Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  madurid Structured version   Visualization version   GIF version

 Description: Multiplying a matrix with its adjunct results in the identity matrix multiplied with the determinant of the matrix. See Proposition 4.16 in [Lang] p. 518. (Contributed by Stefan O'Rear, 16-Jul-2018.)
Hypotheses
Ref Expression
madurid.a 𝐴 = (𝑁 Mat 𝑅)
Assertion
Ref Expression
madurid ((𝑀𝐵𝑅 ∈ CRing) → (𝑀 · (𝐽𝑀)) = ((𝐷𝑀) 1 ))

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . 3 (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
2 eqid 2610 . . 3 (Base‘𝑅) = (Base‘𝑅)
3 eqid 2610 . . 3 (.r𝑅) = (.r𝑅)
4 simpr 476 . . 3 ((𝑀𝐵𝑅 ∈ CRing) → 𝑅 ∈ CRing)
5 madurid.a . . . . . 6 𝐴 = (𝑁 Mat 𝑅)
6 madurid.b . . . . . 6 𝐵 = (Base‘𝐴)
75, 6matrcl 20037 . . . . 5 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
87simpld 474 . . . 4 (𝑀𝐵𝑁 ∈ Fin)
98adantr 480 . . 3 ((𝑀𝐵𝑅 ∈ CRing) → 𝑁 ∈ Fin)
105, 2, 6matbas2i 20047 . . . 4 (𝑀𝐵𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
1110adantr 480 . . 3 ((𝑀𝐵𝑅 ∈ CRing) → 𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
12 madurid.j . . . . . . 7 𝐽 = (𝑁 maAdju 𝑅)
135, 12, 6maduf 20266 . . . . . 6 (𝑅 ∈ CRing → 𝐽:𝐵𝐵)
1413adantl 481 . . . . 5 ((𝑀𝐵𝑅 ∈ CRing) → 𝐽:𝐵𝐵)
15 simpl 472 . . . . 5 ((𝑀𝐵𝑅 ∈ CRing) → 𝑀𝐵)
1614, 15ffvelrnd 6268 . . . 4 ((𝑀𝐵𝑅 ∈ CRing) → (𝐽𝑀) ∈ 𝐵)
175, 2, 6matbas2i 20047 . . . 4 ((𝐽𝑀) ∈ 𝐵 → (𝐽𝑀) ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
1816, 17syl 17 . . 3 ((𝑀𝐵𝑅 ∈ CRing) → (𝐽𝑀) ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
191, 2, 3, 4, 9, 9, 9, 11, 18mamuval 20011 . 2 ((𝑀𝐵𝑅 ∈ CRing) → (𝑀(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)(𝐽𝑀)) = (𝑎𝑁, 𝑏𝑁 ↦ (𝑅 Σg (𝑐𝑁 ↦ ((𝑎𝑀𝑐)(.r𝑅)(𝑐(𝐽𝑀)𝑏))))))
205, 1matmulr 20063 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
218, 20sylan 487 . . . 4 ((𝑀𝐵𝑅 ∈ CRing) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
22 madurid.t . . . 4 · = (.r𝐴)
2321, 22syl6eqr 2662 . . 3 ((𝑀𝐵𝑅 ∈ CRing) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = · )
2423oveqd 6566 . 2 ((𝑀𝐵𝑅 ∈ CRing) → (𝑀(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)(𝐽𝑀)) = (𝑀 · (𝐽𝑀)))
25 madurid.d . . . . . 6 𝐷 = (𝑁 maDet 𝑅)
26 simp1l 1078 . . . . . 6 (((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) → 𝑀𝐵)
27 simp1r 1079 . . . . . 6 (((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) → 𝑅 ∈ CRing)
28 elmapi 7765 . . . . . . . . . 10 (𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
2911, 28syl 17 . . . . . . . . 9 ((𝑀𝐵𝑅 ∈ CRing) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
30293ad2ant1 1075 . . . . . . . 8 (((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
3130adantr 480 . . . . . . 7 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ 𝑐𝑁) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
32 simpl2 1058 . . . . . . 7 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ 𝑐𝑁) → 𝑎𝑁)
33 simpr 476 . . . . . . 7 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ 𝑐𝑁) → 𝑐𝑁)
3431, 32, 33fovrnd 6704 . . . . . 6 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ 𝑐𝑁) → (𝑎𝑀𝑐) ∈ (Base‘𝑅))
35 simp3 1056 . . . . . 6 (((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) → 𝑏𝑁)
365, 12, 6, 25, 3, 2, 26, 27, 34, 35madugsum 20268 . . . . 5 (((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) → (𝑅 Σg (𝑐𝑁 ↦ ((𝑎𝑀𝑐)(.r𝑅)(𝑐(𝐽𝑀)𝑏)))) = (𝐷‘(𝑑𝑁, 𝑐𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))))
37 iftrue 4042 . . . . . . . . 9 (𝑎 = 𝑏 → if(𝑎 = 𝑏, (𝐷𝑀), (0g𝑅)) = (𝐷𝑀))
3837adantl 481 . . . . . . . 8 (((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → if(𝑎 = 𝑏, (𝐷𝑀), (0g𝑅)) = (𝐷𝑀))
39 ffn 5958 . . . . . . . . . . . . 13 (𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅) → 𝑀 Fn (𝑁 × 𝑁))
4029, 39syl 17 . . . . . . . . . . . 12 ((𝑀𝐵𝑅 ∈ CRing) → 𝑀 Fn (𝑁 × 𝑁))
41 fnov 6666 . . . . . . . . . . . 12 (𝑀 Fn (𝑁 × 𝑁) ↔ 𝑀 = (𝑑𝑁, 𝑐𝑁 ↦ (𝑑𝑀𝑐)))
4240, 41sylib 207 . . . . . . . . . . 11 ((𝑀𝐵𝑅 ∈ CRing) → 𝑀 = (𝑑𝑁, 𝑐𝑁 ↦ (𝑑𝑀𝑐)))
4342adantr 480 . . . . . . . . . 10 (((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → 𝑀 = (𝑑𝑁, 𝑐𝑁 ↦ (𝑑𝑀𝑐)))
44 equtr2 1941 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑏𝑑 = 𝑏) → 𝑎 = 𝑑)
4544oveq1d 6564 . . . . . . . . . . . . . 14 ((𝑎 = 𝑏𝑑 = 𝑏) → (𝑎𝑀𝑐) = (𝑑𝑀𝑐))
4645ifeq1da 4066 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)) = if(𝑑 = 𝑏, (𝑑𝑀𝑐), (𝑑𝑀𝑐)))
47 ifid 4075 . . . . . . . . . . . . 13 if(𝑑 = 𝑏, (𝑑𝑀𝑐), (𝑑𝑀𝑐)) = (𝑑𝑀𝑐)
4846, 47syl6eq 2660 . . . . . . . . . . . 12 (𝑎 = 𝑏 → if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)) = (𝑑𝑀𝑐))
4948adantl 481 . . . . . . . . . . 11 (((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)) = (𝑑𝑀𝑐))
5049mpt2eq3dv 6619 . . . . . . . . . 10 (((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → (𝑑𝑁, 𝑐𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐))) = (𝑑𝑁, 𝑐𝑁 ↦ (𝑑𝑀𝑐)))
5143, 50eqtr4d 2647 . . . . . . . . 9 (((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → 𝑀 = (𝑑𝑁, 𝑐𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐))))
5251fveq2d 6107 . . . . . . . 8 (((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → (𝐷𝑀) = (𝐷‘(𝑑𝑁, 𝑐𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))))
5338, 52eqtr2d 2645 . . . . . . 7 (((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → (𝐷‘(𝑑𝑁, 𝑐𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) = if(𝑎 = 𝑏, (𝐷𝑀), (0g𝑅)))
54533ad2antl1 1216 . . . . . 6 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ 𝑎 = 𝑏) → (𝐷‘(𝑑𝑁, 𝑐𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) = if(𝑎 = 𝑏, (𝐷𝑀), (0g𝑅)))
55 eqid 2610 . . . . . . . 8 (0g𝑅) = (0g𝑅)
56 simpl1r 1106 . . . . . . . 8 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑅 ∈ CRing)
5793ad2ant1 1075 . . . . . . . . 9 (((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) → 𝑁 ∈ Fin)
5857adantr 480 . . . . . . . 8 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑁 ∈ Fin)
5930ad2antrr 758 . . . . . . . . 9 (((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑐𝑁) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
60 simpll2 1094 . . . . . . . . 9 (((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑐𝑁) → 𝑎𝑁)
61 simpr 476 . . . . . . . . 9 (((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑐𝑁) → 𝑐𝑁)
6259, 60, 61fovrnd 6704 . . . . . . . 8 (((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑐𝑁) → (𝑎𝑀𝑐) ∈ (Base‘𝑅))
6330adantr 480 . . . . . . . . . 10 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
6463fovrnda 6703 . . . . . . . . 9 (((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ (𝑑𝑁𝑐𝑁)) → (𝑑𝑀𝑐) ∈ (Base‘𝑅))
65643impb 1252 . . . . . . . 8 (((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑑𝑁𝑐𝑁) → (𝑑𝑀𝑐) ∈ (Base‘𝑅))
66 simpl3 1059 . . . . . . . 8 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑏𝑁)
67 simpl2 1058 . . . . . . . 8 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑎𝑁)
68 df-ne 2782 . . . . . . . . . . 11 (𝑎𝑏 ↔ ¬ 𝑎 = 𝑏)
6968biimpri 217 . . . . . . . . . 10 𝑎 = 𝑏𝑎𝑏)
7069necomd 2837 . . . . . . . . 9 𝑎 = 𝑏𝑏𝑎)
7170adantl 481 . . . . . . . 8 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑏𝑎)
7225, 2, 55, 56, 58, 62, 65, 66, 67, 71mdetralt2 20234 . . . . . . 7 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝐷‘(𝑑𝑁, 𝑐𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐))))) = (0g𝑅))
73 ifid 4075 . . . . . . . . . . 11 if(𝑑 = 𝑎, (𝑑𝑀𝑐), (𝑑𝑀𝑐)) = (𝑑𝑀𝑐)
74 oveq1 6556 . . . . . . . . . . . . 13 (𝑑 = 𝑎 → (𝑑𝑀𝑐) = (𝑎𝑀𝑐))
7574adantl 481 . . . . . . . . . . . 12 (((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑑 = 𝑎) → (𝑑𝑀𝑐) = (𝑎𝑀𝑐))
7675ifeq1da 4066 . . . . . . . . . . 11 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) → if(𝑑 = 𝑎, (𝑑𝑀𝑐), (𝑑𝑀𝑐)) = if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))
7773, 76syl5eqr 2658 . . . . . . . . . 10 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝑑𝑀𝑐) = if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))
7877ifeq2d 4055 . . . . . . . . 9 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) → if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)) = if(𝑑 = 𝑏, (𝑎𝑀𝑐), if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐))))
7978mpt2eq3dv 6619 . . . . . . . 8 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝑑𝑁, 𝑐𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐))) = (𝑑𝑁, 𝑐𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))))
8079fveq2d 6107 . . . . . . 7 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝐷‘(𝑑𝑁, 𝑐𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) = (𝐷‘(𝑑𝑁, 𝑐𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐))))))
81 iffalse 4045 . . . . . . . 8 𝑎 = 𝑏 → if(𝑎 = 𝑏, (𝐷𝑀), (0g𝑅)) = (0g𝑅))
8281adantl 481 . . . . . . 7 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) → if(𝑎 = 𝑏, (𝐷𝑀), (0g𝑅)) = (0g𝑅))
8372, 80, 823eqtr4d 2654 . . . . . 6 ((((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝐷‘(𝑑𝑁, 𝑐𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) = if(𝑎 = 𝑏, (𝐷𝑀), (0g𝑅)))
8454, 83pm2.61dan 828 . . . . 5 (((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) → (𝐷‘(𝑑𝑁, 𝑐𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) = if(𝑎 = 𝑏, (𝐷𝑀), (0g𝑅)))
8536, 84eqtrd 2644 . . . 4 (((𝑀𝐵𝑅 ∈ CRing) ∧ 𝑎𝑁𝑏𝑁) → (𝑅 Σg (𝑐𝑁 ↦ ((𝑎𝑀𝑐)(.r𝑅)(𝑐(𝐽𝑀)𝑏)))) = if(𝑎 = 𝑏, (𝐷𝑀), (0g𝑅)))
8685mpt2eq3dva 6617 . . 3 ((𝑀𝐵𝑅 ∈ CRing) → (𝑎𝑁, 𝑏𝑁 ↦ (𝑅 Σg (𝑐𝑁 ↦ ((𝑎𝑀𝑐)(.r𝑅)(𝑐(𝐽𝑀)𝑏))))) = (𝑎𝑁, 𝑏𝑁 ↦ if(𝑎 = 𝑏, (𝐷𝑀), (0g𝑅))))
87 madurid.i . . . . 5 1 = (1r𝐴)
8887oveq2i 6560 . . . 4 ((𝐷𝑀) 1 ) = ((𝐷𝑀) (1r𝐴))
89 crngring 18381 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
9089adantl 481 . . . . 5 ((𝑀𝐵𝑅 ∈ CRing) → 𝑅 ∈ Ring)
9125, 5, 6, 2mdetf 20220 . . . . . . 7 (𝑅 ∈ CRing → 𝐷:𝐵⟶(Base‘𝑅))
9291adantl 481 . . . . . 6 ((𝑀𝐵𝑅 ∈ CRing) → 𝐷:𝐵⟶(Base‘𝑅))
9392, 15ffvelrnd 6268 . . . . 5 ((𝑀𝐵𝑅 ∈ CRing) → (𝐷𝑀) ∈ (Base‘𝑅))
94 madurid.s . . . . . 6 = ( ·𝑠𝐴)
955, 2, 94, 55matsc 20075 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐷𝑀) ∈ (Base‘𝑅)) → ((𝐷𝑀) (1r𝐴)) = (𝑎𝑁, 𝑏𝑁 ↦ if(𝑎 = 𝑏, (𝐷𝑀), (0g𝑅))))
969, 90, 93, 95syl3anc 1318 . . . 4 ((𝑀𝐵𝑅 ∈ CRing) → ((𝐷𝑀) (1r𝐴)) = (𝑎𝑁, 𝑏𝑁 ↦ if(𝑎 = 𝑏, (𝐷𝑀), (0g𝑅))))
9788, 96syl5eq 2656 . . 3 ((𝑀𝐵𝑅 ∈ CRing) → ((𝐷𝑀) 1 ) = (𝑎𝑁, 𝑏𝑁 ↦ if(𝑎 = 𝑏, (𝐷𝑀), (0g𝑅))))
9886, 97eqtr4d 2647 . 2 ((𝑀𝐵𝑅 ∈ CRing) → (𝑎𝑁, 𝑏𝑁 ↦ (𝑅 Σg (𝑐𝑁 ↦ ((𝑎𝑀𝑐)(.r𝑅)(𝑐(𝐽𝑀)𝑏))))) = ((𝐷𝑀) 1 ))
9919, 24, 983eqtr3d 2652 1 ((𝑀𝐵𝑅 ∈ CRing) → (𝑀 · (𝐽𝑀)) = ((𝐷𝑀) 1 ))