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Theorem smadiadetglem2 22021
Description: Lemma 2 for smadiadetg 22022. (Contributed by AV, 14-Feb-2019.)
Hypotheses
Ref Expression
smadiadet.a 𝐴 = (𝑁 Mat 𝑅)
smadiadet.b 𝐵 = (Base‘𝐴)
smadiadet.r 𝑅 ∈ CRing
smadiadet.d 𝐷 = (𝑁 maDet 𝑅)
smadiadet.h 𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)
smadiadetg.x · = (.r𝑅)
Assertion
Ref Expression
smadiadetglem2 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = ((({𝐾} × 𝑁) × {𝑆}) ∘f · ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁))))

Proof of Theorem smadiadetglem2
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 5388 . . . . 5 {𝐾} ∈ V
21a1i 11 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → {𝐾} ∈ V)
3 smadiadet.a . . . . . . 7 𝐴 = (𝑁 Mat 𝑅)
4 smadiadet.b . . . . . . 7 𝐵 = (Base‘𝐴)
53, 4matrcl 21759 . . . . . 6 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
6 elex 3463 . . . . . . 7 (𝑁 ∈ Fin → 𝑁 ∈ V)
76adantr 481 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝑁 ∈ V)
85, 7syl 17 . . . . 5 (𝑀𝐵𝑁 ∈ V)
983ad2ant1 1133 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝑁 ∈ V)
10 simp13 1205 . . . 4 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗𝑁) → 𝑆 ∈ (Base‘𝑅))
11 smadiadet.r . . . . . 6 𝑅 ∈ CRing
12 crngring 19976 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1311, 12mp1i 13 . . . . 5 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗𝑁) → 𝑅 ∈ Ring)
14 eqid 2736 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
15 eqid 2736 . . . . . . 7 (1r𝑅) = (1r𝑅)
1614, 15ringidcl 19989 . . . . . 6 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
17 eqid 2736 . . . . . . 7 (0g𝑅) = (0g𝑅)
1814, 17ring0cl 19990 . . . . . 6 (𝑅 ∈ Ring → (0g𝑅) ∈ (Base‘𝑅))
1916, 18ifcld 4532 . . . . 5 (𝑅 ∈ Ring → if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)) ∈ (Base‘𝑅))
2013, 19syl 17 . . . 4 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗𝑁) → if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)) ∈ (Base‘𝑅))
21 fconstmpo 7473 . . . . 5 (({𝐾} × 𝑁) × {𝑆}) = (𝑖 ∈ {𝐾}, 𝑗𝑁𝑆)
2221a1i 11 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (({𝐾} × 𝑁) × {𝑆}) = (𝑖 ∈ {𝐾}, 𝑗𝑁𝑆))
23 eqidd 2737 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))))
242, 9, 10, 20, 22, 23offval22 8020 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((({𝐾} × 𝑁) × {𝑆}) ∘f · (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)))) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)))))
2511, 12mp1i 13 . . . . . . . . . 10 (𝑆 ∈ (Base‘𝑅) → 𝑅 ∈ Ring)
26 smadiadetg.x . . . . . . . . . . 11 · = (.r𝑅)
2714, 26, 15ringridm 19993 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r𝑅)) = 𝑆)
2825, 27mpancom 686 . . . . . . . . 9 (𝑆 ∈ (Base‘𝑅) → (𝑆 · (1r𝑅)) = 𝑆)
29283ad2ant3 1135 . . . . . . . 8 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r𝑅)) = 𝑆)
3029ad2antrl 726 . . . . . . 7 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · (1r𝑅)) = 𝑆)
31 iftrue 4492 . . . . . . . . 9 (𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)) = (1r𝑅))
3231adantr 481 . . . . . . . 8 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)) = (1r𝑅))
3332oveq2d 7373 . . . . . . 7 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = (𝑆 · (1r𝑅)))
34 iftrue 4492 . . . . . . . 8 (𝑗 = 𝐾 → if(𝑗 = 𝐾, 𝑆, (0g𝑅)) = 𝑆)
3534adantr 481 . . . . . . 7 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → if(𝑗 = 𝐾, 𝑆, (0g𝑅)) = 𝑆)
3630, 33, 353eqtr4d 2786 . . . . . 6 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
3714, 26, 17ringrz 20012 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g𝑅)) = (0g𝑅))
3825, 37mpancom 686 . . . . . . . . 9 (𝑆 ∈ (Base‘𝑅) → (𝑆 · (0g𝑅)) = (0g𝑅))
39383ad2ant3 1135 . . . . . . . 8 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g𝑅)) = (0g𝑅))
4039ad2antrl 726 . . . . . . 7 ((¬ 𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · (0g𝑅)) = (0g𝑅))
41 iffalse 4495 . . . . . . . . 9 𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)) = (0g𝑅))
4241oveq2d 7373 . . . . . . . 8 𝑗 = 𝐾 → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = (𝑆 · (0g𝑅)))
4342adantr 481 . . . . . . 7 ((¬ 𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = (𝑆 · (0g𝑅)))
44 iffalse 4495 . . . . . . . 8 𝑗 = 𝐾 → if(𝑗 = 𝐾, 𝑆, (0g𝑅)) = (0g𝑅))
4544adantr 481 . . . . . . 7 ((¬ 𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → if(𝑗 = 𝐾, 𝑆, (0g𝑅)) = (0g𝑅))
4640, 43, 453eqtr4d 2786 . . . . . 6 ((¬ 𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
4736, 46pm2.61ian 810 . . . . 5 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
48473adant2 1131 . . . 4 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗𝑁) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
4948mpoeq3dva 7434 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)))) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g𝑅))))
5024, 49eqtrd 2776 . 2 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((({𝐾} × 𝑁) × {𝑆}) ∘f · (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)))) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g𝑅))))
51 simp2 1137 . . . . . 6 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝐾𝑁)
52 eqid 2736 . . . . . . 7 (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅)
533, 4, 52, 15, 17minmar1val 21997 . . . . . 6 ((𝑀𝐵𝐾𝑁𝐾𝑁) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
5451, 53syld3an3 1409 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
5554reseq1d 5936 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁)) = ((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))) ↾ ({𝐾} × 𝑁)))
56 snssi 4768 . . . . . 6 (𝐾𝑁 → {𝐾} ⊆ 𝑁)
57563ad2ant2 1134 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → {𝐾} ⊆ 𝑁)
58 ssid 3966 . . . . 5 𝑁𝑁
59 resmpo 7476 . . . . 5 (({𝐾} ⊆ 𝑁𝑁𝑁) → ((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
6057, 58, 59sylancl 586 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
61 mposnif 7472 . . . . 5 (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)))
6261a1i 11 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))))
6355, 60, 623eqtrd 2780 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))))
6463oveq2d 7373 . 2 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((({𝐾} × 𝑁) × {𝑆}) ∘f · ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁))) = ((({𝐾} × 𝑁) × {𝑆}) ∘f · (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)))))
65 3simpb 1149 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑀𝐵𝑆 ∈ (Base‘𝑅)))
66 eqid 2736 . . . . . 6 (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
673, 4, 66, 17marrepval 21911 . . . . 5 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐾𝑁)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))))
6865, 51, 51, 67syl12anc 835 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))))
6968reseq1d 5936 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = ((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))) ↾ ({𝐾} × 𝑁)))
70 resmpo 7476 . . . 4 (({𝐾} ⊆ 𝑁𝑁𝑁) → ((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))))
7157, 58, 70sylancl 586 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))))
72 mposnif 7472 . . . 4 (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
7372a1i 11 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g𝑅))))
7469, 71, 733eqtrd 2780 . 2 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g𝑅))))
7550, 64, 743eqtr4rd 2787 1 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = ((({𝐾} × 𝑁) × {𝑆}) ∘f · ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  cdif 3907  wss 3910  ifcif 4486  {csn 4586   × cxp 5631  cres 5635  cfv 6496  (class class class)co 7357  cmpo 7359  f cof 7615  Fincfn 8883  Basecbs 17083  .rcmulr 17134  0gc0g 17321  1rcur 19913  Ringcrg 19964  CRingccrg 19965   Mat cmat 21754   matRRep cmarrep 21905   maDet cmdat 21933   minMatR1 cminmar1 21982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-mat 21755  df-marrep 21907  df-minmar1 21984
This theorem is referenced by:  smadiadetg  22022
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