐷‘(1r𝐴))))
1933, 4, 30grpsubid 19055 . . . . . . . . 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 2775 . . . . . . 7 (𝜑 → ((𝐷‘(1r𝐴))(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴)))) = 0 )
196185, 195sylan9eqr 2797 . . . . . 6 ((𝜑𝑎 = (1r𝐴)) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = 0 )
1974fvexi 6921 . . . . . . 7 0 ∈ V
198197a1i 11 . . . . . 6 (𝜑0 ∈ V)
199181, 196, 20, 198fvmptd 7023 . . . . 5 (𝜑 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘(1r𝐴)) = 0 )
200 eqid 2735 . . . . 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 22642 . . . 4 (𝜑 → (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))) = (𝐵 × { 0 }))
202201fveq1d 6909 . . 3 (𝜑 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝐹) = ((𝐵 × { 0 })‘𝐹))
203 fveq2 6907 . . . . . 6 (𝑎 = 𝐹 → (𝐷𝑎) = (𝐷𝐹))
204 fveq2 6907 . . . . . . 7 (𝑎 = 𝐹 → (𝐸𝑎) = (𝐸𝐹))
205204oveq2d 7447 . . . . . 6 (𝑎 = 𝐹 → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸𝐹)))
206203, 205oveq12d 7449 . . . . 5 (𝑎 = 𝐹 → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))))
207206adantl 481 . . . 4 ((𝜑𝑎 = 𝐹) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))))
208 mdetuni.f . . . 4 (𝜑𝐹𝐵)
209 ovexd 7466 . . . 4 (𝜑 → ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))) ∈ V)
210181, 207, 208, 209fvmptd 7023 . . 3 (𝜑 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝐹) = ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))))
211197fvconst2 7224 . . . 4 (𝐹𝐵 → ((𝐵 × { 0 })‘𝐹) = 0 )
212208, 211syl 17 . . 3 (𝜑 → ((𝐵 × { 0 })‘𝐹) = 0 )
213202, 210, 2123eqtr3d 2783 . 2 (𝜑 → ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))) = 0 )
21413, 208ffvelcdmd 7105 . . 3 (𝜑 → (𝐷𝐹) ∈ 𝐾)
21526, 208ffvelcdmd 7105 . . . 4 (𝜑 → (𝐸𝐹) ∈ 𝐾)
2163, 7ringcl 20268 . . . 4 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝐹) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · (𝐸𝐹)) ∈ 𝐾)
2179, 21, 215, 216syl3anc 1370 . . 3 (𝜑 → ((𝐷‘(1r𝐴)) · (𝐸𝐹)) ∈ 𝐾)
2183, 4, 30grpsubeq0 19057 . . 3 ((𝑅 ∈ Grp ∧ (𝐷𝐹) ∈ 𝐾 ∧ ((𝐷‘(1r𝐴)) · (𝐸𝐹)) ∈ 𝐾) → (((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))) = 0 ↔ (𝐷𝐹) = ((𝐷‘(1r𝐴)) · (𝐸𝐹))))
21911, 214, 217, 218syl3anc 1370 . 2 (𝜑 → (((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))) = 0 ↔ (𝐷𝐹) = ((𝐷‘(1r𝐴)) · (𝐸𝐹))))
220213, 219mpbid 232 1 (𝜑 → (𝐷𝐹) = ((𝐷‘(1r𝐴)) · (𝐸𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wne 2938  wral 3059  Vcvv 3478  cdif 3960  ifcif 4531  {csn 4631  cmpt 5231   × cxp 5687  cres 5691  wf 6559  cfv 6563  (class class class)co 7431  f cof 7695  m cmap 8865  Fincfn 8984  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  0gc0g 17486  Grpcgrp 18964  -gcsg 18966  CMndccmn 19813  Abelcabl 19814  mulGrpcmgp 20152  1rcur 20199  Ringcrg 20251  CRingccrg 20252   Mat cmat 22427   maDet cmdat 22606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-addf 11232  ax-mulf 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1509  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-dec 12732  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-word 14550  df-lsw 14598  df-concat 14606  df-s1 14631  df-substr 14676  df-pfx 14706  df-splice 14785  df-reverse 14794  df-s2 14884  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-efmnd 18895  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-gim 19290  df-cntz 19348  df-oppg 19377  df-symg 19402  df-pmtr 19475  df-psgn 19524  df-evpm 19525  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-srg 20205  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-dvr 20418  df-rhm 20489  df-subrng 20563  df-subrg 20587  df-drng 20748  df-lmod 20877  df-lss 20948  df-sra 21190  df-rgmod 21191  df-cnfld 21383  df-zring 21476  df-zrh 21532  df-dsmm 21770  df-frlm 21785  df-mamu 22411  df-mat 22428  df-mdet 22607
This theorem is referenced by:  mdetuni  22644  mdetmul  22645
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