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Theorem mdetuni0 20718
Description: Lemma for mdetuni 20719. (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 18833 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
119, 10syl 17 . . . . . . . 8 (𝜑𝑅 ∈ Grp)
1211adantr 472 . . . . . . 7 ((𝜑𝑎𝐵) → 𝑅 ∈ Grp)
13 mdetuni.ff . . . . . . . 8 (𝜑𝐷:𝐵𝐾)
1413ffvelrnda 6553 . . . . . . 7 ((𝜑𝑎𝐵) → (𝐷𝑎) ∈ 𝐾)
159adantr 472 . . . . . . . 8 ((𝜑𝑎𝐵) → 𝑅 ∈ Ring)
168, 9jca 507 . . . . . . . . . . 11 (𝜑 → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
171matring 20539 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
18 eqid 2765 . . . . . . . . . . . 12 (1r𝐴) = (1r𝐴)
192, 18ringidcl 18849 . . . . . . . . . . 11 (𝐴 ∈ Ring → (1r𝐴) ∈ 𝐵)
2016, 17, 193syl 18 . . . . . . . . . 10 (𝜑 → (1r𝐴) ∈ 𝐵)
2113, 20ffvelrnd 6554 . . . . . . . . 9 (𝜑 → (𝐷‘(1r𝐴)) ∈ 𝐾)
2221adantr 472 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝐷‘(1r𝐴)) ∈ 𝐾)
23 mdetuni.cr . . . . . . . . . 10 (𝜑𝑅 ∈ CRing)
24 mdetuni.e . . . . . . . . . . 11 𝐸 = (𝑁 maDet 𝑅)
2524, 1, 2, 3mdetf 20692 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝐸:𝐵𝐾)
2623, 25syl 17 . . . . . . . . 9 (𝜑𝐸:𝐵𝐾)
2726ffvelrnda 6553 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝐸𝑎) ∈ 𝐾)
283, 7ringcl 18842 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑎) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) ∈ 𝐾)
2915, 22, 27, 28syl3anc 1490 . . . . . . 7 ((𝜑𝑎𝐵) → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) ∈ 𝐾)
30 eqid 2765 . . . . . . . 8 (-g𝑅) = (-g𝑅)
313, 30grpsubcl 17776 . . . . . . 7 ((𝑅 ∈ Grp ∧ (𝐷𝑎) ∈ 𝐾 ∧ ((𝐷‘(1r𝐴)) · (𝐸𝑎)) ∈ 𝐾) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) ∈ 𝐾)
3212, 14, 29, 31syl3anc 1490 . . . . . 6 ((𝜑𝑎𝐵) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) ∈ 𝐾)
3332fmpttd 6579 . . . . 5 (𝜑 → (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))):𝐵𝐾)
34 simpr1 1248 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → 𝑏𝐵)
35 fveq2 6379 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝐷𝑎) = (𝐷𝑏))
36 fveq2 6379 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (𝐸𝑎) = (𝐸𝑏))
3736oveq2d 6862 . . . . . . . . . . . 12 (𝑎 = 𝑏 → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸𝑏)))
3835, 37oveq12d 6864 . . . . . . . . . . 11 (𝑎 = 𝑏 → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
39 eqid 2765 . . . . . . . . . . 11 (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))) = (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))
40 ovex 6878 . . . . . . . . . . 11 ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))) ∈ V
4138, 39, 40fvmpt 6475 . . . . . . . . . 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 1162 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))))
44 simp1 1166 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝜑)
45 simp21 1263 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑏𝐵)
46 simp3r 1259 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))
47 oveq2 6854 . . . . . . . . . . . . 13 (𝑒 = 𝑤 → (𝑐𝑏𝑒) = (𝑐𝑏𝑤))
48 oveq2 6854 . . . . . . . . . . . . 13 (𝑒 = 𝑤 → (𝑑𝑏𝑒) = (𝑑𝑏𝑤))
4947, 48eqeq12d 2780 . . . . . . . . . . . 12 (𝑒 = 𝑤 → ((𝑐𝑏𝑒) = (𝑑𝑏𝑒) ↔ (𝑐𝑏𝑤) = (𝑑𝑏𝑤)))
5049cbvralv 3319 . . . . . . . . . . 11 (∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒) ↔ ∀𝑤𝑁 (𝑐𝑏𝑤) = (𝑑𝑏𝑤))
5146, 50sylib 209 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ∀𝑤𝑁 (𝑐𝑏𝑤) = (𝑑𝑏𝑤))
52 simp22 1264 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑐𝑁)
53 simp23 1265 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑑𝑁)
54 simp3l 1258 . . . . . . . . . 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 20709 . . . . . . . . . 10 (((𝜑𝑏𝐵 ∧ ∀𝑤𝑁 (𝑐𝑏𝑤) = (𝑑𝑏𝑤)) ∧ (𝑐𝑁𝑑𝑁𝑐𝑑)) → (𝐷𝑏) = 0 )
5944, 45, 51, 52, 53, 54, 58syl33anc 1504 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → (𝐷𝑏) = 0 )
60233ad2ant1 1163 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑅 ∈ CRing)
6124, 1, 2, 4, 60, 45, 52, 53, 54, 46mdetralt 20705 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → (𝐸𝑏) = 0 )
6261oveq2d 6862 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝐷‘(1r𝐴)) · (𝐸𝑏)) = ((𝐷‘(1r𝐴)) · 0 ))
6359, 62oveq12d 6864 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝐷𝑏)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑏))) = ( 0 (-g𝑅)((𝐷‘(1r𝐴)) · 0 )))
643, 7, 4ringrz 18869 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · 0 ) = 0 )
659, 21, 64syl2anc 579 . . . . . . . . . . 11 (𝜑 → ((𝐷‘(1r𝐴)) · 0 ) = 0 )
6665oveq2d 6862 . . . . . . . . . 10 (𝜑 → ( 0 (-g𝑅)((𝐷‘(1r𝐴)) · 0 )) = ( 0 (-g𝑅) 0 ))
673, 4grpidcl 17731 . . . . . . . . . . . 12 (𝑅 ∈ Grp → 0𝐾)
6811, 67syl 17 . . . . . . . . . . 11 (𝜑0𝐾)
693, 4, 30grpsubid 17780 . . . . . . . . . . 11 ((𝑅 ∈ Grp ∧ 0𝐾) → ( 0 (-g𝑅) 0 ) = 0 )
7011, 68, 69syl2anc 579 . . . . . . . . . 10 (𝜑 → ( 0 (-g𝑅) 0 ) = 0 )
7166, 70eqtrd 2799 . . . . . . . . 9 (𝜑 → ( 0 (-g𝑅)((𝐷‘(1r𝐴)) · 0 )) = 0 )
72713ad2ant1 1163 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ( 0 (-g𝑅)((𝐷‘(1r𝐴)) · 0 )) = 0 )
7343, 63, 723eqtrd 2803 . . . . . . 7 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = 0 )
74733expia 1150 . . . . . 6 ((𝜑 ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → ((𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒)) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = 0 ))
7574ralrimivvva 3119 . . . . 5 (𝜑 → ∀𝑏𝐵𝑐𝑁𝑑𝑁 ((𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒)) → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑏) = 0 ))
76 simp1 1166 . . . . . . . . . . . 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 1321 . . . . . . . . . . . 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 1322 . . . . . . . . . . . 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 1323 . . . . . . . . . . . 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 1324 . . . . . . . . . . . 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 1266 . . . . . . . . . . . 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 1267 . . . . . . . . . . . 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 1268 . . . . . . . . . . . 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 20711 . . . . . . . . . . . 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 1520 . . . . . . . . . . 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 1163 . . . . . . . . . . . . 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 20699 . . . . . . . . . . . 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 6862 . . . . . . . . . . 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 6864 . . . . . . . . . 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 797 . . . . . . . . . . . 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 1162 . . . . . . . . . 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 798 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑐𝐵)
94 fveq2 6379 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑐 → (𝐷𝑎) = (𝐷𝑐))
95 fveq2 6379 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑐 → (𝐸𝑎) = (𝐸𝑐))
9695oveq2d 6862 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑐 → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸𝑐)))
9794, 96oveq12d 6864 . . . . . . . . . . . . . . 15 (𝑎 = 𝑐 → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝑐)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑐))))
98 ovex 6878 . . . . . . . . . . . . . . 15 ((𝐷𝑐)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑐))) ∈ V
9997, 39, 98fvmpt 6475 . . . . . . . . . . . . . 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 799 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑑𝐵)
102 fveq2 6379 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑑 → (𝐷𝑎) = (𝐷𝑑))
103 fveq2 6379 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑑 → (𝐸𝑎) = (𝐸𝑑))
104103oveq2d 6862 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑑 → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸𝑑)))
105102, 104oveq12d 6864 . . . . . . . . . . . . . . 15 (𝑎 = 𝑑 → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑))))
106 ovex 6878 . . . . . . . . . . . . . . 15 ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑))) ∈ V
107105, 39, 106fvmpt 6475 . . . . . . . . . . . . . 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 6864 . . . . . . . . . . . 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 18861 . . . . . . . . . . . . . . 15 (𝑅 ∈ Ring → 𝑅 ∈ Abel)
1119, 110syl 17 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ Abel)
112111adantr 472 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ Abel)
11313adantr 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝐷:𝐵𝐾)
114113, 93ffvelrnd 6554 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷𝑐) ∈ 𝐾)
115113, 101ffvelrnd 6554 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷𝑑) ∈ 𝐾)
1169adantr 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ Ring)
11721adantr 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷‘(1r𝐴)) ∈ 𝐾)
11826adantr 472 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝐸:𝐵𝐾)
119118, 93ffvelrnd 6554 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐸𝑐) ∈ 𝐾)
1203, 7ringcl 18842 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑐) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · (𝐸𝑐)) ∈ 𝐾)
121116, 117, 119, 120syl3anc 1490 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝐷‘(1r𝐴)) · (𝐸𝑐)) ∈ 𝐾)
122118, 101ffvelrnd 6554 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝐸𝑑) ∈ 𝐾)
1233, 7ringcl 18842 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑑) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · (𝐸𝑑)) ∈ 𝐾)
124116, 117, 122, 123syl3anc 1490 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝐷‘(1r𝐴)) · (𝐸𝑑)) ∈ 𝐾)
1253, 6, 30ablsub4 18498 . . . . . . . . . . . . 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 1498 . . . . . . . . . . . 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 18847 . . . . . . . . . . . . . . 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 1491 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑))) = (((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑))))
129128eqcomd 2771 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑))) = ((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑))))
130129oveq2d 6862 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝐷𝑐) + (𝐷𝑑))(-g𝑅)(((𝐷‘(1r𝐴)) · (𝐸𝑐)) + ((𝐷‘(1r𝐴)) · (𝐸𝑑)))) = (((𝐷𝑐) + (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · ((𝐸𝑐) + (𝐸𝑑)))))
131109, 126, 1303eqtr2d 2805 . . . . . . . . . . 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 1162 . . . . . . . . . 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 2809 . . . . . . . . 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 1150 . . . . . . . 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 459 . . . . . . 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 3118 . . . . . 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 3118 . . . . 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 1166 . . . . . . . . . . . 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 1321 . . . . . . . . . . . 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 1322 . . . . . . . . . . . 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 1323 . . . . . . . . . . . 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 1324 . . . . . . . . . . . 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 1258 . . . . . . . . . . . 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 1259 . . . . . . . . . . . 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 20712 . . . . . . . . . . . 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 1512 . . . . . . . . . . 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 1163 . . . . . . . . . . . . 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 20700 . . . . . . . . . . . 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 6862 . . . . . . . . . . 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 6864 . . . . . . . . . 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 797 . . . . . . . . . . . 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 1162 . . . . . . . . . 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 799 . . . . . . . . . . . . . 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 6862 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 · ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑)) = (𝑐 · ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑)))))
1579adantr 472 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ Ring)
158 simprlr 798 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝑐𝐾)
15913adantr 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝐷:𝐵𝐾)
160159, 154ffvelrnd 6554 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷𝑑) ∈ 𝐾)
16121adantr 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝐷‘(1r𝐴)) ∈ 𝐾)
16226adantr 472 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → 𝐸:𝐵𝐾)
163162, 154ffvelrnd 6554 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝐸𝑑) ∈ 𝐾)
164157, 161, 163, 123syl3anc 1490 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → ((𝐷‘(1r𝐴)) · (𝐸𝑑)) ∈ 𝐾)
1653, 7, 30, 157, 158, 160, 164ringsubdi 18880 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 · ((𝐷𝑑)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑑)))) = ((𝑐 · (𝐷𝑑))(-g𝑅)(𝑐 · ((𝐷‘(1r𝐴)) · (𝐸𝑑)))))
166 eqid 2765 . . . . . . . . . . . . . . . . 17 (mulGrp‘𝑅) = (mulGrp‘𝑅)
167166crngmgp 18836 . . . . . . . . . . . . . . . 16 (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
16823, 167syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
169168adantr 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (mulGrp‘𝑅) ∈ CMnd)
170166, 3mgpbas 18776 . . . . . . . . . . . . . . 15 𝐾 = (Base‘(mulGrp‘𝑅))
171166, 7mgpplusg 18774 . . . . . . . . . . . . . . 15 · = (+g‘(mulGrp‘𝑅))
172170, 171cmn12 18493 . . . . . . . . . . . . . 14 (((mulGrp‘𝑅) ∈ CMnd ∧ (𝑐𝐾 ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝑑) ∈ 𝐾)) → (𝑐 · ((𝐷‘(1r𝐴)) · (𝐸𝑑))) = ((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑))))
173169, 158, 161, 163, 172syl13anc 1491 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 · ((𝐷‘(1r𝐴)) · (𝐸𝑑))) = ((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑))))
174173oveq2d 6862 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑐 · (𝐷𝑑))(-g𝑅)(𝑐 · ((𝐷‘(1r𝐴)) · (𝐸𝑑)))) = ((𝑐 · (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑)))))
175156, 165, 1743eqtrd 2803 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑏𝐵𝑐𝐾) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 · ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝑑)) = ((𝑐 · (𝐷𝑑))(-g𝑅)((𝐷‘(1r𝐴)) · (𝑐 · (𝐸𝑑)))))
1761753adant3 1162 . . . . . . . . . 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 2809 . . . . . . . . 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 1150 . . . . . . . 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 459 . . . . . . 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 3118 . . . . . 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 3118 . . . . 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 2766 . . . . . 6 (𝜑 → (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))) = (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))))
183 fveq2 6379 . . . . . . . 8 (𝑎 = (1r𝐴) → (𝐷𝑎) = (𝐷‘(1r𝐴)))
184 fveq2 6379 . . . . . . . . 9 (𝑎 = (1r𝐴) → (𝐸𝑎) = (𝐸‘(1r𝐴)))
185184oveq2d 6862 . . . . . . . 8 (𝑎 = (1r𝐴) → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴))))
186183, 185oveq12d 6864 . . . . . . 7 (𝑎 = (1r𝐴) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷‘(1r𝐴))(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴)))))
18724, 1, 18, 5mdet1 20698 . . . . . . . . . . . 12 ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐸‘(1r𝐴)) = 1 )
18823, 8, 187syl2anc 579 . . . . . . . . . . 11 (𝜑 → (𝐸‘(1r𝐴)) = 1 )
189188oveq2d 6862 . . . . . . . . . 10 (𝜑 → ((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴))) = ((𝐷‘(1r𝐴)) · 1 ))
1903, 7, 5ringridm 18853 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · 1 ) = (𝐷‘(1r𝐴)))
1919, 21, 190syl2anc 579 . . . . . . . . . 10 (𝜑 → ((𝐷‘(1r𝐴)) · 1 ) = (𝐷‘(1r𝐴)))
192189, 191eqtrd 2799 . . . . . . . . 9 (𝜑 → ((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴))) = (𝐷‘(1r𝐴)))
193192oveq2d 6862 . . . . . . . 8 (𝜑 → ((𝐷‘(1r𝐴))(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴)))) = ((𝐷‘(1r𝐴))(-g𝑅)(𝐷‘(1r𝐴))))
1943, 4, 30grpsubid 17780 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ (𝐷‘(1r𝐴)) ∈ 𝐾) → ((𝐷‘(1r𝐴))(-g𝑅)(𝐷‘(1r𝐴))) = 0 )
19511, 21, 194syl2anc 579 . . . . . . . 8 (𝜑 → ((𝐷‘(1r𝐴))(-g𝑅)(𝐷‘(1r𝐴))) = 0 )
196193, 195eqtrd 2799 . . . . . . 7 (𝜑 → ((𝐷‘(1r𝐴))(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸‘(1r𝐴)))) = 0 )
197186, 196sylan9eqr 2821 . . . . . 6 ((𝜑𝑎 = (1r𝐴)) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = 0 )
1984fvexi 6393 . . . . . . 7 0 ∈ V
199198a1i 11 . . . . . 6 (𝜑0 ∈ V)
200182, 197, 20, 199fvmptd 6481 . . . . 5 (𝜑 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘(1r𝐴)) = 0 )
201 eqid 2765 . . . . 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 )}
2021, 2, 3, 4, 5, 6, 7, 8, 9, 33, 75, 137, 181, 200, 201mdetunilem9 20717 . . . 4 (𝜑 → (𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎)))) = (𝐵 × { 0 }))
203202fveq1d 6381 . . 3 (𝜑 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝐹) = ((𝐵 × { 0 })‘𝐹))
204 fveq2 6379 . . . . . 6 (𝑎 = 𝐹 → (𝐷𝑎) = (𝐷𝐹))
205 fveq2 6379 . . . . . . 7 (𝑎 = 𝐹 → (𝐸𝑎) = (𝐸𝐹))
206205oveq2d 6862 . . . . . 6 (𝑎 = 𝐹 → ((𝐷‘(1r𝐴)) · (𝐸𝑎)) = ((𝐷‘(1r𝐴)) · (𝐸𝐹)))
207204, 206oveq12d 6864 . . . . 5 (𝑎 = 𝐹 → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))))
208207adantl 473 . . . 4 ((𝜑𝑎 = 𝐹) → ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))) = ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))))
209 mdetuni.f . . . 4 (𝜑𝐹𝐵)
210 ovexd 6880 . . . 4 (𝜑 → ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))) ∈ V)
211182, 208, 209, 210fvmptd 6481 . . 3 (𝜑 → ((𝑎𝐵 ↦ ((𝐷𝑎)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝑎))))‘𝐹) = ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))))
212198fvconst2 6666 . . . 4 (𝐹𝐵 → ((𝐵 × { 0 })‘𝐹) = 0 )
213209, 212syl 17 . . 3 (𝜑 → ((𝐵 × { 0 })‘𝐹) = 0 )
214203, 211, 2133eqtr3d 2807 . 2 (𝜑 → ((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))) = 0 )
21513, 209ffvelrnd 6554 . . 3 (𝜑 → (𝐷𝐹) ∈ 𝐾)
21626, 209ffvelrnd 6554 . . . 4 (𝜑 → (𝐸𝐹) ∈ 𝐾)
2173, 7ringcl 18842 . . . 4 ((𝑅 ∈ Ring ∧ (𝐷‘(1r𝐴)) ∈ 𝐾 ∧ (𝐸𝐹) ∈ 𝐾) → ((𝐷‘(1r𝐴)) · (𝐸𝐹)) ∈ 𝐾)
2189, 21, 216, 217syl3anc 1490 . . 3 (𝜑 → ((𝐷‘(1r𝐴)) · (𝐸𝐹)) ∈ 𝐾)
2193, 4, 30grpsubeq0 17782 . . 3 ((𝑅 ∈ Grp ∧ (𝐷𝐹) ∈ 𝐾 ∧ ((𝐷‘(1r𝐴)) · (𝐸𝐹)) ∈ 𝐾) → (((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))) = 0 ↔ (𝐷𝐹) = ((𝐷‘(1r𝐴)) · (𝐸𝐹))))
22011, 215, 218, 219syl3anc 1490 . 2 (𝜑 → (((𝐷𝐹)(-g𝑅)((𝐷‘(1r𝐴)) · (𝐸𝐹))) = 0 ↔ (𝐷𝐹) = ((𝐷‘(1r𝐴)) · (𝐸𝐹))))
221214, 220mpbid 223 1 (𝜑 → (𝐷𝐹) = ((𝐷‘(1r𝐴)) · (𝐸𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  {cab 2751  wne 2937  wral 3055  Vcvv 3350  cdif 3731  ifcif 4245  {csn 4336  cmpt 4890   × cxp 5277  cres 5281  wf 6066  cfv 6070  (class class class)co 6846  𝑓 cof 7097  𝑚 cmap 8064  Fincfn 8164  Basecbs 16144  +gcplusg 16228  .rcmulr 16229  0gc0g 16380  Grpcgrp 17703  -gcsg 17705  CMndccmn 18473  Abelcabl 18474  mulGrpcmgp 18770  1rcur 18782  Ringcrg 18828  CRingccrg 18829   Mat cmat 20503   maDet cmdat 20681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-addf 10272  ax-mulf 10273
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-xor 1634  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-ot 4345  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-of 7099  df-om 7268  df-1st 7370  df-2nd 7371  df-supp 7502  df-tpos 7559  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-oadd 7772  df-er 7951  df-map 8066  df-pm 8067  df-ixp 8118  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-fsupp 8487  df-sup 8559  df-oi 8626  df-card 9020  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-div 10943  df-nn 11279  df-2 11339  df-3 11340  df-4 11341  df-5 11342  df-6 11343  df-7 11344  df-8 11345  df-9 11346  df-n0 11543  df-xnn0 11615  df-z 11629  df-dec 11746  df-uz 11892  df-rp 12034  df-fz 12539  df-fzo 12679  df-seq 13014  df-exp 13073  df-hash 13327  df-word 13492  df-lsw 13540  df-concat 13548  df-s1 13573  df-substr 13623  df-pfx 13672  df-splice 13779  df-reverse 13797  df-s2 13891  df-struct 16146  df-ndx 16147  df-slot 16148  df-base 16150  df-sets 16151  df-ress 16152  df-plusg 16241  df-mulr 16242  df-starv 16243  df-sca 16244  df-vsca 16245  df-ip 16246  df-tset 16247  df-ple 16248  df-ds 16250  df-unif 16251  df-hom 16252  df-cco 16253  df-0g 16382  df-gsum 16383  df-prds 16388  df-pws 16390  df-mre 16526  df-mrc 16527  df-acs 16529  df-mgm 17522  df-sgrp 17564  df-mnd 17575  df-mhm 17615  df-submnd 17616  df-grp 17706  df-minusg 17707  df-sbg 17708  df-mulg 17822  df-subg 17869  df-ghm 17936  df-gim 17979  df-cntz 18027  df-oppg 18053  df-symg 18075  df-pmtr 18139  df-psgn 18188  df-evpm 18189  df-cmn 18475  df-abl 18476  df-mgp 18771  df-ur 18783  df-srg 18787  df-ring 18830  df-cring 18831  df-oppr 18904  df-dvdsr 18922  df-unit 18923  df-invr 18953  df-dvr 18964  df-rnghom 18998  df-drng 19032  df-subrg 19061  df-lmod 19148  df-lss 19216  df-sra 19460  df-rgmod 19461  df-cnfld 20034  df-zring 20106  df-zrh 20139  df-dsmm 20366  df-frlm 20381  df-mamu 20480  df-mat 20504  df-mdet 20682
This theorem is referenced by:  mdetuni  20719  mdetmul  20720
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