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Theorem qustgphaus 23627
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 2733 . . . . . . . 8 (0gβ€˜πΊ) = (0gβ€˜πΊ)
31, 2qus0 19068 . . . . . . 7 (π‘Œ ∈ (NrmSGrpβ€˜πΊ) β†’ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) = (0gβ€˜π»))
433ad2ant2 1135 . . . . . 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 23582 . . . . . . . . 9 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ Grp)
653ad2ant1 1134 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ 𝐺 ∈ Grp)
7 eqid 2733 . . . . . . . . 9 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
87, 2grpidcl 18850 . . . . . . . 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 7442 . . . . . . . 8 (𝐺 ~QG π‘Œ) ∈ V
1110ecelqsi 8767 . . . . . . 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 2835 . . . . 5 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ (0gβ€˜π») ∈ ((Baseβ€˜πΊ) / (𝐺 ~QG π‘Œ)))
1413snssd 4813 . . . 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 2733 . . . . . . 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 19038 . . . . . . . . . . 11 (π‘Œ ∈ (NrmSGrpβ€˜πΊ) β†’ π‘Œ ∈ (SubGrpβ€˜πΊ))
18173ad2ant2 1135 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ π‘Œ ∈ (SubGrpβ€˜πΊ))
19 eqid 2733 . . . . . . . . . . 11 (𝐺 ~QG π‘Œ) = (𝐺 ~QG π‘Œ)
207, 19, 2eqgid 19060 . . . . . . . . . 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 19007 . . . . . . . . . 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 4021 . . . . . . . 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 4216 . . . . . . . 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 19058 . . . . . . . . . . . . 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 8752 . . . . . . . . . . 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 482 . . . . . . . . . 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 482 . . . . . . . . . . 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 2735 . . . . . . . . . 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 3479 . . . . . . . . . 10 π‘₯ ∈ V
35 fvex 6905 . . . . . . . . . 10 (0gβ€˜πΊ) ∈ V
3634, 35elec 8747 . . . . . . . . 9 (π‘₯ ∈ [(0gβ€˜πΊ)](𝐺 ~QG π‘Œ) ↔ (0gβ€˜πΊ)(𝐺 ~QG π‘Œ)π‘₯)
37 fvex 6905 . . . . . . . . . . 11 (0gβ€˜π») ∈ V
3837elsn2 4668 . . . . . . . . . 10 ([π‘₯](𝐺 ~QG π‘Œ) ∈ {(0gβ€˜π»)} ↔ [π‘₯](𝐺 ~QG π‘Œ) = (0gβ€˜π»))
39 eqcom 2740 . . . . . . . . . 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 4218 . . . . . . 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 2781 . . . . . 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 2785 . . . . 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 1139 . . . . 5 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ π‘Œ ∈ (Clsdβ€˜π½))
4644, 45eqeltrd 2834 . . . 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 23586 . . . . . 6 (𝐺 ∈ TopGrp β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
49483ad2ant1 1134 . . . . 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 2734 . . . . . 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 1137 . . . . . 6 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ 𝐺 ∈ TopGrp)
5450, 51, 15, 52, 53quslem 17489 . . . . 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 23217 . . . . 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 585 . . . 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 712 . . 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 17488 . . . . 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 23224 . . . 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 6896 . . 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 2837 . 2 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ {(0gβ€˜π»)} ∈ (Clsdβ€˜πΎ))
631qustgp 23626 . . . 4 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ)) β†’ 𝐻 ∈ TopGrp)
64633adant3 1133 . . 3 ((𝐺 ∈ TopGrp ∧ π‘Œ ∈ (NrmSGrpβ€˜πΊ) ∧ π‘Œ ∈ (Clsdβ€˜π½)) β†’ 𝐻 ∈ TopGrp)
65 eqid 2733 . . . 4 (0gβ€˜π») = (0gβ€˜π»)
6665, 59tgphaus 23621 . . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475   ∩ cin 3948   βŠ† wss 3949  {csn 4629   class class class wbr 5149   ↦ cmpt 5232  β—‘ccnv 5676   β€œ cima 5680  β€“ontoβ†’wfo 6542  β€˜cfv 6544  (class class class)co 7409   Er wer 8700  [cec 8701   / cqs 8702  Basecbs 17144  TopOpenctopn 17367  0gc0g 17385   qTop cqtop 17449   /s cqus 17451  Grpcgrp 18819  SubGrpcsubg 19000  NrmSGrpcnsg 19001   ~QG cqg 19002  TopOnctopon 22412  Clsdccld 22520  Hauscha 22812  TopGrpctgp 23575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-tpos 8211  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-ec 8705  df-qs 8709  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-fz 13485  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-rest 17368  df-topn 17369  df-0g 17387  df-topgen 17389  df-qtop 17453  df-imas 17454  df-qus 17455  df-plusf 18560  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-sbg 18824  df-subg 19003  df-nsg 19004  df-eqg 19005  df-oppg 19210  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-cld 22523  df-cn 22731  df-cnp 22732  df-t1 22818  df-haus 22819  df-tx 23066  df-hmeo 23259  df-tmd 23576  df-tgp 23577
This theorem is referenced by: (None)
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