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Theorem mdetmul 20845
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 2778 . . 3 (Base‘𝑅) = (Base‘𝑅)
4 eqid 2778 . . 3 (0g𝑅) = (0g𝑅)
5 eqid 2778 . . 3 (1r𝑅) = (1r𝑅)
6 eqid 2778 . . 3 (+g𝑅) = (+g𝑅)
7 mdetmul.t1 . . 3 · = (.r𝑅)
81, 2matrcl 20633 . . . . 5 (𝐹𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
98simpld 490 . . . 4 (𝐹𝐵𝑁 ∈ Fin)
1093ad2ant2 1125 . . 3 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → 𝑁 ∈ Fin)
11 crngring 18956 . . . 4 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12113ad2ant1 1124 . . 3 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → 𝑅 ∈ Ring)
13 mdetmul.d . . . . . . . 8 𝐷 = (𝑁 maDet 𝑅)
1413, 1, 2, 3mdetf 20817 . . . . . . 7 (𝑅 ∈ CRing → 𝐷:𝐵⟶(Base‘𝑅))
15143ad2ant1 1124 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → 𝐷:𝐵⟶(Base‘𝑅))
1615adantr 474 . . . . 5 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑎𝐵) → 𝐷:𝐵⟶(Base‘𝑅))
171matring 20664 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
1810, 12, 17syl2anc 579 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → 𝐴 ∈ Ring)
1918adantr 474 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑎𝐵) → 𝐴 ∈ Ring)
20 simpr 479 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑎𝐵) → 𝑎𝐵)
21 simpl3 1203 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑎𝐵) → 𝐺𝐵)
22 mdetmul.t2 . . . . . . 7 = (.r𝐴)
232, 22ringcl 18959 . . . . . 6 ((𝐴 ∈ Ring ∧ 𝑎𝐵𝐺𝐵) → (𝑎 𝐺) ∈ 𝐵)
2419, 20, 21, 23syl3anc 1439 . . . . 5 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑎𝐵) → (𝑎 𝐺) ∈ 𝐵)
2516, 24ffvelrnd 6626 . . . 4 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑎𝐵) → (𝐷‘(𝑎 𝐺)) ∈ (Base‘𝑅))
2625fmpttd 6651 . . 3 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺))):𝐵⟶(Base‘𝑅))
27 simp21 1220 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑏𝐵)
28 fvoveq1 6947 . . . . . . . 8 (𝑎 = 𝑏 → (𝐷‘(𝑎 𝐺)) = (𝐷‘(𝑏 𝐺)))
29 eqid 2778 . . . . . . . 8 (𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺))) = (𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))
30 fvex 6461 . . . . . . . 8 (𝐷‘(𝑏 𝐺)) ∈ V
3128, 29, 30fvmpt 6544 . . . . . . 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 1217 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑅 ∈ CRing)
3418adantr 474 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → 𝐴 ∈ Ring)
35 simpr1 1205 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → 𝑏𝐵)
36 simpl3 1203 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → 𝐺𝐵)
372, 22ringcl 18959 . . . . . . . . 9 ((𝐴 ∈ Ring ∧ 𝑏𝐵𝐺𝐵) → (𝑏 𝐺) ∈ 𝐵)
3834, 35, 36, 37syl3anc 1439 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → (𝑏 𝐺) ∈ 𝐵)
39383adant3 1123 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → (𝑏 𝐺) ∈ 𝐵)
40 simp22 1221 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑐𝑁)
41 simp23 1222 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑑𝑁)
42 simp3l 1215 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑐𝑑)
43 simpl3r 1260 . . . . . . . . . 10 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) ∧ 𝑎𝑁) → ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))
44 eqid 2778 . . . . . . . . . . . 12 𝑁 = 𝑁
45 oveq1 6931 . . . . . . . . . . . . 13 ((𝑐𝑏𝑒) = (𝑑𝑏𝑒) → ((𝑐𝑏𝑒) · (𝑒𝐺𝑎)) = ((𝑑𝑏𝑒) · (𝑒𝐺𝑎)))
4645ralimi 3134 . . . . . . . . . . . 12 (∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒) → ∀𝑒𝑁 ((𝑐𝑏𝑒) · (𝑒𝐺𝑎)) = ((𝑑𝑏𝑒) · (𝑒𝐺𝑎)))
47 mpteq12 4973 . . . . . . . . . . . 12 ((𝑁 = 𝑁 ∧ ∀𝑒𝑁 ((𝑐𝑏𝑒) · (𝑒𝐺𝑎)) = ((𝑑𝑏𝑒) · (𝑒𝐺𝑎))) → (𝑒𝑁 ↦ ((𝑐𝑏𝑒) · (𝑒𝐺𝑎))) = (𝑒𝑁 ↦ ((𝑑𝑏𝑒) · (𝑒𝐺𝑎))))
4844, 46, 47sylancr 581 . . . . . . . . . . 11 (∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒) → (𝑒𝑁 ↦ ((𝑐𝑏𝑒) · (𝑒𝐺𝑎))) = (𝑒𝑁 ↦ ((𝑑𝑏𝑒) · (𝑒𝐺𝑎))))
4948oveq2d 6940 . . . . . . . . . 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 1127 . . . . . . . . . . . . . . 15 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → 𝑅 ∈ CRing)
52 eqid 2778 . . . . . . . . . . . . . . . . 17 (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
531, 52matmulr 20659 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
5453, 22syl6eqr 2832 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = )
5510, 51, 54syl2anc 579 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = )
5655ad2antrr 716 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = )
5756oveqd 6941 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → (𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) = (𝑏 𝐺))
5857oveqd 6941 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → (𝑐(𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺)𝑎) = (𝑐(𝑏 𝐺)𝑎))
59 simpll1 1226 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → 𝑅 ∈ CRing)
6010ad2antrr 716 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → 𝑁 ∈ Fin)
61 simplr1 1232 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → 𝑏𝐵)
621, 3, 2matbas2i 20643 . . . . . . . . . . . . 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 20643 . . . . . . . . . . . . . 14 (𝐺𝐵𝐺 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
65643ad2ant3 1126 . . . . . . . . . . . . 13 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → 𝐺 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
6665ad2antrr 716 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → 𝐺 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
67 simplr2 1234 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → 𝑐𝑁)
68 simpr 479 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → 𝑎𝑁)
6952, 3, 7, 59, 60, 60, 60, 63, 66, 67, 68mamufv 20608 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → (𝑐(𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺)𝑎) = (𝑅 Σg (𝑒𝑁 ↦ ((𝑐𝑏𝑒) · (𝑒𝐺𝑎)))))
7058, 69eqtr3d 2816 . . . . . . . . . 10 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → (𝑐(𝑏 𝐺)𝑎) = (𝑅 Σg (𝑒𝑁 ↦ ((𝑐𝑏𝑒) · (𝑒𝐺𝑎)))))
71703adantl3 1170 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) ∧ 𝑎𝑁) → (𝑐(𝑏 𝐺)𝑎) = (𝑅 Σg (𝑒𝑁 ↦ ((𝑐𝑏𝑒) · (𝑒𝐺𝑎)))))
7257oveqd 6941 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → (𝑑(𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺)𝑎) = (𝑑(𝑏 𝐺)𝑎))
73 simplr3 1236 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → 𝑑𝑁)
7452, 3, 7, 59, 60, 60, 60, 63, 66, 73, 68mamufv 20608 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → (𝑑(𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺)𝑎) = (𝑅 Σg (𝑒𝑁 ↦ ((𝑑𝑏𝑒) · (𝑒𝐺𝑎)))))
7572, 74eqtr3d 2816 . . . . . . . . . 10 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) ∧ 𝑎𝑁) → (𝑑(𝑏 𝐺)𝑎) = (𝑅 Σg (𝑒𝑁 ↦ ((𝑑𝑏𝑒) · (𝑒𝐺𝑎)))))
76753adantl3 1170 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) ∧ 𝑎𝑁) → (𝑑(𝑏 𝐺)𝑎) = (𝑅 Σg (𝑒𝑁 ↦ ((𝑑𝑏𝑒) · (𝑒𝐺𝑎)))))
7750, 71, 763eqtr4d 2824 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) ∧ 𝑎𝑁) → (𝑐(𝑏 𝐺)𝑎) = (𝑑(𝑏 𝐺)𝑎))
7877ralrimiva 3148 . . . . . . 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 20830 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → (𝐷‘(𝑏 𝐺)) = (0g𝑅))
8032, 79eqtrd 2814 . . . . 5 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁) ∧ (𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (0g𝑅))
81803expia 1111 . . . 4 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ (𝑏𝐵𝑐𝑁𝑑𝑁)) → ((𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒)) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (0g𝑅)))
8281ralrimivvva 3154 . . 3 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → ∀𝑏𝐵𝑐𝑁𝑑𝑁 ((𝑐𝑑 ∧ ∀𝑒𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒)) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝑏) = (0g𝑅)))
83 simp11 1217 . . . . . . . . 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 474 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝐴 ∈ Ring)
85 simprll 769 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑏𝐵)
86 simpl3 1203 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝐺𝐵)
8784, 85, 86, 37syl3anc 1439 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑏 𝐺) ∈ 𝐵)
88873adant3 1123 . . . . . . . . 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 770 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑐𝐵)
902, 22ringcl 18959 . . . . . . . . . . 11 ((𝐴 ∈ Ring ∧ 𝑐𝐵𝐺𝐵) → (𝑐 𝐺) ∈ 𝐵)
9184, 89, 86, 90syl3anc 1439 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 𝐺) ∈ 𝐵)
92913adant3 1123 . . . . . . . . 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 771 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑑𝐵)
942, 22ringcl 18959 . . . . . . . . . . 11 ((𝐴 ∈ Ring ∧ 𝑑𝐵𝐺𝐵) → (𝑑 𝐺) ∈ 𝐵)
9584, 93, 86, 94syl3anc 1439 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑑 𝐺) ∈ 𝐵)
96953adant3 1123 . . . . . . . . 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 1281 . . . . . . . . 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 1223 . . . . . . . . . . . 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 6939 . . . . . . . . . . 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 474 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ Ring)
101 eqid 2778 . . . . . . . . . . . . 13 (𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)
102 snfi 8328 . . . . . . . . . . . . . 14 {𝑒} ∈ Fin
103102a1i 11 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → {𝑒} ∈ Fin)
10410adantr 474 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑁 ∈ Fin)
1051, 3, 2matbas2i 20643 . . . . . . . . . . . . . . 15 (𝑐𝐵𝑐 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
10689, 105syl 17 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑐 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
107 simprrr 772 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑒𝑁)
108107snssd 4573 . . . . . . . . . . . . . . 15 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → {𝑒} ⊆ 𝑁)
109 xpss1 5376 . . . . . . . . . . . . . . 15 ({𝑒} ⊆ 𝑁 → ({𝑒} × 𝑁) ⊆ (𝑁 × 𝑁))
110108, 109syl 17 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ({𝑒} × 𝑁) ⊆ (𝑁 × 𝑁))
111 elmapssres 8167 . . . . . . . . . . . . . 14 ((𝑐 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)) ∧ ({𝑒} × 𝑁) ⊆ (𝑁 × 𝑁)) → (𝑐 ↾ ({𝑒} × 𝑁)) ∈ ((Base‘𝑅) ↑𝑚 ({𝑒} × 𝑁)))
112106, 110, 111syl2anc 579 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐 ↾ ({𝑒} × 𝑁)) ∈ ((Base‘𝑅) ↑𝑚 ({𝑒} × 𝑁)))
1131, 3, 2matbas2i 20643 . . . . . . . . . . . . . . 15 (𝑑𝐵𝑑 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
11493, 113syl 17 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝑑 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
115 elmapssres 8167 . . . . . . . . . . . . . 14 ((𝑑 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)) ∧ ({𝑒} × 𝑁) ⊆ (𝑁 × 𝑁)) → (𝑑 ↾ ({𝑒} × 𝑁)) ∈ ((Base‘𝑅) ↑𝑚 ({𝑒} × 𝑁)))
116114, 110, 115syl2anc 579 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑑 ↾ ({𝑒} × 𝑁)) ∈ ((Base‘𝑅) ↑𝑚 ({𝑒} × 𝑁)))
11765adantr 474 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → 𝐺 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
1183, 100, 101, 103, 104, 104, 6, 112, 116, 117mamudi 20624 . . . . . . . . . . . 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 1123 . . . . . . . . . . 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 2814 . . . . . . . . . 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 474 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = )
122121oveqd 6941 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) = (𝑏 𝐺))
123122reseq1d 5643 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑏 𝐺) ↾ ({𝑒} × 𝑁)))
124 simpl1 1199 . . . . . . . . . . . . 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 20611 . . . . . . . . . . . 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 2816 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏 𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑏 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
1281273adant3 1123 . . . . . . . . . 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 6941 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑐(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) = (𝑐 𝐺))
130129reseq1d 5643 . . . . . . . . . . . . 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 20611 . . . . . . . . . . . . 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 2816 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑐 𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
133121oveqd 6941 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑑(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) = (𝑑 𝐺))
134133reseq1d 5643 . . . . . . . . . . . . 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 20611 . . . . . . . . . . . . 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 2816 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑑 𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑑 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
137132, 136oveq12d 6942 . . . . . . . . . . 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 1123 . . . . . . . . . 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 2824 . . . . . . . . 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 1224 . . . . . . . . . . 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 6939 . . . . . . . . . 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 5643 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑏 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))
143 eqid 2778 . . . . . . . . . . . . 13 (𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩) = (𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)
144 difssd 3961 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → (𝑁 ∖ {𝑒}) ⊆ 𝑁)
14552, 143, 3, 124, 104, 104, 104, 144, 125, 117mamures 20611 . . . . . . . . . . . 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 2816 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
1471463adant3 1123 . . . . . . . . . 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 5643 . . . . . . . . . . . 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 20611 . . . . . . . . . . . 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 2816 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑐 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
1511503adant3 1123 . . . . . . . . . 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 2824 . . . . . . . . 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 1225 . . . . . . . . . . 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 6939 . . . . . . . . . 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 5643 . . . . . . . . . . . 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 20611 . . . . . . . . . . . 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 2816 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐𝐵) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑑 𝐺) ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = ((𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))(𝑅 maMul ⟨(𝑁 ∖ {𝑒}), 𝑁, 𝑁⟩)𝐺))
1581573adant3 1123 . . . . . . . . . 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 2824 . . . . . . . . 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 20824 . . . . . . . 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 1123 . . . . . . . 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 6947 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (𝐷‘(𝑎 𝐺)) = (𝐷‘(𝑐 𝐺)))
164 fvex 6461 . . . . . . . . . . . 12 (𝐷‘(𝑐 𝐺)) ∈ V
165163, 29, 164fvmpt 6544 . . . . . . . . . . 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 6947 . . . . . . . . . . . 12 (𝑎 = 𝑑 → (𝐷‘(𝑎 𝐺)) = (𝐷‘(𝑑 𝐺)))
168 fvex 6461 . . . . . . . . . . . 12 (𝐷‘(𝑑 𝐺)) ∈ V
169167, 29, 168fvmpt 6544 . . . . . . . . . . 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 6942 . . . . . . . . 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 1123 . . . . . . . 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 2824 . . . . . . 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 1111 . . . . . 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 461 . . . . 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 3153 . . . 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 3153 . . 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 1217 . . . . . . . . 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 474 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝐴 ∈ Ring)
180 simprll 769 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝑏𝐵)
181 simpl3 1203 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝐺𝐵)
182179, 180, 181, 37syl3anc 1439 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → (𝑏 𝐺) ∈ 𝐵)
1831823adant3 1123 . . . . . . . . 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 1279 . . . . . . . . 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 771 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝑑𝐵)
186179, 185, 181, 94syl3anc 1439 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → (𝑑 𝐺) ∈ 𝐵)
1871863adant3 1123 . . . . . . . . 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 1281 . . . . . . . . 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 1215 . . . . . . . . . . 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 6939 . . . . . . . . . 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 474 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = )
192191oveqd 6941 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → (𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) = (𝑏 𝐺))
193192reseq1d 5643 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑏 𝐺) ↾ ({𝑒} × 𝑁)))
194 simpl1 1199 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ CRing)
19510adantr 474 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝑁 ∈ Fin)
196 simprrr 772 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝑒𝑁)
197196snssd 4573 . . . . . . . . . . . . 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 474 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝐺 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
20052, 101, 3, 194, 195, 195, 195, 197, 198, 199mamures 20611 . . . . . . . . . . . 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 2816 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑏 𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑏 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
2022013adant3 1123 . . . . . . . . . 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 6941 . . . . . . . . . . . . . . 15 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → (𝑑(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐺) = (𝑑 𝐺))
204203reseq1d 5643 . . . . . . . . . . . . . 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 20611 . . . . . . . . . . . . . 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 2816 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → ((𝑑 𝐺) ↾ ({𝑒} × 𝑁)) = ((𝑑 ↾ ({𝑒} × 𝑁))(𝑅 maMul ⟨{𝑒}, 𝑁, 𝑁⟩)𝐺))
208207oveq2d 6940 . . . . . . . . . . . 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 474 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → 𝑅 ∈ Ring)
210102a1i 11 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → {𝑒} ∈ Fin)
211 simprlr 770 . . . . . . . . . . . . 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 579 . . . . . . . . . . . . 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 20626 . . . . . . . . . . . 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 2817 . . . . . . . . . . 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 1123 . . . . . . . . . 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 2824 . . . . . . . . 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 1216 . . . . . . . . . . 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 6939 . . . . . . . . . 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 5643 . . . . . . . . . . . 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 3961 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) ∧ ((𝑏𝐵𝑐 ∈ (Base‘𝑅)) ∧ (𝑑𝐵𝑒𝑁))) → (𝑁 ∖ {𝑒}) ⊆ 𝑁)
22252, 143, 3, 194, 195, 195, 195, 221, 198, 199mamures 20611 . . . . . . . . . . . 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 2816 . . . . . . . . . . 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 1123 . . . . . . . . . 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 5643 . . . . . . . . . . . 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 20611 . . . . . . . . . . . 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 2816 . . . . . . . . . . 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 1123 . . . . . . . . . 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 2824 . . . . . . . . 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 20825 . . . . . . . 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 1278 . . . . . . . . 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 1280 . . . . . . . . 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 6940 . . . . . . . . 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 2824 . . . . . . 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 1111 . . . . . 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 461 . . . . 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 3153 . . . 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 3153 . . 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 1128 . . 3 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → 𝐹𝐵)
2421, 2, 3, 4, 5, 6, 7, 10, 12, 26, 82, 177, 240, 13, 51, 241mdetuni0 20843 . 2 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝐹) = (((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘(1r𝐴)) · (𝐷𝐹)))
243 fvoveq1 6947 . . . 4 (𝑎 = 𝐹 → (𝐷‘(𝑎 𝐺)) = (𝐷‘(𝐹 𝐺)))
244 fvex 6461 . . . 4 (𝐷‘(𝐹 𝐺)) ∈ V
245243, 29, 244fvmpt 6544 . . 3 (𝐹𝐵 → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝐹) = (𝐷‘(𝐹 𝐺)))
2462453ad2ant2 1125 . 2 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘𝐹) = (𝐷‘(𝐹 𝐺)))
247 eqid 2778 . . . . . . 7 (1r𝐴) = (1r𝐴)
2482, 247ringidcl 18966 . . . . . 6 (𝐴 ∈ Ring → (1r𝐴) ∈ 𝐵)
249 fvoveq1 6947 . . . . . . 7 (𝑎 = (1r𝐴) → (𝐷‘(𝑎 𝐺)) = (𝐷‘((1r𝐴) 𝐺)))
250 fvex 6461 . . . . . . 7 (𝐷‘((1r𝐴) 𝐺)) ∈ V
251249, 29, 250fvmpt 6544 . . . . . 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 1129 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → 𝐺𝐵)
2542, 22, 247ringlidm 18969 . . . . . . 7 ((𝐴 ∈ Ring ∧ 𝐺𝐵) → ((1r𝐴) 𝐺) = 𝐺)
25518, 253, 254syl2anc 579 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → ((1r𝐴) 𝐺) = 𝐺)
256255fveq2d 6452 . . . . 5 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (𝐷‘((1r𝐴) 𝐺)) = (𝐷𝐺))
257252, 256eqtrd 2814 . . . 4 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → ((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘(1r𝐴)) = (𝐷𝐺))
258257oveq1d 6939 . . 3 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘(1r𝐴)) · (𝐷𝐹)) = ((𝐷𝐺) · (𝐷𝐹)))
25915, 253ffvelrnd 6626 . . . 4 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (𝐷𝐺) ∈ (Base‘𝑅))
26015, 241ffvelrnd 6626 . . . 4 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (𝐷𝐹) ∈ (Base‘𝑅))
2613, 7crngcom 18960 . . . 4 ((𝑅 ∈ CRing ∧ (𝐷𝐺) ∈ (Base‘𝑅) ∧ (𝐷𝐹) ∈ (Base‘𝑅)) → ((𝐷𝐺) · (𝐷𝐹)) = ((𝐷𝐹) · (𝐷𝐺)))
26251, 259, 260, 261syl3anc 1439 . . 3 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → ((𝐷𝐺) · (𝐷𝐹)) = ((𝐷𝐹) · (𝐷𝐺)))
263258, 262eqtrd 2814 . 2 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (((𝑎𝐵 ↦ (𝐷‘(𝑎 𝐺)))‘(1r𝐴)) · (𝐷𝐹)) = ((𝐷𝐹) · (𝐷𝐺)))
264242, 246, 2633eqtr3d 2822 1 ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (𝐷‘(𝐹 𝐺)) = ((𝐷𝐹) · (𝐷𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107  wne 2969  wral 3090  Vcvv 3398  cdif 3789  wss 3792  {csn 4398  cotp 4406  cmpt 4967   × cxp 5355  cres 5359  wf 6133  cfv 6137  (class class class)co 6924  𝑓 cof 7174  𝑚 cmap 8142  Fincfn 8243  Basecbs 16266  +gcplusg 16349  .rcmulr 16350  0gc0g 16497   Σg cgsu 16498  1rcur 18899  Ringcrg 18945  CRingccrg 18946   maMul cmmul 20604   Mat cmat 20628   maDet cmdat 20806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-inf2 8837  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351  ax-addf 10353  ax-mulf 10354
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-xor 1583  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-ot 4407  df-uni 4674  df-int 4713  df-iun 4757  df-iin 4758  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-se 5317  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-isom 6146  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-of 7176  df-om 7346  df-1st 7447  df-2nd 7448  df-supp 7579  df-tpos 7636  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-2o 7846  df-oadd 7849  df-er 8028  df-map 8144  df-pm 8145  df-ixp 8197  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-fsupp 8566  df-sup 8638  df-oi 8706  df-card 9100  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-div 11036  df-nn 11380  df-2 11443  df-3 11444  df-4 11445  df-5 11446  df-6 11447  df-7 11448  df-8 11449  df-9 11450  df-n0 11648  df-xnn0 11720  df-z 11734  df-dec 11851  df-uz 11998  df-rp 12143  df-fz 12649  df-fzo 12790  df-seq 13125  df-exp 13184  df-hash 13442  df-word 13606  df-lsw 13659  df-concat 13667  df-s1 13692  df-substr 13737  df-pfx 13786  df-splice 13893  df-reverse 13911  df-s2 14005  df-struct 16268  df-ndx 16269  df-slot 16270  df-base 16272  df-sets 16273  df-ress 16274  df-plusg 16362  df-mulr 16363  df-starv 16364  df-sca 16365  df-vsca 16366  df-ip 16367  df-tset 16368  df-ple 16369  df-ds 16371  df-unif 16372  df-hom 16373  df-cco 16374  df-0g 16499  df-gsum 16500  df-prds 16505  df-pws 16507  df-mre 16643  df-mrc 16644  df-acs 16646  df-mgm 17639  df-sgrp 17681  df-mnd 17692  df-mhm 17732  df-submnd 17733  df-grp 17823  df-minusg 17824  df-sbg 17825  df-mulg 17939  df-subg 17986  df-ghm 18053  df-gim 18096  df-cntz 18144  df-oppg 18170  df-symg 18192  df-pmtr 18256  df-psgn 18305  df-evpm 18306  df-cmn 18592  df-abl 18593  df-mgp 18888  df-ur 18900  df-srg 18904  df-ring 18947  df-cring 18948  df-oppr 19021  df-dvdsr 19039  df-unit 19040  df-invr 19070  df-dvr 19081  df-rnghom 19115  df-drng 19152  df-subrg 19181  df-lmod 19268  df-lss 19336  df-sra 19580  df-rgmod 19581  df-cnfld 20154  df-zring 20226  df-zrh 20259  df-dsmm 20486  df-frlm 20501  df-mamu 20605  df-mat 20629  df-mdet 20807
This theorem is referenced by:  matunit  20901  cramerimplem3  20909  matunitlindflem2  34041
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