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Theorem mdetuni0 20475
Description: Lemma for mdetuni 20476. (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 18598 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
119, 10syl 17 . . . . . . . 8 (𝜑𝑅 ∈ Grp)
1211adantr 480 . . . . . . 7 ((𝜑𝑎𝐵) → 𝑅 ∈ Grp)
13 mdetuni.ff . . . . . . . 8 (𝜑𝐷:𝐵𝐾)
1413ffvelrnda 6399 . . . . . . 7 ((𝜑𝑎𝐵) → (𝐷𝑎) ∈ 𝐾)
159adantr 480 . . . . . . . 8 ((𝜑𝑎𝐵) → 𝑅 ∈ Ring)
168, 9jca 553 . . . . . . . . . . 11 (𝜑 → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
171matring 20297 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
18 eqid 2651 . . . . . . . . . . . 12 (1r𝐴) = (1r𝐴)
192, 18ringidcl 18614 . . . . . . . . . . 11 (𝐴 ∈ Ring → (1r𝐴) ∈ 𝐵)
2016, 17, 193syl 18 . . . . . . . . . 10 (𝜑 → (1r𝐴) ∈ 𝐵)
2113, 20ffvelrnd 6400 . . . . . . . . 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 20449 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝐸:𝐵𝐾)
2623, 25syl 17 . . . . . . . . 9 (𝜑𝐸:𝐵𝐾)
2726ffvelrnda 6399 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝐸𝑎) ∈ 𝐾)
283, 7ringcl 18607 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑎) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) ∈ 𝐾)
2915, 22, 27, 28syl3anc 1366 . . . . . . 7 ((𝜑𝑎𝐵) → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) ∈ 𝐾)
30 eqid 2651 . . . . . . . 8 (-g𝑅) = (-g𝑅)
313, 30grpsubcl 17542 . . . . . . 7 ((𝑅 ∈ Grp ∧ (𝐷𝑎) ∈ 𝐾 ∧ ((𝐷‘(1r𝐴)) · (𝐸𝑎)) ∈ 𝐾) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) ∈ 𝐾)
3212, 14, 29, 31syl3anc 1366 . . . . . 6 ((𝜑𝑎𝐵) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) ∈ 𝐾)
33 eqid 2651 . . . . . 6 (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))) = (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))
3432, 33fmptd 6425 . . . . 5 (𝜑 → (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))):𝐵𝐾)
35 simpr1 1087 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → 𝑏𝐵)
36 fveq2 6229 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝐷𝑎) = (𝐷𝑏))
37 fveq2 6229 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (𝐸𝑎) = (𝐸𝑏))
3837oveq2d 6706 . . . . . . . . . . . 12 (𝑎 = 𝑏 → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸𝑏)))
3936, 38oveq12d 6708 . . . . . . . . . . 11 (𝑎 = 𝑏 → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
40 ovex 6718 . . . . . . . . . . 11 ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))) ∈ V
4139, 33, 40fvmpt 6321 . . . . . . . . . 10 (𝑏𝐵 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
4235, 41syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
43423adant3 1101 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
44 simp1 1081 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝜑)
45 simp21 1114 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑏𝐵)
46 simp3r 1110 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))
47 oveq2 6698 . . . . . . . . . . . . 13 (𝑒 = 𝑤 → (𝑐𝑏𝑒) = (𝑐𝑏𝑤))
48 oveq2 6698 . . . . . . . . . . . . 13 (𝑒 = 𝑤 → (𝑑𝑏𝑒) = (𝑑𝑏𝑤))
4947, 48eqeq12d 2666 . . . . . . . . . . . 12 (𝑒 = 𝑤 → ((𝑐𝑏𝑒) = (𝑑𝑏𝑒) ↔ (𝑐𝑏𝑤) = (𝑑𝑏𝑤)))
5049cbvralv 3201 . . . . . . . . . . 11 (∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒) ↔ ∀𝑤𝑁 (𝑐𝑏𝑤) = (𝑑𝑏𝑤))
5146, 50sylib 208 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ∀𝑤𝑁 (𝑐𝑏𝑤) = (𝑑𝑏𝑤))
52 simp22 1115 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑐𝑁)
53 simp23 1116 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑑𝑁)
54 simp3l 1109 . . . . . . . . . 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 20466 . . . . . . . . . 10 (((𝜑𝑏𝐵 ∧ ∀𝑤𝑁 (𝑐𝑏𝑤) = (𝑑𝑏𝑤)) ∧ (𝑐𝑁𝑑𝑁𝑐𝑑)) → (𝐷𝑏) = 0 )
5944, 45, 51, 52, 53, 54, 58syl33anc 1381 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → (𝐷𝑏) = 0 )
60233ad2ant1 1102 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑅 ∈ CRing)
6124, 1, 2, 4, 60, 45, 52, 53, 54, 46mdetralt 20462 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → (𝐸𝑏) = 0 )
6261oveq2d 6706 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝐷‘(1r𝐴)) · (𝐸𝑏)) = ((𝐷‘(1r𝐴)) · 0 ))
6359, 62oveq12d 6708 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))) = ( 0 (-g𝑅)((𝐷‘(1r𝐴)) · 0 )))
643, 7, 4ringrz 18634 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · 0 ) = 0 )
659, 21, 64syl2anc 694 . . . . . . . . . . 11 (𝜑 → ((𝐷‘(1r𝐴)) · 0 ) = 0 )
6665oveq2d 6706 . . . . . . . . . 10 (𝜑 → ( 0 (-g𝑅)((𝐷‘(1r𝐴)) · 0 )) = ( 0 (-g𝑅) 0 ))
673, 4grpidcl 17497 . . . . . . . . . . . 12 (𝑅 ∈ Grp → 0𝐾)
6811, 67syl 17 . . . . . . . . . . 11 (𝜑0𝐾)
693, 4, 30grpsubid 17546 . . . . . . . . . . 11 ((𝑅 ∈ Grp ∧ 0𝐾) → ( 0 (-g𝑅) 0 ) = 0 )
7011, 68, 69syl2anc 694 . . . . . . . . . 10 (𝜑 → ( 0 (-g𝑅) 0 ) = 0 )
7166, 70eqtrd 2685 . . . . . . . . 9 (𝜑 → ( 0 (-g𝑅)((𝐷‘(1r𝐴)) · 0 )) = 0 )
72713ad2ant1 1102 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ( 0 (-g𝑅)((𝐷‘(1r𝐴)) · 0 )) = 0 )
7343, 63, 723eqtrd 2689 . . . . . . 7 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = 0 )
74733expia 1286 . . . . . 6 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → ((𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒)) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = 0 ))
7574ralrimivvva 3001 . . . . 5 (𝜑 → ∀𝑏𝐵𝑐𝑁𝑑𝑁 ((𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒)) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = 0 ))
76 simp1 1081 . . . . . . . . . . . 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)
77 simp2ll 1148 . . . . . . . . . . . 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)
78 simp2lr 1149 . . . . . . . . . . . 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)
79 simp2rl 1150 . . . . . . . . . . . 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)
80 simp2rr 1151 . . . . . . . . . . . 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)
81 simp31 1117 . . . . . . . . . . . 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))))
82 simp32 1118 . . . . . . . . . . . 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)))
83 simp33 1119 . . . . . . . . . . . 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)))
841, 2, 3, 4, 5, 6, 7, 8, 9, 13, 55, 56, 57mdetunilem3 20468 . . . . . . . . . . . 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)))
8576, 77, 78, 79, 80, 81, 82, 83, 84syl332anc 1397 . . . . . . . . . . 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)))
86233ad2ant1 1102 . . . . . . . . . . . . 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)
8724, 1, 2, 6, 86, 77, 78, 79, 80, 81, 82, 83mdetrlin 20456 . . . . . . . . . . . 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)))
8887oveq2d 6706 . . . . . . . . . . 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))))
8985, 88oveq12d 6708 . . . . . . . . . 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)))))
90 simprll 819 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑏𝐵)
9190, 41syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
92913adant3 1101 . . . . . . . . . 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))))
93 simprlr 820 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑐𝐵)
94 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑐 → (𝐷𝑎) = (𝐷𝑐))
95 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑐 → (𝐸𝑎) = (𝐸𝑐))
9695oveq2d 6706 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑐 → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸𝑐)))
9794, 96oveq12d 6708 . . . . . . . . . . . . . . 15 (𝑎 = 𝑐 → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝑐)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑐))))
98 ovex 6718 . . . . . . . . . . . . . . 15 ((𝐷𝑐)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑐))) ∈ V
9997, 33, 98fvmpt 6321 . . . . . . . . . . . . . 14 (𝑐𝐵 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑐) = ((𝐷𝑐)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑐))))
10093, 99syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑐) = ((𝐷𝑐)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑐))))
101 simprrl 821 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑑𝐵)
102 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑑 → (𝐷𝑎) = (𝐷𝑑))
103 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑑 → (𝐸𝑎) = (𝐸𝑑))
104103oveq2d 6706 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑑 → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸𝑑)))
105102, 104oveq12d 6708 . . . . . . . . . . . . . . 15 (𝑎 = 𝑑 → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑))))
106 ovex 6718 . . . . . . . . . . . . . . 15 ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑))) ∈ V
107105, 33, 106fvmpt 6321 . . . . . . . . . . . . . 14 (𝑑𝐵 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑) = ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑))))
108101, 107syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑) = ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑))))
109100, 108oveq12d 6708 . . . . . . . . . . . 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)))))
110 ringabl 18626 . . . . . . . . . . . . . . 15 (𝑅 ∈ Ring → 𝑅 ∈ Abel)
1119, 110syl 17 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ Abel)
112111adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ Abel)
11313adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝐷:𝐵𝐾)
114113, 93ffvelrnd 6400 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷𝑐) ∈ 𝐾)
115113, 101ffvelrnd 6400 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷𝑑) ∈ 𝐾)
1169adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ Ring)
11721adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷‘(1r𝐴)) ∈ 𝐾)
11826adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝐸:𝐵𝐾)
119118, 93ffvelrnd 6400 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐸𝑐) ∈ 𝐾)
1203, 7ringcl 18607 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑐) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · (𝐸𝑐)) ∈ 𝐾)
121116, 117, 119, 120syl3anc 1366 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝐷‘(1r𝐴)) · (𝐸𝑐)) ∈ 𝐾)
122118, 101ffvelrnd 6400 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐸𝑑) ∈ 𝐾)
1233, 7ringcl 18607 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑑) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · (𝐸𝑑)) ∈ 𝐾)
124116, 117, 122, 123syl3anc 1366 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝐷‘(1r𝐴)) · (𝐸𝑑)) ∈ 𝐾)
1253, 6, 30ablsub4 18264 . . . . . . . . . . . . 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)))))
126112, 114, 115, 121, 124, 125syl122anc 1375 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝐷𝑐) + (𝐷𝑑))(-g𝑅)(((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑)))) = (((𝐷𝑐)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑐))) + ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑)))))
1273, 6, 7ringdi 18612 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ ((𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑐) ∈ 𝐾 ∧ (𝐸𝑑) ∈ 𝐾)) → ((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑))) = (((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑))))
128116, 117, 119, 122, 127syl13anc 1368 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑))) = (((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑))))
129128eqcomd 2657 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑))) = ((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑))))
130129oveq2d 6706 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝐷𝑐) + (𝐷𝑑))(-g𝑅)(((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑)))) = (((𝐷𝑐) + (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑)))))
131109, 126, 1303eqtr2d 2691 . . . . . . . . . . 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)))))
1321313adant3 1101 . . . . . . . . . 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)))))
13389, 92, 1323eqtr4d 2695 . . . . . . . . 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)))
1341333expia 1286 . . . . . . . 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))))
135134anassrs 681 . . . . . . 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))))
136135ralrimivva 3000 . . . . . 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))))
137136ralrimivva 3000 . . . . 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))))
138 simp1 1081 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝜑)
139 simp2ll 1148 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑏𝐵)
140 simp2lr 1149 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑐𝐾)
141 simp2rl 1150 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑑𝐵)
142 simp2rr 1151 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑒𝑁)
143 simp3l 1109 . . . . . . . . . . . 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))))
144 simp3r 1110 . . . . . . . . . . . 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)))
1451, 2, 3, 4, 5, 6, 7, 8, 9, 13, 55, 56, 57mdetunilem4 20469 . . . . . . . . . . . 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)))
146138, 139, 140, 141, 142, 143, 144, 145syl133anc 1389 . . . . . . . . . . 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)))
147233ad2ant1 1102 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘𝑓 · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑅 ∈ CRing)
14824, 1, 2, 3, 7, 147, 139, 140, 141, 142, 143, 144mdetrsca 20457 . . . . . . . . . . . 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)))
149148oveq2d 6706 . . . . . . . . . . 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))))
150146, 149oveq12d 6708 . . . . . . . . . 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)))))
151 simprll 819 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝑏𝐵)
152151, 41syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
1531523adant3 1101 . . . . . . . . . 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))))
154 simprrl 821 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝑑𝐵)
155154, 107syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑) = ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑))))
156155oveq2d 6706 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 · ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑)) = (𝑐 · ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑)))))
1579adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ Ring)
158 simprlr 820 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝑐𝐾)
15913adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝐷:𝐵𝐾)
160159, 154ffvelrnd 6400 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷𝑑) ∈ 𝐾)
16121adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷‘(1r𝐴)) ∈ 𝐾)
16226adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝐸:𝐵𝐾)
163162, 154ffvelrnd 6400 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝐸𝑑) ∈ 𝐾)
164157, 161, 163, 123syl3anc 1366 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → ((𝐷‘(1r𝐴)) · (𝐸𝑑)) ∈ 𝐾)
1653, 7, 30, 157, 158, 160, 164ringsubdi 18645 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 · ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑)))) = ((𝑐 · (𝐷𝑑))(-g𝑅)(𝑐 · ((𝐷‘(1r𝐴)) · (𝐸𝑑)))))
166 eqid 2651 . . . . . . . . . . . . . . . . 17 (mulGrp‘𝑅) = (mulGrp‘𝑅)
167166crngmgp 18601 . . . . . . . . . . . . . . . 16 (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
16823, 167syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
169168adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (mulGrp‘𝑅) ∈ CMnd)
170166, 3mgpbas 18541 . . . . . . . . . . . . . . 15 𝐾 = (Base‘(mulGrp‘𝑅))
171166, 7mgpplusg 18539 . . . . . . . . . . . . . . 15 · = (+g‘(mulGrp‘𝑅))
172170, 171cmn12 18259 . . . . . . . . . . . . . 14 (((mulGrp‘𝑅) ∈ CMnd ∧ (𝑐𝐾 ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑑) ∈ 𝐾)) → (𝑐 · ((𝐷‘(1r𝐴)) · (𝐸𝑑))) = ((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑))))
173169, 158, 161, 163, 172syl13anc 1368 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 · ((𝐷‘(1r𝐴)) · (𝐸𝑑))) = ((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑))))
174173oveq2d 6706 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑐 · (𝐷𝑑))(-g𝑅)(𝑐 · ((𝐷‘(1r𝐴)) · (𝐸𝑑)))) = ((𝑐 · (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑)))))
175156, 165, 1743eqtrd 2689 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 · ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑)) = ((𝑐 · (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑)))))
1761753adant3 1101 . . . . . . . . . 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)))))
177150, 153, 1763eqtr4d 2695 . . . . . . . . 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)))
1781773expia 1286 . . . . . . . 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))))
179178anassrs 681 . . . . . . 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))))
180179ralrimivva 3000 . . . . . 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))))
181180ralrimivva 3000 . . . . 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))))
182 eqidd 2652 . . . . . 6 (𝜑 → (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))) = (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))))
183 fveq2 6229 . . . . . . . 8 (𝑎 = (1r𝐴) → (𝐷𝑎) = (𝐷‘(1r𝐴)))
184 fveq2 6229 . . . . . . . . 9 (𝑎 = (1r𝐴) → (𝐸𝑎) = (𝐸‘(1r𝐴)))
185184oveq2d 6706 . . . . . . . 8 (𝑎 = (1r𝐴) → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴))))
186183, 185oveq12d 6708 . . . . . . 7 (𝑎 = (1r𝐴) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷‘(1r𝐴))(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴)))))
18724, 1, 18, 5mdet1 20455 . . . . . . . . . . . 12 ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐸‘(1r𝐴)) = 1 )
18823, 8, 187syl2anc 694 . . . . . . . . . . 11 (𝜑 → (𝐸‘(1r𝐴)) = 1 )
189188oveq2d 6706 . . . . . . . . . 10 (𝜑 → ((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴))) = ((𝐷‘(1r𝐴)) · 1 ))
1903, 7, 5ringridm 18618 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · 1 ) = (𝐷‘(1r𝐴)))
1919, 21, 190syl2anc 694 . . . . . . . . . 10 (𝜑 → ((𝐷‘(1r𝐴)) · 1 ) = (𝐷‘(1r𝐴)))
192189, 191eqtrd 2685 . . . . . . . . 9 (𝜑 → ((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴))) = (𝐷‘(1r𝐴)))
193192oveq2d 6706 . . . . . . . 8 (𝜑 → ((𝐷‘(1r𝐴))(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴)))) = ((𝐷‘(1r𝐴))(-g𝑅)(𝐷‘(1r𝐴))))
1943, 4, 30grpsubid 17546 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ (𝐷‘(1r𝐴)) ∈ 𝐾) → ((𝐷‘(1r𝐴))(-g𝑅)(𝐷‘(1r𝐴))) = 0 )
19511, 21, 194syl2anc 694 . . . . . . . 8 (𝜑 → ((𝐷‘(1r𝐴))(-g𝑅)(𝐷‘(1r𝐴))) = 0 )
196193, 195eqtrd 2685 . . . . . . 7 (𝜑 → ((𝐷‘(1r𝐴))(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴)))) = 0 )
197186, 196sylan9eqr 2707 . . . . . 6 ((𝜑𝑎 = (1r𝐴)) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = 0 )
198 fvex 6239 . . . . . . . 8 (0g𝑅) ∈ V
1994, 198eqeltri 2726 . . . . . . 7 0 ∈ V
200199a1i 11 . . . . . 6 (𝜑0 ∈ V)
201182, 197, 20, 200fvmptd 6327 . . . . 5 (𝜑 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘(1r𝐴)) = 0 )
202 eqid 2651 . . . . 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 )}
2031, 2, 3, 4, 5, 6, 7, 8, 9, 34, 75, 137, 181, 201, 202mdetunilem9 20474 . . . 4 (𝜑 → (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))) = (𝐵 × { 0 }))
204203fveq1d 6231 . . 3 (𝜑 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝐹) = ((𝐵 × { 0 })‘𝐹))
205 fveq2 6229 . . . . . 6 (𝑎 = 𝐹 → (𝐷𝑎) = (𝐷𝐹))
206 fveq2 6229 . . . . . . 7 (𝑎 = 𝐹 → (𝐸𝑎) = (𝐸𝐹))
207206oveq2d 6706 . . . . . 6 (𝑎 = 𝐹 → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸𝐹)))
208205, 207oveq12d 6708 . . . . 5 (𝑎 = 𝐹 → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))))
209208adantl 481 . . . 4 ((𝜑𝑎 = 𝐹) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))))
210 mdetuni.f . . . 4 (𝜑𝐹𝐵)
211 ovexd 6720 . . . 4 (𝜑 → ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))) ∈ V)
212182, 209, 210, 211fvmptd 6327 . . 3 (𝜑 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝐹) = ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))))
213199fvconst2 6510 . . . 4 (𝐹𝐵 → ((𝐵 × { 0 })‘𝐹) = 0 )
214210, 213syl 17 . . 3 (𝜑 → ((𝐵 × { 0 })‘𝐹) = 0 )
215204, 212, 2143eqtr3d 2693 . 2 (𝜑 → ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))) = 0 )
21613, 210ffvelrnd 6400 . . 3 (𝜑 → (𝐷𝐹) ∈ 𝐾)
21726, 210ffvelrnd 6400 . . . 4 (𝜑 → (𝐸𝐹) ∈ 𝐾)
2183, 7ringcl 18607 . . . 4 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝐹) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · (𝐸𝐹)) ∈ 𝐾)
2199, 21, 217, 218syl3anc 1366 . . 3 (𝜑 → ((𝐷‘(1r𝐴)) · (𝐸𝐹)) ∈ 𝐾)
2203, 4, 30grpsubeq0 17548 . . 3 ((𝑅 ∈ Grp ∧ (𝐷𝐹) ∈ 𝐾 ∧ ((𝐷‘(1r𝐴)) · (𝐸𝐹)) ∈ 𝐾) → (((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))) = 0 ↔ (𝐷𝐹) = ((𝐷‘(1r𝐴)) · (𝐸𝐹))))
22111, 216, 219, 220syl3anc 1366 . 2 (𝜑 → (((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))) = 0 ↔ (𝐷𝐹) = ((𝐷‘(1r𝐴)) · (𝐸𝐹))))
222215, 221mpbid 222 1 (𝜑 → (𝐷𝐹) = ((𝐷‘(1r𝐴)) · (𝐸𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  Vcvv 3231  cdif 3604  ifcif 4119  {csn 4210  cmpt 4762   × cxp 5141  cres 5145  wf 5922  cfv 5926  (class class class)co 6690  𝑓 cof 6937  𝑚 cmap 7899  Fincfn 7997  Basecbs 15904  +gcplusg 15988  .rcmulr 15989  0gc0g 16147  Grpcgrp 17469  -gcsg 17471  CMndccmn 18239  Abelcabl 18240  mulGrpcmgp 18535  1rcur 18547  Ringcrg 18593  CRingccrg 18594   Mat cmat 20261   maDet cmdat 20438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-xor 1505  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-ot 4219  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-xnn0 11402  df-z 11416  df-dec 11532  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-word 13331  df-lsw 13332  df-concat 13333  df-s1 13334  df-substr 13335  df-splice 13336  df-reverse 13337  df-s2 13639  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-0g 16149  df-gsum 16150  df-prds 16155  df-pws 16157  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-ghm 17705  df-gim 17748  df-cntz 17796  df-oppg 17822  df-symg 17844  df-pmtr 17908  df-psgn 17957  df-evpm 17958  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-srg 18552  df-ring 18595  df-cring 18596  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-dvr 18729  df-rnghom 18763  df-drng 18797  df-subrg 18826  df-lmod 18913  df-lss 18981  df-sra 19220  df-rgmod 19221  df-cnfld 19795  df-zring 19867  df-zrh 19900  df-dsmm 20124  df-frlm 20139  df-mamu 20238  df-mat 20262  df-mdet 20439
This theorem is referenced by:  mdetuni  20476  mdetmul  20477
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