MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  qustgphaus Structured version   Visualization version   GIF version
Theorem qustgphaus 24040
Description: The quotient of a topological group by a closed normal subgroup is a Hausdorff topological group. In particular, the quotient by the closure of the identity is a Hausdorff topological group, isomorphic to both the Kolmogorov quotient and the Hausdorff quotient operations on topological spaces (because T0 and Hausdorff coincide for topological groups). (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
qustgp.h 𝐻 = (𝐺 /s (𝐺 ~QG π‘Œ))
qustgphaus.j 𝐽 = (TopOpenβ€˜πΊ)
qustgphaus.k 𝐾 = (TopOpenβ€˜π»)
Assertion
Ref Expression
qustgphaus ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ 𝐾 ∈ Haus)
Proof of Theorem qustgphaus
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 qustgp.h . . . . . . . 8 𝐻 = (𝐺 /s (𝐺 ~QG π‘Œ))
2 eqid 2725 . . . . . . . 8 (0gβ€˜πΊ) = (0gβ€˜πΊ)
31, 2qus0 19143 . . . . . . 7 (π‘Œ ∈ (NrmSGrpβ€˜πΊ) β†’ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) = (0gβ€˜π»))
433ad2ant2 1131 . . . . . 6 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) = (0gβ€˜π»))
5 tgpgrp 23995 . . . . . . . . 9 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ Grp)
653ad2ant1 1130 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ 𝐺 ∈ Grp)
7 eqid 2725 . . . . . . . . 9 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
87, 2grpidcl 18921 . . . . . . . 8 (𝐺 ∈ Grp β†’ (0gβ€˜πΊ) ∈ (Baseβ€˜πΊ))
96, 8syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ (0gβ€˜πΊ) ∈ (Baseβ€˜πΊ))
10 ovex 7446 . . . . . . . 8 (𝐺 ~QG π‘Œ) ∈ V
1110ecelqsi 8785 . . . . . . 7 ((0gβ€˜πΊ) ∈ (Baseβ€˜πΊ) β†’ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) ∈ ((Baseβ€˜πΊ) / (𝐺 ~QG π‘Œ)))
129, 11syl 17 . . . . . 6 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) ∈ ((Baseβ€˜πΊ) / (𝐺 ~QG π‘Œ)))
134, 12eqeltrrd 2826 . . . . 5 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ (0gβ€˜π») ∈ ((Baseβ€˜πΊ) / (𝐺 ~QG π‘Œ)))
1413snssd 4809 . . . 4 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ {(0gβ€˜π»)} βŠ† ((Baseβ€˜πΊ) / (𝐺 ~QG π‘Œ)))
15 eqid 2725 . . . . . . 7 (π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ)) = (π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ))
1615mptpreima 6238 . . . . . 6 (β—‘(π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ)) β€œ {(0gβ€˜π»)}) = {π‘₯ ∈ (Baseβ€˜πΊ) ∣ [π‘₯](𝐺 ~QG π‘Œ) ∈ {(0gβ€˜π»)}}
17 nsgsubg 19112 . . . . . . . . . . 11 (π‘Œ ∈ (NrmSGrpβ€˜πΊ) β†’ π‘Œ ∈ (SubGrpβ€˜πΊ))
18173ad2ant2 1131 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ π‘Œ ∈ (SubGrpβ€˜πΊ))
19 eqid 2725 . . . . . . . . . . 11 (𝐺 ~QG π‘Œ) = (𝐺 ~QG π‘Œ)
207, 19, 2eqgid 19134 . . . . . . . . . 10 (π‘Œ ∈ (SubGrpβ€˜πΊ) β†’ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) = π‘Œ)
2118, 20syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) = π‘Œ)
227subgss 19081 . . . . . . . . . 10 (π‘Œ ∈ (SubGrpβ€˜πΊ) β†’ π‘Œ βŠ† (Baseβ€˜πΊ))
2318, 22syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ π‘Œ βŠ† (Baseβ€˜πΊ))
2421, 23eqsstrd 4012 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) βŠ† (Baseβ€˜πΊ))
25 sseqin2 4210 . . . . . . . 8 ([(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) βŠ† (Baseβ€˜πΊ) ↔ ((Baseβ€˜πΊ) ∩ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ)) = [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ))
2624, 25sylib 217 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ ((Baseβ€˜πΊ) ∩ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ)) = [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ))
277, 19eqger 19132 . . . . . . . . . . . . 13 (π‘Œ ∈ (SubGrpβ€˜πΊ) β†’ (𝐺 ~QG π‘Œ) Er (Baseβ€˜πΊ))
2818, 27syl 17 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ (𝐺 ~QG π‘Œ) Er (Baseβ€˜πΊ))
2928, 9erth 8768 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ ((0gβ€˜πΊ)(𝐺 ~QG π‘Œ)π‘₯ ↔ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) = [π‘₯](𝐺 ~QG π‘Œ)))
3029adantr 479 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((0gβ€˜πΊ)(𝐺 ~QG π‘Œ)π‘₯ ↔ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) = [π‘₯](𝐺 ~QG π‘Œ)))
314adantr 479 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) = (0gβ€˜π»))
3231eqeq1d 2727 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ([(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) = [π‘₯](𝐺 ~QG π‘Œ) ↔ (0gβ€˜π») = [π‘₯](𝐺 ~QG π‘Œ)))
3330, 32bitrd 278 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((0gβ€˜πΊ)(𝐺 ~QG π‘Œ)π‘₯ ↔ (0gβ€˜π») = [π‘₯](𝐺 ~QG π‘Œ)))
34 vex 3467 . . . . . . . . . 10 π‘₯ ∈ V
35 fvex 6903 . . . . . . . . . 10 (0gβ€˜πΊ) ∈ V
3634, 35elec 8763 . . . . . . . . 9 (π‘₯ ∈ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) ↔ (0gβ€˜πΊ)(𝐺 ~QG π‘Œ)π‘₯)
37 fvex 6903 . . . . . . . . . . 11 (0gβ€˜π») ∈ V
3837elsn2 4664 . . . . . . . . . 10 ([π‘₯](𝐺 ~QG π‘Œ) ∈ {(0gβ€˜π»)} ↔ [π‘₯](𝐺 ~QG π‘Œ) = (0gβ€˜π»))
39 eqcom 2732 . . . . . . . . . 10 ([π‘₯](𝐺 ~QG π‘Œ) = (0gβ€˜π») ↔ (0gβ€˜π») = [π‘₯](𝐺 ~QG π‘Œ))
4038, 39bitri 274 . . . . . . . . 9 ([π‘₯](𝐺 ~QG π‘Œ) ∈ {(0gβ€˜π»)} ↔ (0gβ€˜π») = [π‘₯](𝐺 ~QG π‘Œ))
4133, 36, 403bitr4g 313 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ (π‘₯ ∈ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) ↔ [π‘₯](𝐺 ~QG π‘Œ) ∈ {(0gβ€˜π»)}))
4241rabbi2dva 4213 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ ((Baseβ€˜πΊ) ∩ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ)) = {π‘₯ ∈ (Baseβ€˜πΊ) ∣ [π‘₯](𝐺 ~QG π‘Œ) ∈ {(0gβ€˜π»)}})
4326, 42, 213eqtr3d 2773 . . . . . 6 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ {π‘₯ ∈ (Baseβ€˜πΊ) ∣ [π‘₯](𝐺 ~QG π‘Œ) ∈ {(0gβ€˜π»)}} = π‘Œ)
4416, 43eqtrid 2777 . . . . 5 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ (β—‘(π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ)) β€œ {(0gβ€˜π»)}) = π‘Œ)
45 simp3 1135 . . . . 5 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ π‘Œ ∈ (Clsdβ€˜π½))
4644, 45eqeltrd 2825 . . . 4 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ (β—‘(π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ)) β€œ {(0gβ€˜π»)}) ∈ (Clsdβ€˜π½))
47 qustgphaus.j . . . . . . 7 𝐽 = (TopOpenβ€˜πΊ)
4847, 7tgptopon 23999 . . . . . 6 (𝐺 ∈ TopGrp β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
49483ad2ant1 1130 . . . . 5 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
501a1i 11 . . . . . 6 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ 𝐻 = (𝐺 /s (𝐺 ~QG π‘Œ)))
51 eqidd 2726 . . . . . 6 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ (Baseβ€˜πΊ) = (Baseβ€˜πΊ))
5210a1i 11 . . . . . 6 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ (𝐺 ~QG π‘Œ) ∈ V)
53 simp1 1133 . . . . . 6 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ 𝐺 ∈ TopGrp)
5450, 51, 15, 52, 53quslem 17519 . . . . 5 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ (π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ)):(Baseβ€˜πΊ)–ontoβ†’((Baseβ€˜πΊ) / (𝐺 ~QG π‘Œ)))
55 qtopcld 23630 . . . . 5 ((𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)) ∧ (π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ)):(Baseβ€˜πΊ)–ontoβ†’((Baseβ€˜πΊ) / (𝐺 ~QG π‘Œ))) β†’ ({(0gβ€˜π»)} ∈ (Clsdβ€˜(𝐽 qTop (π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ)))) ↔ ({(0gβ€˜π»)} βŠ† ((Baseβ€˜πΊ) / (𝐺 ~QG π‘Œ)) ∧ (β—‘(π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ)) β€œ {(0gβ€˜π»)}) ∈ (Clsdβ€˜π½))))
5649, 54, 55syl2anc 582 . . . 4 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ ({(0gβ€˜π»)} ∈ (Clsdβ€˜(𝐽 qTop (π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ)))) ↔ ({(0gβ€˜π»)} βŠ† ((Baseβ€˜πΊ) / (𝐺 ~QG π‘Œ)) ∧ (β—‘(π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ)) β€œ {(0gβ€˜π»)}) ∈ (Clsdβ€˜π½))))
5714, 46, 56mpbir2and 711 . . 3 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ {(0gβ€˜π»)} ∈ (Clsdβ€˜(𝐽 qTop (π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ)))))
5850, 51, 15, 52, 53qusval 17518 . . . . 5 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ 𝐻 = ((π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ)) β€œs 𝐺))
59 qustgphaus.k . . . . 5 𝐾 = (TopOpenβ€˜π»)
6058, 51, 54, 53, 47, 59imastopn 23637 . . . 4 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ 𝐾 = (𝐽 qTop (π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ))))
6160fveq2d 6894 . . 3 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ (Clsdβ€˜πΎ) = (Clsdβ€˜(𝐽 qTop (π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ)))))
6257, 61eleqtrrd 2828 . 2 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ {(0gβ€˜π»)} ∈ (Clsdβ€˜πΎ))
631qustgp 24039 . . . 4 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ)) β†’ 𝐻 ∈ TopGrp)
64633adant3 1129 . . 3 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ 𝐻 ∈ TopGrp)
65 eqid 2725 . . . 4 (0gβ€˜π») = (0gβ€˜π»)
6665, 59tgphaus 24034 . . 3 (𝐻 ∈ TopGrp β†’ (𝐾 ∈ Haus ↔ {(0gβ€˜π»)} ∈ (Clsdβ€˜πΎ)))
6764, 66syl 17 . 2 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ (𝐾 ∈ Haus ↔ {(0gβ€˜π»)} ∈ (Clsdβ€˜πΎ)))
6862, 67mpbird 256 1 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ 𝐾 ∈ Haus)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {crab 3419  Vcvv 3463   ∩ cin 3940   βŠ† wss 3941  {csn 4625   class class class wbr 5144   ↦ cmpt 5227  β—‘ccnv 5672   β€œ cima 5676  β€“ontoβ†’wfo 6541  β€˜cfv 6543  (class class class)co 7413   Er wer 8715  [cec 8716   / cqs 8717  Basecbs 17174  TopOpenctopn 17397  0gc0g 17415   qTop cqtop 17479   /s cqus 17481  Grpcgrp 18889  SubGrpcsubg 19074  NrmSGrpcnsg 19075   ~QG cqg 19076  TopOnctopon 22825  Clsdccld 22933  Hauscha 23225  TopGrpctgp 23988
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 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-ec 8720  df-qs 8724  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9460  df-inf 9461  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-z 12584  df-dec 12703  df-uz 12848  df-fz 13512  df-struct 17110  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-ress 17204  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-rest 17398  df-topn 17399  df-0g 17417  df-topgen 17419  df-qtop 17483  df-imas 17484  df-qus 17485  df-plusf 18593  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-grp 18892  df-minusg 18893  df-sbg 18894  df-subg 19077  df-nsg 19078  df-eqg 19079  df-oppg 19296  df-top 22809  df-topon 22826  df-topsp 22848  df-bases 22862  df-cld 22936  df-cn 23144  df-cnp 23145  df-t1 23231  df-haus 23232  df-tx 23479  df-hmeo 23672  df-tmd 23989  df-tgp 23990
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator