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Theorem mdetmul 22645
Description: Multiplicativity of the determinant function: the determinant of a matrix product of square matrices equals the product of their determinants. Proposition 4.15 in [Lang] p. 517. (Contributed by Stefan O'Rear, 16-Jul-2018.)
Hypotheses
Ref Expression
mdetmul.a 𝐴 = (𝑁 Mat 𝑅)
mdetmul.b 𝐵 = (Base‘𝐴)
mdetmul.d 𝐷 = (𝑁 maDet 𝑅)
mdetmul.t1 · = (.r𝑅)
mdetmul.t2 = (.r𝐴)
Assertion
Ref Expression
mdetmul ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (𝐷‘(𝐹 𝐺)) = ((𝐷𝐹) · (𝐷𝐺)))

Proof of Theorem mdetmul
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mdetmul.a . . 3 𝐴 = (𝑁 Mat 𝑅)
2 mdetmul.b . . 3 𝐵 = (Base‘𝐴)
3 eqid 2735 . . 3 (Base‘𝑅) = (Base‘𝑅)
4 eqid 2735 . . 3 (0g𝑅) = (0g𝑅)
5 eqid 2735 . . 3 (1r𝑅) = (1r𝑅)
6 eqid 2735 . . 3 (+g𝑅) = (+g𝑅)
7 mdetmul.t1 . . 3 · = (.r𝑅)
81, 2matrcl 22432 . . . . 5 (𝐹𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
98simpld 494 . . . 4 (𝐹𝐵𝑁 ∈ Fin)
1093ad2ant2 1133 . . 3 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → 𝑁 ∈ Fin)
11 crngring 20263 . . . 4 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12113ad2ant1 1132 . . 3 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → 𝑅 ∈ Ring)
13 mdetmul.d . . . . . . . 8 𝐷 = (𝑁 maDet 𝑅)
1413, 1, 2, 3mdetf 22617 . . . . . . 7 (𝑅 ∈ CRing → 𝐷:𝐵⟶(Base‘𝑅))
15143ad2ant1 1132 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → 𝐷:𝐵⟶(Base‘𝑅))
1615adantr 480 . . . . 5 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑎𝐵) → 𝐷:𝐵⟶(Base‘𝑅))
171matring 22465 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
1810, 12, 17syl2anc 584 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → 𝐴 ∈ Ring)
1918adantr 480 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑎𝐵) → 𝐴 ∈ Ring)
20 simpr 484 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑎𝐵) → 𝑎𝐵)
21 simpl3 1192 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑎𝐵) → 𝐺𝐵)
22 mdetmul.t2 . . . . . . 7 = (.r𝐴)
232, 22ringcl 20268 . . . . . 6 ((𝐴 ∈ Ring ∧ 𝑎𝐵𝐺𝐵) → (𝑎 𝐺) ∈ 𝐵)
2419, 20, 21, 23syl3anc 1370 . . . . 5 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑎𝐵) → (𝑎 𝐺) ∈ 𝐵)
2516, 24ffvelcdmd 7105 . . . 4 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑎𝐵) → (𝐷‘(𝑎 𝐺)) ∈ (Base‘𝑅))
2625fmpttd 7135 . . 3 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺))):𝐵⟶(Base‘𝑅))
27 simp21 1205 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑏𝐵)
28 fvoveq1 7454 . . . . . . . 8 (𝑎 = 𝑏 → (𝐷‘(𝑎 𝐺)) = (𝐷‘(𝑏 𝐺)))
29 eqid 2735 . . . . . . . 8 (𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺))) = (𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))
30 fvex 6920 . . . . . . . 8 (𝐷‘(𝑏 𝐺)) ∈ V
3128, 29, 30fvmpt 7016 . . . . . . 7 (𝑏𝐵 → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (𝐷‘(𝑏 𝐺)))
3227, 31syl 17 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (𝐷‘(𝑏 𝐺)))
33 simp11 1202 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑅 ∈ CRing)
3418adantr 480 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → 𝐴 ∈ Ring)
35 simpr1 1193 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → 𝑏𝐵)
36 simpl3 1192 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → 𝐺𝐵)
372, 22ringcl 20268 . . . . . . . . 9 ((𝐴 ∈ Ring ∧ 𝑏𝐵𝐺𝐵) → (𝑏 𝐺) ∈ 𝐵)
3834, 35, 36, 37syl3anc 1370 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → (𝑏 𝐺) ∈ 𝐵)
39383adant3 1131 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → (𝑏 𝐺) ∈ 𝐵)
40 simp22 1206 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑐𝑁)
41 simp23 1207 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑑𝑁)
42 simp3l 1200 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑐𝑑)
43 simpl3r 1228 . . . . . . . . . 10 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) ∧ 𝑎𝑁) → ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))
44 eqid 2735 . . . . . . . . . . . 12 𝑁 = 𝑁
45 oveq1 7438 . . . . . . . . . . . . 13 ((𝑐𝑏𝑒) = (𝑑𝑏𝑒) → ((𝑐𝑏𝑒) · (𝑒𝐺𝑎)) = ((𝑑𝑏𝑒) · (𝑒𝐺𝑎)))
4645ralimi 3081 . . . . . . . . . . . 12 (∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒) → ∀𝑒𝑁 ((𝑐𝑏𝑒) · (𝑒𝐺𝑎)) = ((𝑑𝑏𝑒) · (𝑒𝐺𝑎)))
47 mpteq12 5240 . . . . . . . . . . . 12 ((𝑁 = 𝑁 ∧ ∀𝑒𝑁 ((𝑐𝑏𝑒) · (𝑒𝐺𝑎)) = ((𝑑𝑏𝑒) · (𝑒𝐺𝑎))) → (𝑒𝑁 ↦ ((𝑐𝑏𝑒) · (𝑒𝐺𝑎))) = (𝑒𝑁 ↦ ((𝑑𝑏𝑒) · (𝑒𝐺𝑎))))
4844, 46, 47sylancr 587 . . . . . . . . . . 11 (∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒) → (𝑒𝑁 ↦ ((𝑐𝑏𝑒) · (𝑒𝐺𝑎))) = (𝑒𝑁 ↦ ((𝑑𝑏𝑒) · (𝑒𝐺𝑎))))
4948oveq2d 7447 . . . . . . . . . 10 (∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒) → (𝑅 Σg (𝑒𝑁 ↦ ((𝑐𝑏𝑒) · (𝑒𝐺𝑎)))) = (𝑅 Σg (𝑒𝑁 ↦ ((𝑑𝑏𝑒) · (𝑒𝐺𝑎)))))
5043, 49syl 17 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) ∧ 𝑎𝑁) → (𝑅 Σg (𝑒𝑁 ↦ ((𝑐𝑏𝑒) · (𝑒𝐺𝑎)))) = (𝑅 Σg (𝑒𝑁 ↦ ((𝑑𝑏𝑒) · (𝑒𝐺𝑎)))))
51 simp1 1135 . . . . . . . . . . . . . . 15 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → 𝑅 ∈ CRing)
52 eqid 2735 . . . . . . . . . . . . . . . . 17 (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
531, 52matmulr 22460 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
5453, 22eqtr4di 2793 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = )
5510, 51, 54syl2anc 584 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = )
5655ad2antrr 726 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = )
5756oveqd 7448 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → (𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) = (𝑏 𝐺))
5857oveqd 7448 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → (𝑐(𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺)𝑎) = (𝑐(𝑏 𝐺)𝑎))
59 simpll1 1211 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → 𝑅 ∈ CRing)
6010ad2antrr 726 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → 𝑁 ∈ Fin)
61 simplr1 1214 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → 𝑏𝐵)
621, 3, 2matbas2i 22444 . . . . . . . . . . . . 13 (𝑏𝐵𝑏 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
6361, 62syl 17 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → 𝑏 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
641, 3, 2matbas2i 22444 . . . . . . . . . . . . . 14 (𝐺𝐵𝐺 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
65643ad2ant3 1134 . . . . . . . . . . . . 13 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → 𝐺 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
6665ad2antrr 726 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → 𝐺 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
67 simplr2 1215 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → 𝑐𝑁)
68 simpr 484 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → 𝑎𝑁)
6952, 3, 7, 59, 60, 60, 60, 63, 66, 67, 68mamufv 22414 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → (𝑐(𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺)𝑎) = (𝑅 Σg (𝑒𝑁 ↦ ((𝑐𝑏𝑒) · (𝑒𝐺𝑎)))))
7058, 69eqtr3d 2777 . . . . . . . . . 10 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → (𝑐(𝑏 𝐺)𝑎) = (𝑅 Σg (𝑒𝑁 ↦ ((𝑐𝑏𝑒) · (𝑒𝐺𝑎)))))
71703adantl3 1167 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) ∧ 𝑎𝑁) → (𝑐(𝑏 𝐺)𝑎) = (𝑅 Σg (𝑒𝑁 ↦ ((𝑐𝑏𝑒) · (𝑒𝐺𝑎)))))
7257oveqd 7448 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → (𝑑(𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺)𝑎) = (𝑑(𝑏 𝐺)𝑎))
73 simplr3 1216 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → 𝑑𝑁)
7452, 3, 7, 59, 60, 60, 60, 63, 66, 73, 68mamufv 22414 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → (𝑑(𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺)𝑎) = (𝑅 Σg (𝑒𝑁 ↦ ((𝑑𝑏𝑒) · (𝑒𝐺𝑎)))))
7572, 74eqtr3d 2777 . . . . . . . . . 10 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → (𝑑(𝑏 𝐺)𝑎) = (𝑅 Σg (𝑒𝑁 ↦ ((𝑑𝑏𝑒) · (𝑒𝐺𝑎)))))
76753adantl3 1167 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) ∧ 𝑎𝑁) → (𝑑(𝑏 𝐺)𝑎) = (𝑅 Σg (𝑒𝑁 ↦ ((𝑑𝑏𝑒) · (𝑒𝐺𝑎)))))
7750, 71, 763eqtr4d 2785 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) ∧ 𝑎𝑁) → (𝑐(𝑏 𝐺)𝑎) = (𝑑(𝑏 𝐺)𝑎))
7877ralrimiva 3144 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ∀𝑎𝑁 (𝑐(𝑏 𝐺)𝑎) = (𝑑(𝑏 𝐺)𝑎))
7913, 1, 2, 4, 33, 39, 40, 41, 42, 78mdetralt 22630 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → (𝐷‘(𝑏 𝐺)) = (0g𝑅))
8032, 79eqtrd 2775 . . . . 5 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (0g𝑅))
81803expia 1120 . . . 4 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → ((𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒)) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (0g𝑅)))
8281ralrimivvva 3203 . . 3 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → ∀𝑏𝐵𝑐𝑁𝑑𝑁 ((𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒)) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (0g𝑅)))
83 simp11 1202 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑅 ∈ CRing)
8418adantr 480 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝐴 ∈ Ring)
85 simprll 779 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑏𝐵)
86 simpl3 1192 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝐺𝐵)
8784, 85, 86, 37syl3anc 1370 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑏 𝐺) ∈ 𝐵)
88873adant3 1131 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑏 𝐺) ∈ 𝐵)
89 simprlr 780 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑐𝐵)
902, 22ringcl 20268 . . . . . . . . . . 11 ((𝐴 ∈ Ring ∧ 𝑐𝐵𝐺𝐵) → (𝑐 𝐺) ∈ 𝐵)
9184, 89, 86, 90syl3anc 1370 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 𝐺) ∈ 𝐵)
92913adant3 1131 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑐 𝐺) ∈ 𝐵)
93 simprrl 781 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑑𝐵)
942, 22ringcl 20268 . . . . . . . . . . 11 ((𝐴 ∈ Ring ∧ 𝑑𝐵𝐺𝐵) → (𝑑 𝐺) ∈ 𝐵)
9584, 93, 86, 94syl3anc 1370 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑑 𝐺) ∈ 𝐵)
96953adant3 1131 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑑 𝐺) ∈ 𝐵)
97 simp2rr 1242 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑒𝑁)
98 simp31 1208 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(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 (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))))
9998oveq1d 7446 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑏 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺) = (((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁)))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
10012adantr 480 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ Ring)
101 eqid 2735 . . . . . . . . . . . . 13 (𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)
102 snfi 9082 . . . . . . . . . . . . . 14 {𝑒} ∈ Fin
103102a1i 11 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → {𝑒} ∈ Fin)
10410adantr 480 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑁 ∈ Fin)
1051, 3, 2matbas2i 22444 . . . . . . . . . . . . . . 15 (𝑐𝐵𝑐 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
10689, 105syl 17 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑐 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
107 simprrr 782 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑒𝑁)
108107snssd 4814 . . . . . . . . . . . . . . 15 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → {𝑒} ⊆ 𝑁)
109 xpss1 5708 . . . . . . . . . . . . . . 15 ({𝑒} ⊆ 𝑁 → ({𝑒} × 𝑁) ⊆ (𝑁 × 𝑁))
110108, 109syl 17 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ({𝑒} × 𝑁) ⊆ (𝑁 × 𝑁))
111 elmapssres 8906 . . . . . . . . . . . . . 14 ((𝑐 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) ∧ ({𝑒} × 𝑁) ⊆ (𝑁 × 𝑁)) → (𝑐 ↾ ({𝑒} × 𝑁)) ∈ ((Base‘𝑅) ↑m ({𝑒} × 𝑁)))
112106, 110, 111syl2anc 584 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 ↾ ({𝑒} × 𝑁)) ∈ ((Base‘𝑅) ↑m ({𝑒} × 𝑁)))
1131, 3, 2matbas2i 22444 . . . . . . . . . . . . . . 15 (𝑑𝐵𝑑 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
11493, 113syl 17 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑑 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
115 elmapssres 8906 . . . . . . . . . . . . . 14 ((𝑑 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) ∧ ({𝑒} × 𝑁) ⊆ (𝑁 × 𝑁)) → (𝑑 ↾ ({𝑒} × 𝑁)) ∈ ((Base‘𝑅) ↑m ({𝑒} × 𝑁)))
116114, 110, 115syl2anc 584 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑑 ↾ ({𝑒} × 𝑁)) ∈ ((Base‘𝑅) ↑m ({𝑒} × 𝑁)))
11765adantr 480 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝐺 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
1183, 100, 101, 103, 104, 104, 6, 112, 116, 117mamudi 22423 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁)))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺) = (((𝑐 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺) ∘f (+g𝑅)((𝑑 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺)))
1191183adant3 1131 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁)))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺) = (((𝑐 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺) ∘f (+g𝑅)((𝑑 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺)))
12099, 119eqtrd 2775 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑏 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺) = (((𝑐 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺) ∘f (+g𝑅)((𝑑 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺)))
12155adantr 480 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = )
122121oveqd 7448 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) = (𝑏 𝐺))
123122reseq1d 5999 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑏 𝐺) ↾ ({𝑒} × 𝑁)))
124 simpl1 1190 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ CRing)
12585, 62syl 17 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑏 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
12652, 101, 3, 124, 104, 104, 104, 108, 125, 117mamures 22417 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑏 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
127123, 126eqtr3d 2777 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏 𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑏 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
1281273adant3 1131 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑏 𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑏 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
129121oveqd 7448 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) = (𝑐 𝐺))
130129reseq1d 5999 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑐(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑐 𝐺) ↾ ({𝑒} × 𝑁)))
13152, 101, 3, 124, 104, 104, 104, 108, 106, 117mamures 22417 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑐(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
132130, 131eqtr3d 2777 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑐 𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
133121oveqd 7448 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑑(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) = (𝑑 𝐺))
134133reseq1d 5999 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑑(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑑 𝐺) ↾ ({𝑒} × 𝑁)))
13552, 101, 3, 124, 104, 104, 104, 108, 114, 117mamures 22417 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑑(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑑 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
136134, 135eqtr3d 2777 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑑 𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑑 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
137132, 136oveq12d 7449 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝑐 𝐺) ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)((𝑑 𝐺) ↾ ({𝑒} × 𝑁))) = (((𝑐 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺) ∘f (+g𝑅)((𝑑 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺)))
1381373adant3 1131 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (((𝑐 𝐺) ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)((𝑑 𝐺) ↾ ({𝑒} × 𝑁))) = (((𝑐 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺) ∘f (+g𝑅)((𝑑 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺)))
139120, 128, 1383eqtr4d 2785 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑏 𝐺) ↾ ({𝑒} × 𝑁)) = (((𝑐 𝐺) ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)((𝑑 𝐺) ↾ ({𝑒} × 𝑁))))
140 simp32 1209 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(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)))
141140oveq1d 7446 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺) = ((𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
142122reseq1d 5999 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑏 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))
143 eqid 2735 . . . . . . . . . . . . 13 (𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩) = (𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)
144 difssd 4147 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑁 ∖ {𝑒}) ⊆ 𝑁)
14552, 143, 3, 124, 104, 104, 104, 144, 125, 117mamures 22417 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
146142, 145eqtr3d 2777 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
1471463adant3 1131 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑏 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
148129reseq1d 5999 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑐(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑐 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))
14952, 143, 3, 124, 104, 104, 104, 144, 106, 117mamures 22417 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑐(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
150148, 149eqtr3d 2777 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑐 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
1511503adant3 1131 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑐 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
152141, 147, 1513eqtr4d 2785 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑏 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑐 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))
153 simp33 1210 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(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)))
154153oveq1d 7446 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺) = ((𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
155133reseq1d 5999 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑑(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑑 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))
15652, 143, 3, 124, 104, 104, 104, 144, 114, 117mamures 22417 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑑(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
157155, 156eqtr3d 2777 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑑 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
1581573adant3 1131 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑑 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
159154, 147, 1583eqtr4d 2785 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑏 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑑 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))
16013, 1, 2, 6, 83, 88, 92, 96, 97, 139, 152, 159mdetrlin 22624 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝐷‘(𝑏 𝐺)) = ((𝐷‘(𝑐 𝐺))(+g𝑅)(𝐷‘(𝑑 𝐺))))
16185, 31syl 17 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (𝐷‘(𝑏 𝐺)))
1621613adant3 1131 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (𝐷‘(𝑏 𝐺)))
163 fvoveq1 7454 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (𝐷‘(𝑎 𝐺)) = (𝐷‘(𝑐 𝐺)))
164 fvex 6920 . . . . . . . . . . . 12 (𝐷‘(𝑐 𝐺)) ∈ V
165163, 29, 164fvmpt 7016 . . . . . . . . . . 11 (𝑐𝐵 → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑐) = (𝐷‘(𝑐 𝐺)))
16689, 165syl 17 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑐) = (𝐷‘(𝑐 𝐺)))
167 fvoveq1 7454 . . . . . . . . . . . 12 (𝑎 = 𝑑 → (𝐷‘(𝑎 𝐺)) = (𝐷‘(𝑑 𝐺)))
168 fvex 6920 . . . . . . . . . . . 12 (𝐷‘(𝑑 𝐺)) ∈ V
169167, 29, 168fvmpt 7016 . . . . . . . . . . 11 (𝑑𝐵 → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑑) = (𝐷‘(𝑑 𝐺)))
17093, 169syl 17 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑑) = (𝐷‘(𝑑 𝐺)))
171166, 170oveq12d 7449 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑐)(+g𝑅)((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑑)) = ((𝐷‘(𝑐 𝐺))(+g𝑅)(𝐷‘(𝑑 𝐺))))
1721713adant3 1131 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑐)(+g𝑅)((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑑)) = ((𝐷‘(𝑐 𝐺))(+g𝑅)(𝐷‘(𝑑 𝐺))))
173160, 162, 1723eqtr4d 2785 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑐)(+g𝑅)((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑑)))
1741733expia 1120 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑐)(+g𝑅)((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑑))))
175174anassrs 467 . . . . 5 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝐵)) ∧ (𝑑𝐵𝑒𝑁)) → (((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑐)(+g𝑅)((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑑))))
176175ralrimivva 3200 . . . 4 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝐵)) → ∀𝑑𝐵𝑒𝑁 (((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑐)(+g𝑅)((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑑))))
177176ralrimivva 3200 . . 3 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → ∀𝑏𝐵𝑐𝐵𝑑𝐵𝑒𝑁 (((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f (+g𝑅)(𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑐)(+g𝑅)((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑑))))
178 simp11 1202 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑅 ∈ CRing)
17918adantr 480 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝐴 ∈ Ring)
180 simprll 779 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝑏𝐵)
181 simpl3 1192 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝐺𝐵)
182179, 180, 181, 37syl3anc 1370 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → (𝑏 𝐺) ∈ 𝐵)
1831823adant3 1131 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑏 𝐺) ∈ 𝐵)
184 simp2lr 1240 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑐 ∈ (Base‘𝑅))
185 simprrl 781 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝑑𝐵)
186179, 185, 181, 94syl3anc 1370 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → (𝑑 𝐺) ∈ 𝐵)
1871863adant3 1131 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑑 𝐺) ∈ 𝐵)
188 simp2rr 1242 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑒𝑁)
189 simp3l 1200 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (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))))
190189oveq1d 7446 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑏 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺) = (((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁)))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
19155adantr 480 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = )
192191oveqd 7448 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → (𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) = (𝑏 𝐺))
193192reseq1d 5999 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑏 𝐺) ↾ ({𝑒} × 𝑁)))
194 simpl1 1190 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ CRing)
19510adantr 480 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝑁 ∈ Fin)
196 simprrr 782 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝑒𝑁)
197196snssd 4814 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → {𝑒} ⊆ 𝑁)
198180, 62syl 17 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝑏 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
19965adantr 480 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝐺 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
20052, 101, 3, 194, 195, 195, 195, 197, 198, 199mamures 22417 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑏 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
201193, 200eqtr3d 2777 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏 𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑏 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
2022013adant3 1131 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑏 𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑏 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
203191oveqd 7448 . . . . . . . . . . . . . . 15 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → (𝑑(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) = (𝑑 𝐺))
204203reseq1d 5999 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑑(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑑 𝐺) ↾ ({𝑒} × 𝑁)))
205185, 113syl 17 . . . . . . . . . . . . . . 15 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝑑 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
20652, 101, 3, 194, 195, 195, 195, 197, 205, 199mamures 22417 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑑(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑑 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
207204, 206eqtr3d 2777 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑑 𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑑 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
208207oveq2d 7447 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((({𝑒} × 𝑁) × {𝑐}) ∘f · ((𝑑 𝐺) ↾ ({𝑒} × 𝑁))) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · ((𝑑 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺)))
20912adantr 480 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ Ring)
210102a1i 11 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → {𝑒} ∈ Fin)
211 simprlr 780 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝑐 ∈ (Base‘𝑅))
212197, 109syl 17 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ({𝑒} × 𝑁) ⊆ (𝑁 × 𝑁))
213205, 212, 115syl2anc 584 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → (𝑑 ↾ ({𝑒} × 𝑁)) ∈ ((Base‘𝑅) ↑m ({𝑒} × 𝑁)))
2143, 209, 101, 210, 195, 195, 7, 211, 213, 199mamuvs1 22425 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → (((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁)))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · ((𝑑 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺)))
215208, 214eqtr4d 2778 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((({𝑒} × 𝑁) × {𝑐}) ∘f · ((𝑑 𝐺) ↾ ({𝑒} × 𝑁))) = (((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁)))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
2162153adant3 1131 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((({𝑒} × 𝑁) × {𝑐}) ∘f · ((𝑑 𝐺) ↾ ({𝑒} × 𝑁))) = (((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁)))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
217190, 202, 2163eqtr4d 2785 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑏 𝐺) ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · ((𝑑 𝐺) ↾ ({𝑒} × 𝑁))))
218 simp3r 1201 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (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)))
219218oveq1d 7446 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺) = ((𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
220192reseq1d 5999 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑏 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))
221 difssd 4147 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → (𝑁 ∖ {𝑒}) ⊆ 𝑁)
22252, 143, 3, 194, 195, 195, 195, 221, 198, 199mamures 22417 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
223220, 222eqtr3d 2777 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
2242233adant3 1131 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑏 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
225203reseq1d 5999 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑑(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑑 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))
22652, 143, 3, 194, 195, 195, 195, 221, 205, 199mamures 22417 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑑(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
227225, 226eqtr3d 2777 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑑 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
2282273adant3 1131 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑑 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
229219, 224, 2283eqtr4d 2785 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑏 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑑 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))
23013, 1, 2, 3, 7, 178, 183, 184, 187, 188, 217, 229mdetrsca 22625 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝐷‘(𝑏 𝐺)) = (𝑐 · (𝐷‘(𝑑 𝐺))))
231 simp2ll 1239 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑏𝐵)
232231, 31syl 17 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (𝐷‘(𝑏 𝐺)))
233 simp2rl 1241 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑑𝐵)
234169oveq2d 7447 . . . . . . . . 9 (𝑑𝐵 → (𝑐 · ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑑)) = (𝑐 · (𝐷‘(𝑑 𝐺))))
235233, 234syl 17 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑐 · ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑑)) = (𝑐 · (𝐷‘(𝑑 𝐺))))
236230, 232, 2353eqtr4d 2785 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (𝑐 · ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑑)))
2372363expia 1120 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → (((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (𝑐 · ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑑))))
238237anassrs 467 . . . . 5 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐 ∈ (Base‘𝑅))) ∧ (𝑑𝐵𝑒𝑁)) → (((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (𝑐 · ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑑))))
239238ralrimivva 3200 . . . 4 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐 ∈ (Base‘𝑅))) → ∀𝑑𝐵𝑒𝑁 (((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (𝑐 · ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑑))))
240239ralrimivva 3200 . . 3 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → ∀𝑏𝐵𝑐 ∈ (Base‘𝑅)∀𝑑𝐵𝑒𝑁 (((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (𝑐 · ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑑))))
241 simp2 1136 . . 3 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → 𝐹𝐵)
2421, 2, 3, 4, 5, 6, 7, 10, 12, 26, 82, 177, 240, 13, 51, 241mdetuni0 22643 . 2 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝐹) = (((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘(1r𝐴)) · (𝐷𝐹)))
243 fvoveq1 7454 . . . 4 (𝑎 = 𝐹 → (𝐷‘(𝑎 𝐺)) = (𝐷‘(𝐹 𝐺)))
244 fvex 6920 . . . 4 (𝐷‘(𝐹 𝐺)) ∈ V
245243, 29, 244fvmpt 7016 . . 3 (𝐹𝐵 → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝐹) = (𝐷‘(𝐹 𝐺)))
2462453ad2ant2 1133 . 2 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝐹) = (𝐷‘(𝐹 𝐺)))
247 eqid 2735 . . . . . . 7 (1r𝐴) = (1r𝐴)
2482, 247ringidcl 20280 . . . . . 6 (𝐴 ∈ Ring → (1r𝐴) ∈ 𝐵)
249 fvoveq1 7454 . . . . . . 7 (𝑎 = (1r𝐴) → (𝐷‘(𝑎 𝐺)) = (𝐷‘((1r𝐴) 𝐺)))
250 fvex 6920 . . . . . . 7 (𝐷‘((1r𝐴) 𝐺)) ∈ V
251249, 29, 250fvmpt 7016 . . . . . 6 ((1r𝐴) ∈ 𝐵 → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘(1r𝐴)) = (𝐷‘((1r𝐴) 𝐺)))
25218, 248, 2513syl 18 . . . . 5 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘(1r𝐴)) = (𝐷‘((1r𝐴) 𝐺)))
253 simp3 1137 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → 𝐺𝐵)
2542, 22, 247ringlidm 20283 . . . . . . 7 ((𝐴 ∈ Ring ∧ 𝐺𝐵) → ((1r𝐴) 𝐺) = 𝐺)
25518, 253, 254syl2anc 584 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → ((1r𝐴) 𝐺) = 𝐺)
256255fveq2d 6911 . . . . 5 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (𝐷‘((1r𝐴) 𝐺)) = (𝐷𝐺))
257252, 256eqtrd 2775 . . . 4 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘(1r𝐴)) = (𝐷𝐺))
258257oveq1d 7446 . . 3 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘(1r𝐴)) · (𝐷𝐹)) = ((𝐷𝐺) · (𝐷𝐹)))
25915, 253ffvelcdmd 7105 . . . 4 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (𝐷𝐺) ∈ (Base‘𝑅))
26015, 241ffvelcdmd 7105 . . . 4 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (𝐷𝐹) ∈ (Base‘𝑅))
2613, 7crngcom 20269 . . . 4 ((𝑅 ∈ CRing ∧ (𝐷𝐺) ∈ (Base‘𝑅) ∧ (𝐷𝐹) ∈ (Base‘𝑅)) → ((𝐷𝐺) · (𝐷𝐹)) = ((𝐷𝐹) · (𝐷𝐺)))
26251, 259, 260, 261syl3anc 1370 . . 3 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → ((𝐷𝐺) · (𝐷𝐹)) = ((𝐷𝐹) · (𝐷𝐺)))
263258, 262eqtrd 2775 . 2 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘(1r𝐴)) · (𝐷𝐹)) = ((𝐷𝐹) · (𝐷𝐺)))
264242, 246, 2633eqtr3d 2783 1 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (𝐷‘(𝐹 𝐺)) = ((𝐷𝐹) · (𝐷𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  Vcvv 3478  cdif 3960  wss 3963  {csn 4631  cotp 4639  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   Σg cgsu 17487  1rcur 20199  Ringcrg 20251  CRingccrg 20252   maMul cmmul 22410   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:  matunit  22700  cramerimplem3  22707  matunitlindflem2  37604
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