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Theorem qustgphaus 24043
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 19146 . . . . . . 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 23998 . . . . . . . . 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 18924 . . . . . . . 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 7448 . . . . . . . 8 (𝐺 ~QG π‘Œ) ∈ V
1110ecelqsi 8788 . . . . . . 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 4808 . . . 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 6237 . . . . . 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 19115 . . . . . . . . . . 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 19137 . . . . . . . . . 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 19084 . . . . . . . . . 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 4011 . . . . . . . 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 4209 . . . . . . . 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 19135 . . . . . . . . . . . . 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 8771 . . . . . . . . . . 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 6904 . . . . . . . . . 10 (0gβ€˜πΊ) ∈ V
3634, 35elec 8766 . . . . . . . . 9 (π‘₯ ∈ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) ↔ (0gβ€˜πΊ)(𝐺 ~QG π‘Œ)π‘₯)
37 fvex 6904 . . . . . . . . . . 11 (0gβ€˜π») ∈ V
3837elsn2 4663 . . . . . . . . . 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 4212 . . . . . . 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 24002 . . . . . 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 17522 . . . . 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 23633 . . . . 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 17521 . . . . 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 23640 . . . 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 6895 . . 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 24042 . . . 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 24037 . . 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 3939   βŠ† wss 3940  {csn 4624   class class class wbr 5143   ↦ cmpt 5226  β—‘ccnv 5671   β€œ cima 5675  β€“ontoβ†’wfo 6540  β€˜cfv 6542  (class class class)co 7415   Er wer 8718  [cec 8719   / cqs 8720  Basecbs 17177  TopOpenctopn 17400  0gc0g 17418   qTop cqtop 17482   /s cqus 17484  Grpcgrp 18892  SubGrpcsubg 19077  NrmSGrpcnsg 19078   ~QG cqg 19079  TopOnctopon 22828  Clsdccld 22936  Hauscha 23228  TopGrpctgp 23991
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 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
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 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-tpos 8228  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-er 8721  df-ec 8723  df-qs 8727  df-map 8843  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-sup 9463  df-inf 9464  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12501  df-z 12587  df-dec 12706  df-uz 12851  df-fz 13515  df-struct 17113  df-sets 17130  df-slot 17148  df-ndx 17160  df-base 17178  df-ress 17207  df-plusg 17243  df-mulr 17244  df-sca 17246  df-vsca 17247  df-ip 17248  df-tset 17249  df-ple 17250  df-ds 17252  df-rest 17401  df-topn 17402  df-0g 17420  df-topgen 17422  df-qtop 17486  df-imas 17487  df-qus 17488  df-plusf 18596  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-grp 18895  df-minusg 18896  df-sbg 18897  df-subg 19080  df-nsg 19081  df-eqg 19082  df-oppg 19299  df-top 22812  df-topon 22829  df-topsp 22851  df-bases 22865  df-cld 22939  df-cn 23147  df-cnp 23148  df-t1 23234  df-haus 23235  df-tx 23482  df-hmeo 23675  df-tmd 23992  df-tgp 23993
This theorem is referenced by: (None)
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