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Theorem smadiadetglem2 20392
Description: Lemma 2 for smadiadetg 20393. (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 4874 . . . . 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 20132 . . . . . 6 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
6 elex 3203 . . . . . . 7 (𝑁 ∈ Fin → 𝑁 ∈ V)
76adantr 481 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝑁 ∈ V)
85, 7syl 17 . . . . 5 (𝑀𝐵𝑁 ∈ V)
983ad2ant1 1080 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝑁 ∈ V)
10 simp13 1091 . . . 4 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗𝑁) → 𝑆 ∈ (Base‘𝑅))
11 smadiadet.r . . . . . 6 𝑅 ∈ CRing
12 crngring 18474 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1311, 12mp1i 13 . . . . 5 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗𝑁) → 𝑅 ∈ Ring)
14 eqid 2626 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
15 eqid 2626 . . . . . . 7 (1r𝑅) = (1r𝑅)
1614, 15ringidcl 18484 . . . . . 6 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
17 eqid 2626 . . . . . . 7 (0g𝑅) = (0g𝑅)
1814, 17ring0cl 18485 . . . . . 6 (𝑅 ∈ Ring → (0g𝑅) ∈ (Base‘𝑅))
1916, 18ifcld 4108 . . . . 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 6709 . . . . 5 (({𝐾} × 𝑁) × {𝑆}) = (𝑖 ∈ {𝐾}, 𝑗𝑁𝑆)
2221a1i 11 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (({𝐾} × 𝑁) × {𝑆}) = (𝑖 ∈ {𝐾}, 𝑗𝑁𝑆))
23 eqidd 2627 . . . 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 7199 . . 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 18488 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r𝑅)) = 𝑆)
2825, 27mpancom 702 . . . . . . . . 9 (𝑆 ∈ (Base‘𝑅) → (𝑆 · (1r𝑅)) = 𝑆)
29283ad2ant3 1082 . . . . . . . 8 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r𝑅)) = 𝑆)
3029ad2antrl 763 . . . . . . 7 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · (1r𝑅)) = 𝑆)
31 iftrue 4069 . . . . . . . . 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 6621 . . . . . . 7 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = (𝑆 · (1r𝑅)))
34 iftrue 4069 . . . . . . . 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 2670 . . . . . 6 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
3714, 26, 17ringrz 18504 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g𝑅)) = (0g𝑅))
3825, 37mpancom 702 . . . . . . . . 9 (𝑆 ∈ (Base‘𝑅) → (𝑆 · (0g𝑅)) = (0g𝑅))
39383ad2ant3 1082 . . . . . . . 8 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g𝑅)) = (0g𝑅))
4039ad2antrl 763 . . . . . . 7 ((¬ 𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · (0g𝑅)) = (0g𝑅))
41 iffalse 4072 . . . . . . . . 9 𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)) = (0g𝑅))
4241oveq2d 6621 . . . . . . . 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 4072 . . . . . . . 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 2670 . . . . . 6 ((¬ 𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
4736, 46pm2.61ian 830 . . . . 5 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
48473adant2 1078 . . . 4 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗𝑁) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
4948mpt2eq3dva 6673 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)))) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g𝑅))))
5024, 49eqtrd 2660 . 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 1060 . . . . . 6 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝐾𝑁)
52 eqid 2626 . . . . . . 7 (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅)
533, 4, 52, 15, 17minmar1val 20368 . . . . . 6 ((𝑀𝐵𝐾𝑁𝐾𝑁) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
5451, 53syld3an3 1368 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
5554reseq1d 5359 . . . 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 4313 . . . . . 6 (𝐾𝑁 → {𝐾} ⊆ 𝑁)
57563ad2ant2 1081 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → {𝐾} ⊆ 𝑁)
58 ssid 3608 . . . . 5 𝑁𝑁
59 resmpt2 6712 . . . . 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 693 . . . 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 6708 . . . . 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 2664 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))))
6463oveq2d 6621 . 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 1057 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑀𝐵𝑆 ∈ (Base‘𝑅)))
66 eqid 2626 . . . . . 6 (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
673, 4, 66, 17marrepval 20282 . . . . 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 1321 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))))
6968reseq1d 5359 . . 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 6712 . . . 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 693 . . 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 6708 . . . 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 2664 . 2 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g𝑅))))
7550, 64, 743eqtr4rd 2671 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 384  w3a 1036   = wceq 1480  wcel 1992  Vcvv 3191  cdif 3557  wss 3560  ifcif 4063  {csn 4153   × cxp 5077  cres 5081  cfv 5850  (class class class)co 6605  cmpt2 6607  𝑓 cof 6849  Fincfn 7900  Basecbs 15776  .rcmulr 15858  0gc0g 16016  1rcur 18417  Ringcrg 18463  CRingccrg 18464   Mat cmat 20127   matRRep cmarrep 20276   maDet cmdat 20304   minMatR1 cminmar1 20353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-plusg 15870  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-grp 17341  df-mgp 18406  df-ur 18418  df-ring 18465  df-cring 18466  df-mat 20128  df-marrep 20278  df-minmar1 20355
This theorem is referenced by:  smadiadetg  20393
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