MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  smadiadetglem2 Structured version   Visualization version   GIF version

Theorem smadiadetglem2 20297
Description: Lemma 2 for smadiadetg 20298. (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 4835 . . . . 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 20037 . . . . . 6 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
6 elex 3185 . . . . . . 7 (𝑁 ∈ Fin → 𝑁 ∈ V)
76adantr 480 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝑁 ∈ V)
85, 7syl 17 . . . . 5 (𝑀𝐵𝑁 ∈ V)
983ad2ant1 1075 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝑁 ∈ V)
10 simp13 1086 . . . 4 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗𝑁) → 𝑆 ∈ (Base‘𝑅))
11 smadiadet.r . . . . . 6 𝑅 ∈ CRing
12 crngring 18381 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1311, 12mp1i 13 . . . . 5 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗𝑁) → 𝑅 ∈ Ring)
14 eqid 2610 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
15 eqid 2610 . . . . . . 7 (1r𝑅) = (1r𝑅)
1614, 15ringidcl 18391 . . . . . 6 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
17 eqid 2610 . . . . . . 7 (0g𝑅) = (0g𝑅)
1814, 17ring0cl 18392 . . . . . 6 (𝑅 ∈ Ring → (0g𝑅) ∈ (Base‘𝑅))
1916, 18ifcld 4081 . . . . 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 6653 . . . . 5 (({𝐾} × 𝑁) × {𝑆}) = (𝑖 ∈ {𝐾}, 𝑗𝑁𝑆)
2221a1i 11 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (({𝐾} × 𝑁) × {𝑆}) = (𝑖 ∈ {𝐾}, 𝑗𝑁𝑆))
23 eqidd 2611 . . . 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 7140 . . 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 18395 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r𝑅)) = 𝑆)
2825, 27mpancom 700 . . . . . . . . 9 (𝑆 ∈ (Base‘𝑅) → (𝑆 · (1r𝑅)) = 𝑆)
29283ad2ant3 1077 . . . . . . . 8 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑆 · (1r𝑅)) = 𝑆)
3029ad2antrl 760 . . . . . . 7 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · (1r𝑅)) = 𝑆)
31 iftrue 4042 . . . . . . . . 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 6565 . . . . . . 7 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = (𝑆 · (1r𝑅)))
34 iftrue 4042 . . . . . . . 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 2654 . . . . . 6 ((𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
3714, 26, 17ringrz 18411 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g𝑅)) = (0g𝑅))
3825, 37mpancom 700 . . . . . . . . 9 (𝑆 ∈ (Base‘𝑅) → (𝑆 · (0g𝑅)) = (0g𝑅))
39383ad2ant3 1077 . . . . . . . 8 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑆 · (0g𝑅)) = (0g𝑅))
4039ad2antrl 760 . . . . . . 7 ((¬ 𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · (0g𝑅)) = (0g𝑅))
41 iffalse 4045 . . . . . . . . 9 𝑗 = 𝐾 → if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)) = (0g𝑅))
4241oveq2d 6565 . . . . . . . 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 4045 . . . . . . . 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 2654 . . . . . 6 ((¬ 𝑗 = 𝐾 ∧ ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁)) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
4736, 46pm2.61ian 827 . . . . 5 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑗𝑁) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
48473adant2 1073 . . . 4 (((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) ∧ 𝑖 ∈ {𝐾} ∧ 𝑗𝑁) → (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐾, 𝑆, (0g𝑅)))
4948mpt2eq3dva 6617 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ (𝑆 · if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)))) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g𝑅))))
5024, 49eqtrd 2644 . 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 1055 . . . . . 6 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝐾𝑁)
52 eqid 2610 . . . . . . 7 (𝑁 minMatR1 𝑅) = (𝑁 minMatR1 𝑅)
533, 4, 52, 15, 17minmar1val 20273 . . . . . 6 ((𝑀𝐵𝐾𝑁𝐾𝑁) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
5451, 53syld3an3 1363 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
5554reseq1d 5316 . . . 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 4280 . . . . . 6 (𝐾𝑁 → {𝐾} ⊆ 𝑁)
57563ad2ant2 1076 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → {𝐾} ⊆ 𝑁)
58 ssid 3587 . . . . 5 𝑁𝑁
59 resmpt2 6656 . . . . 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 6652 . . . . 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 2648 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (1r𝑅), (0g𝑅))))
6463oveq2d 6565 . 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 1052 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑀𝐵𝑆 ∈ (Base‘𝑅)))
66 eqid 2610 . . . . . 6 (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
673, 4, 66, 17marrepval 20187 . . . . 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 1316 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))))
6968reseq1d 5316 . . 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 6656 . . . 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 6652 . . . 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 2648 . 2 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = (𝑖 ∈ {𝐾}, 𝑗𝑁 ↦ if(𝑗 = 𝐾, 𝑆, (0g𝑅))))
7550, 64, 743eqtr4rd 2655 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 1031   = wceq 1475  wcel 1977  Vcvv 3173  cdif 3537  wss 3540  ifcif 4036  {csn 4125   × cxp 5036  cres 5040  cfv 5804  (class class class)co 6549  cmpt2 6551  𝑓 cof 6793  Fincfn 7841  Basecbs 15695  .rcmulr 15769  0gc0g 15923  1rcur 18324  Ringcrg 18370  CRingccrg 18371   Mat cmat 20032   matRRep cmarrep 20181   maDet cmdat 20209   minMatR1 cminmar1 20258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-mat 20033  df-marrep 20183  df-minmar1 20260
This theorem is referenced by:  smadiadetg  20298
  Copyright terms: Public domain W3C validator