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 Description: Creating the adjunct of matrices is a function from the set of matrices into the set of matrices. (Contributed by Stefan O'Rear, 11-Jul-2018.)
Hypotheses
Ref Expression
maduf.a 𝐴 = (𝑁 Mat 𝑅)
Assertion
Ref Expression
maduf (𝑅 ∈ CRing → 𝐽:𝐵𝐵)

Dummy variables 𝑖 𝑗 𝑘 𝑙 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 maduf.a . . 3 𝐴 = (𝑁 Mat 𝑅)
2 eqid 2760 . . 3 (Base‘𝑅) = (Base‘𝑅)
3 maduf.b . . 3 𝐵 = (Base‘𝐴)
41, 3matrcl 20440 . . . . 5 (𝑚𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
54adantl 473 . . . 4 ((𝑅 ∈ CRing ∧ 𝑚𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
65simpld 477 . . 3 ((𝑅 ∈ CRing ∧ 𝑚𝐵) → 𝑁 ∈ Fin)
7 simpl 474 . . 3 ((𝑅 ∈ CRing ∧ 𝑚𝐵) → 𝑅 ∈ CRing)
8 eqid 2760 . . . . . . 7 (𝑁 maDet 𝑅) = (𝑁 maDet 𝑅)
98, 1, 3, 2mdetf 20623 . . . . . 6 (𝑅 ∈ CRing → (𝑁 maDet 𝑅):𝐵⟶(Base‘𝑅))
109adantr 472 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑚𝐵) → (𝑁 maDet 𝑅):𝐵⟶(Base‘𝑅))
11103ad2ant1 1128 . . . 4 (((𝑅 ∈ CRing ∧ 𝑚𝐵) ∧ 𝑖𝑁𝑗𝑁) → (𝑁 maDet 𝑅):𝐵⟶(Base‘𝑅))
1263ad2ant1 1128 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑚𝐵) ∧ 𝑖𝑁𝑗𝑁) → 𝑁 ∈ Fin)
13 simp1l 1240 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑚𝐵) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ CRing)
14 simp11l 1369 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝑚𝐵) ∧ 𝑖𝑁𝑗𝑁) ∧ 𝑘𝑁𝑙𝑁) → 𝑅 ∈ CRing)
15 crngring 18778 . . . . . . 7 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
16 eqid 2760 . . . . . . . . 9 (1r𝑅) = (1r𝑅)
172, 16ringidcl 18788 . . . . . . . 8 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
18 eqid 2760 . . . . . . . . 9 (0g𝑅) = (0g𝑅)
192, 18ring0cl 18789 . . . . . . . 8 (𝑅 ∈ Ring → (0g𝑅) ∈ (Base‘𝑅))
2017, 19ifcld 4275 . . . . . . 7 (𝑅 ∈ Ring → if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)) ∈ (Base‘𝑅))
2114, 15, 203syl 18 . . . . . 6 ((((𝑅 ∈ CRing ∧ 𝑚𝐵) ∧ 𝑖𝑁𝑗𝑁) ∧ 𝑘𝑁𝑙𝑁) → if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)) ∈ (Base‘𝑅))
22 simp2 1132 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝑚𝐵) ∧ 𝑖𝑁𝑗𝑁) ∧ 𝑘𝑁𝑙𝑁) → 𝑘𝑁)
23 simp3 1133 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝑚𝐵) ∧ 𝑖𝑁𝑗𝑁) ∧ 𝑘𝑁𝑙𝑁) → 𝑙𝑁)
24 simp11r 1370 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝑚𝐵) ∧ 𝑖𝑁𝑗𝑁) ∧ 𝑘𝑁𝑙𝑁) → 𝑚𝐵)
251, 2, 3, 22, 23, 24matecld 20454 . . . . . 6 ((((𝑅 ∈ CRing ∧ 𝑚𝐵) ∧ 𝑖𝑁𝑗𝑁) ∧ 𝑘𝑁𝑙𝑁) → (𝑘𝑚𝑙) ∈ (Base‘𝑅))
2621, 25ifcld 4275 . . . . 5 ((((𝑅 ∈ CRing ∧ 𝑚𝐵) ∧ 𝑖𝑁𝑗𝑁) ∧ 𝑘𝑁𝑙𝑁) → if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙)) ∈ (Base‘𝑅))
271, 2, 3, 12, 13, 26matbas2d 20451 . . . 4 (((𝑅 ∈ CRing ∧ 𝑚𝐵) ∧ 𝑖𝑁𝑗𝑁) → (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙))) ∈ 𝐵)
2811, 27ffvelrnd 6524 . . 3 (((𝑅 ∈ CRing ∧ 𝑚𝐵) ∧ 𝑖𝑁𝑗𝑁) → ((𝑁 maDet 𝑅)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙)))) ∈ (Base‘𝑅))
291, 2, 3, 6, 7, 28matbas2d 20451 . 2 ((𝑅 ∈ CRing ∧ 𝑚𝐵) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑁 maDet 𝑅)‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑅), (0g𝑅)), (𝑘𝑚𝑙))))) ∈ 𝐵)