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Theorem qustgphaus 23982
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 2726 . . . . . . . 8 (0gβ€˜πΊ) = (0gβ€˜πΊ)
31, 2qus0 19115 . . . . . . 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 23937 . . . . . . . . 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 2726 . . . . . . . . 9 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
87, 2grpidcl 18895 . . . . . . . 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 7438 . . . . . . . 8 (𝐺 ~QG π‘Œ) ∈ V
1110ecelqsi 8769 . . . . . . 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 2828 . . . . 5 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ (0gβ€˜π») ∈ ((Baseβ€˜πΊ) / (𝐺 ~QG π‘Œ)))
1413snssd 4807 . . . 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 2726 . . . . . . 7 (π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ)) = (π‘₯ ∈ (Baseβ€˜πΊ) ↦ [π‘₯](𝐺 ~QG π‘Œ))
1615mptpreima 6231 . . . . . 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 19085 . . . . . . . . . . 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 2726 . . . . . . . . . . 11 (𝐺 ~QG π‘Œ) = (𝐺 ~QG π‘Œ)
207, 19, 2eqgid 19107 . . . . . . . . . 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 19054 . . . . . . . . . 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 4015 . . . . . . . 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 19105 . . . . . . . . . . . . 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 8754 . . . . . . . . . . 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 480 . . . . . . . . . 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 480 . . . . . . . . . . 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 2728 . . . . . . . . . 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 279 . . . . . . . . 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 3472 . . . . . . . . . 10 π‘₯ ∈ V
35 fvex 6898 . . . . . . . . . 10 (0gβ€˜πΊ) ∈ V
3634, 35elec 8749 . . . . . . . . 9 (π‘₯ ∈ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) ↔ (0gβ€˜πΊ)(𝐺 ~QG π‘Œ)π‘₯)
37 fvex 6898 . . . . . . . . . . 11 (0gβ€˜π») ∈ V
3837elsn2 4662 . . . . . . . . . 10 ([π‘₯](𝐺 ~QG π‘Œ) ∈ {(0gβ€˜π»)} ↔ [π‘₯](𝐺 ~QG π‘Œ) = (0gβ€˜π»))
39 eqcom 2733 . . . . . . . . . 10 ([π‘₯](𝐺 ~QG π‘Œ) = (0gβ€˜π») ↔ (0gβ€˜π») = [π‘₯](𝐺 ~QG π‘Œ))
4038, 39bitri 275 . . . . . . . . 9 ([π‘₯](𝐺 ~QG π‘Œ) ∈ {(0gβ€˜π»)} ↔ (0gβ€˜π») = [π‘₯](𝐺 ~QG π‘Œ))
4133, 36, 403bitr4g 314 . . . . . . . 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 2774 . . . . . 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 2778 . . . . 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 2827 . . . 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 23941 . . . . . 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 2727 . . . . . 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 17498 . . . . 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 23572 . . . . 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 583 . . . 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 710 . . 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 17497 . . . . 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 23579 . . . 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 6889 . . 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 2830 . 2 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ {(0gβ€˜π»)} ∈ (Clsdβ€˜πΎ))
631qustgp 23981 . . . 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 2726 . . . 4 (0gβ€˜π») = (0gβ€˜π»)
6665, 59tgphaus 23976 . . 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 257 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {crab 3426  Vcvv 3468   ∩ cin 3942   βŠ† wss 3943  {csn 4623   class class class wbr 5141   ↦ cmpt 5224  β—‘ccnv 5668   β€œ cima 5672  β€“ontoβ†’wfo 6535  β€˜cfv 6537  (class class class)co 7405   Er wer 8702  [cec 8703   / cqs 8704  Basecbs 17153  TopOpenctopn 17376  0gc0g 17394   qTop cqtop 17458   /s cqus 17460  Grpcgrp 18863  SubGrpcsubg 19047  NrmSGrpcnsg 19048   ~QG cqg 19049  TopOnctopon 22767  Clsdccld 22875  Hauscha 23167  TopGrpctgp 23930
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-tpos 8212  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-ec 8707  df-qs 8711  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13491  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-sca 17222  df-vsca 17223  df-ip 17224  df-tset 17225  df-ple 17226  df-ds 17228  df-rest 17377  df-topn 17378  df-0g 17396  df-topgen 17398  df-qtop 17462  df-imas 17463  df-qus 17464  df-plusf 18572  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867  df-sbg 18868  df-subg 19050  df-nsg 19051  df-eqg 19052  df-oppg 19262  df-top 22751  df-topon 22768  df-topsp 22790  df-bases 22804  df-cld 22878  df-cn 23086  df-cnp 23087  df-t1 23173  df-haus 23174  df-tx 23421  df-hmeo 23614  df-tmd 23931  df-tgp 23932
This theorem is referenced by: (None)
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