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Theorem smadiadetglem2 22498
Description: Lemma 2 for smadiadetg 22499. (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 5422 . . . . 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 22236 . . . . . 6 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
6 elex 3485 . . . . . . 7 (𝑁 ∈ Fin → 𝑁 ∈ V)
76adantr 480 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝑁 ∈ V)
85, 7syl 17 . . . . 5 (𝑀𝐵𝑁 ∈ V)
983ad2ant1 1130 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝑁 ∈ V)
10 simp13 1202 . . . 4 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗𝑁) → 𝑆 ∈ (Base‘𝑅))
11 smadiadet.r . . . . . 6 𝑅 ∈ CRing
12 crngring 20142 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1311, 12mp1i 13 . . . . 5 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗𝑁) → 𝑅 ∈ Ring)
14 eqid 2724 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
15 eqid 2724 . . . . . . 7 (1r𝑅) = (1r𝑅)
1614, 15ringidcl 20157 . . . . . 6 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
17 eqid 2724 . . . . . . 7 (0g𝑅) = (0g𝑅)
1814, 17ring0cl 20158 . . . . . 6 (𝑅 ∈ Ring → (0g𝑅) ∈ (Base‘𝑅))
1916, 18ifcld 4567 . . . . 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 7518 . . . . 5 (({𝐾} × 𝑁) × {𝑆}) = (𝑖 ∈ {𝐾}, 𝑗𝑁𝑆)
2221a1i 11 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (({𝐾} × 𝑁) × {𝑆}) = (𝑖 ∈ {𝐾}, 𝑗𝑁𝑆))
23 eqidd 2725 . . . 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 8069 . . 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 20161 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r𝑅)) = 𝑆)
2825, 27mpancom 685 . . . . . . . . 9 (𝑆 ∈ (Base‘𝑅) → (𝑆 · (1r𝑅)) = 𝑆)
29283ad2ant3 1132 . . . . . . . 8 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r𝑅)) = 𝑆)
3029ad2antrl 725 . . . . . . 7 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · (1r𝑅)) = 𝑆)
31 iftrue 4527 . . . . . . . . 9 (𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)) = (1r𝑅))
3231adantr 480 . . . . . . . 8 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)) = (1r𝑅))
3332oveq2d 7418 . . . . . . 7 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = (𝑆 · (1r𝑅)))
34 iftrue 4527 . . . . . . . 8 (𝑗 = 𝐾 → if(𝑗 = 𝐾, 𝑆, (0g𝑅)) = 𝑆)
3534adantr 480 . . . . . . 7 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → if(𝑗 = 𝐾, 𝑆, (0g𝑅)) = 𝑆)
3630, 33, 353eqtr4d 2774 . . . . . 6 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
3714, 26, 17ringrz 20185 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g𝑅)) = (0g𝑅))
3825, 37mpancom 685 . . . . . . . . 9 (𝑆 ∈ (Base‘𝑅) → (𝑆 · (0g𝑅)) = (0g𝑅))
39383ad2ant3 1132 . . . . . . . 8 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g𝑅)) = (0g𝑅))
4039ad2antrl 725 . . . . . . 7 ((¬ 𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · (0g𝑅)) = (0g𝑅))
41 iffalse 4530 . . . . . . . . 9 𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)) = (0g𝑅))
4241oveq2d 7418 . . . . . . . 8 𝑗 = 𝐾 → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = (𝑆 · (0g𝑅)))
4342adantr 480 . . . . . . 7 ((¬ 𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = (𝑆 · (0g𝑅)))
44 iffalse 4530 . . . . . . . 8 𝑗 = 𝐾 → if(𝑗 = 𝐾, 𝑆, (0g𝑅)) = (0g𝑅))
4544adantr 480 . . . . . . 7 ((¬ 𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → if(𝑗 = 𝐾, 𝑆, (0g𝑅)) = (0g𝑅))
4640, 43, 453eqtr4d 2774 . . . . . 6 ((¬ 𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
4736, 46pm2.61ian 809 . . . . 5 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
48473adant2 1128 . . . 4 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗𝑁) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
4948mpoeq3dva 7479 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)))) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g𝑅))))
5024, 49eqtrd 2764 . 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 1134 . . . . . 6 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝐾𝑁)
52 eqid 2724 . . . . . . 7 (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅)
533, 4, 52, 15, 17minmar1val 22474 . . . . . 6 ((𝑀𝐵𝐾𝑁𝐾𝑁) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
5451, 53syld3an3 1406 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
5554reseq1d 5971 . . . 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 4804 . . . . . 6 (𝐾𝑁 → {𝐾} ⊆ 𝑁)
57563ad2ant2 1131 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → {𝐾} ⊆ 𝑁)
58 ssid 3997 . . . . 5 𝑁𝑁
59 resmpo 7521 . . . . 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 585 . . . 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 7517 . . . . 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 2768 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))))
6463oveq2d 7418 . 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 1146 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑀𝐵𝑆 ∈ (Base‘𝑅)))
66 eqid 2724 . . . . . 6 (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
673, 4, 66, 17marrepval 22388 . . . . 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 834 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))))
6968reseq1d 5971 . . 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 7521 . . . 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 585 . . 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 7517 . . . 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 2768 . 2 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g𝑅))))
7550, 64, 743eqtr4rd 2775 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 395  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3466  cdif 3938  wss 3941  ifcif 4521  {csn 4621   × cxp 5665  cres 5669  cfv 6534  (class class class)co 7402  cmpo 7404  f cof 7662  Fincfn 8936  Basecbs 17145  .rcmulr 17199  0gc0g 17386  1rcur 20078  Ringcrg 20130  CRingccrg 20131   Mat cmat 22231   matRRep cmarrep 22382   maDet cmdat 22410   minMatR1 cminmar1 22459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-of 7664  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11444  df-neg 11445  df-nn 12211  df-2 12273  df-sets 17098  df-slot 17116  df-ndx 17128  df-base 17146  df-plusg 17211  df-0g 17388  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-cmn 19694  df-abl 19695  df-mgp 20032  df-rng 20050  df-ur 20079  df-ring 20132  df-cring 20133  df-mat 22232  df-marrep 22384  df-minmar1 22461
This theorem is referenced by:  smadiadetg  22499
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