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 Description: Lemma 2 for smadiadetg 20701. (Contributed by AV, 14-Feb-2019.)
Hypotheses
Ref Expression
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))))

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 5057 . . . . 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 20440 . . . . . 6 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
6 elex 3352 . . . . . . 7 (𝑁 ∈ Fin → 𝑁 ∈ V)
76adantr 472 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝑁 ∈ V)
85, 7syl 17 . . . . 5 (𝑀𝐵𝑁 ∈ V)
983ad2ant1 1128 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝑁 ∈ V)
10 simp13 1248 . . . 4 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗𝑁) → 𝑆 ∈ (Base‘𝑅))
11 smadiadet.r . . . . . 6 𝑅 ∈ CRing
12 crngring 18778 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1311, 12mp1i 13 . . . . 5 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗𝑁) → 𝑅 ∈ Ring)
14 eqid 2760 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
15 eqid 2760 . . . . . . 7 (1r𝑅) = (1r𝑅)
1614, 15ringidcl 18788 . . . . . 6 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
17 eqid 2760 . . . . . . 7 (0g𝑅) = (0g𝑅)
1814, 17ring0cl 18789 . . . . . 6 (𝑅 ∈ Ring → (0g𝑅) ∈ (Base‘𝑅))
1916, 18ifcld 4275 . . . . 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 fconstmpt2 6921 . . . . 5 (({𝐾} × 𝑁) × {𝑆}) = (𝑖 ∈ {𝐾}, 𝑗𝑁𝑆)
2221a1i 11 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (({𝐾} × 𝑁) × {𝑆}) = (𝑖 ∈ {𝐾}, 𝑗𝑁𝑆))
23 eqidd 2761 . . . 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 7422 . . 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 18792 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r𝑅)) = 𝑆)
2825, 27mpancom 706 . . . . . . . . 9 (𝑆 ∈ (Base‘𝑅) → (𝑆 · (1r𝑅)) = 𝑆)
29283ad2ant3 1130 . . . . . . . 8 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r𝑅)) = 𝑆)
3029ad2antrl 766 . . . . . . 7 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · (1r𝑅)) = 𝑆)
31 iftrue 4236 . . . . . . . . 9 (𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)) = (1r𝑅))
3231adantr 472 . . . . . . . 8 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)) = (1r𝑅))
3332oveq2d 6830 . . . . . . 7 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = (𝑆 · (1r𝑅)))
34 iftrue 4236 . . . . . . . 8 (𝑗 = 𝐾 → if(𝑗 = 𝐾, 𝑆, (0g𝑅)) = 𝑆)
3534adantr 472 . . . . . . 7 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → if(𝑗 = 𝐾, 𝑆, (0g𝑅)) = 𝑆)
3630, 33, 353eqtr4d 2804 . . . . . 6 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
3714, 26, 17ringrz 18808 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g𝑅)) = (0g𝑅))
3825, 37mpancom 706 . . . . . . . . 9 (𝑆 ∈ (Base‘𝑅) → (𝑆 · (0g𝑅)) = (0g𝑅))
39383ad2ant3 1130 . . . . . . . 8 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g𝑅)) = (0g𝑅))
4039ad2antrl 766 . . . . . . 7 ((¬ 𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · (0g𝑅)) = (0g𝑅))
41 iffalse 4239 . . . . . . . . 9 𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)) = (0g𝑅))
4241oveq2d 6830 . . . . . . . 8 𝑗 = 𝐾 → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = (𝑆 · (0g𝑅)))
4342adantr 472 . . . . . . 7 ((¬ 𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = (𝑆 · (0g𝑅)))
44 iffalse 4239 . . . . . . . 8 𝑗 = 𝐾 → if(𝑗 = 𝐾, 𝑆, (0g𝑅)) = (0g𝑅))
4544adantr 472 . . . . . . 7 ((¬ 𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → if(𝑗 = 𝐾, 𝑆, (0g𝑅)) = (0g𝑅))
4640, 43, 453eqtr4d 2804 . . . . . 6 ((¬ 𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
4736, 46pm2.61ian 866 . . . . 5 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
48473adant2 1126 . . . 4 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗𝑁) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
4948mpt2eq3dva 6885 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)))) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g𝑅))))
5024, 49eqtrd 2794 . 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 1132 . . . . . 6 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝐾𝑁)
52 eqid 2760 . . . . . . 7 (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅)
533, 4, 52, 15, 17minmar1val 20676 . . . . . 6 ((𝑀𝐵𝐾𝑁𝐾𝑁) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
5451, 53syld3an3 1516 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
5554reseq1d 5550 . . . 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 4484 . . . . . 6 (𝐾𝑁 → {𝐾} ⊆ 𝑁)
57563ad2ant2 1129 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → {𝐾} ⊆ 𝑁)
58 ssid 3765 . . . . 5 𝑁𝑁
59 resmpt2 6924 . . . . 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 697 . . . 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 mpt2snif 6920 . . . . 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 2798 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))))
6463oveq2d 6830 . 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 1145 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑀𝐵𝑆 ∈ (Base‘𝑅)))
66 eqid 2760 . . . . . 6 (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
673, 4, 66, 17marrepval 20590 . . . . 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 1475 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))))
6968reseq1d 5550 . . 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 resmpt2 6924 . . . 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 697 . . 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 mpt2snif 6920 . . . 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 2798 . 2 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g𝑅))))
7550, 64, 743eqtr4rd 2805 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 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ∖ cdif 3712   ⊆ wss 3715  ifcif 4230  {csn 4321   × cxp 5264   ↾ cres 5268  ‘cfv 6049  (class class class)co 6814   ↦ cmpt2 6816   ∘𝑓 cof 7061  Fincfn 8123  Basecbs 16079  .rcmulr 16164  0gc0g 16322  1rcur 18721  Ringcrg 18767  CRingccrg 18768   Mat cmat 20435   matRRep cmarrep 20584   maDet cmdat 20612   minMatR1 cminmar1 20661 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-ndx 16082  df-slot 16083  df-base 16085  df-sets 16086  df-plusg 16176  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-grp 17646  df-mgp 18710  df-ur 18722  df-ring 18769  df-cring 18770  df-mat 20436  df-marrep 20586  df-minmar1 20663 This theorem is referenced by:  smadiadetg  20701
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