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Theorem opnsubg 23832
Description: An open subgroup of a topological group is also closed. (Contributed by Mario Carneiro, 17-Sep-2015.)
Hypothesis
Ref Expression
subgntr.h 𝐽 = (TopOpenβ€˜πΊ)
Assertion
Ref Expression
opnsubg ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) β†’ 𝑆 ∈ (Clsdβ€˜π½))
Proof of Theorem opnsubg
Dummy variables π‘₯ 𝑒 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2730 . . . . 5 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
21subgss 19043 . . . 4 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ 𝑆 βŠ† (Baseβ€˜πΊ))
323ad2ant2 1132 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) β†’ 𝑆 βŠ† (Baseβ€˜πΊ))
4 subgntr.h . . . . . 6 𝐽 = (TopOpenβ€˜πΊ)
54, 1tgptopon 23806 . . . . 5 (𝐺 ∈ TopGrp β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
653ad2ant1 1131 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
7 toponuni 22636 . . . 4 (𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)) β†’ (Baseβ€˜πΊ) = βˆͺ 𝐽)
86, 7syl 17 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) β†’ (Baseβ€˜πΊ) = βˆͺ 𝐽)
93, 8sseqtrd 4021 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) β†’ 𝑆 βŠ† βˆͺ 𝐽)
108difeq1d 4120 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) β†’ ((Baseβ€˜πΊ) βˆ– 𝑆) = (βˆͺ 𝐽 βˆ– 𝑆))
11 df-ima 5688 . . . . . . . 8 ((𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)𝑦)) β€œ 𝑆) = ran ((𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)𝑦)) β†Ύ 𝑆)
123adantr 479 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ 𝑆 βŠ† (Baseβ€˜πΊ))
1312resmptd 6039 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)𝑦)) β†Ύ 𝑆) = (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)))
1413rneqd 5936 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ ran ((𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)𝑦)) β†Ύ 𝑆) = ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)))
1511, 14eqtrid 2782 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)𝑦)) β€œ 𝑆) = ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)))
16 simpl1 1189 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ 𝐺 ∈ TopGrp)
17 eldifi 4125 . . . . . . . . . 10 (π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆) β†’ π‘₯ ∈ (Baseβ€˜πΊ))
1817adantl 480 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ π‘₯ ∈ (Baseβ€˜πΊ))
19 eqid 2730 . . . . . . . . . 10 (𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)𝑦)) = (𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)𝑦))
20 eqid 2730 . . . . . . . . . 10 (+gβ€˜πΊ) = (+gβ€˜πΊ)
2119, 1, 20, 4tgplacthmeo 23827 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ (𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)𝑦)) ∈ (𝐽Homeo𝐽))
2216, 18, 21syl2anc 582 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ (𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)𝑦)) ∈ (𝐽Homeo𝐽))
23 simpl3 1191 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ 𝑆 ∈ 𝐽)
24 hmeoima 23489 . . . . . . . 8 (((𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)𝑦)) ∈ (𝐽Homeo𝐽) ∧ 𝑆 ∈ 𝐽) β†’ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)𝑦)) β€œ 𝑆) ∈ 𝐽)
2522, 23, 24syl2anc 582 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)𝑦)) β€œ 𝑆) ∈ 𝐽)
2615, 25eqeltrrd 2832 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)) ∈ 𝐽)
27 tgpgrp 23802 . . . . . . . . 9 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ Grp)
2816, 27syl 17 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ 𝐺 ∈ Grp)
29 eqid 2730 . . . . . . . . 9 (0gβ€˜πΊ) = (0gβ€˜πΊ)
301, 20, 29grprid 18889 . . . . . . . 8 ((𝐺 ∈ Grp ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ (π‘₯(+gβ€˜πΊ)(0gβ€˜πΊ)) = π‘₯)
3128, 18, 30syl2anc 582 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ (π‘₯(+gβ€˜πΊ)(0gβ€˜πΊ)) = π‘₯)
32 simpl2 1190 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))
3329subg0cl 19050 . . . . . . . . 9 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ (0gβ€˜πΊ) ∈ 𝑆)
3432, 33syl 17 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ (0gβ€˜πΊ) ∈ 𝑆)
35 ovex 7444 . . . . . . . 8 (π‘₯(+gβ€˜πΊ)(0gβ€˜πΊ)) ∈ V
36 eqid 2730 . . . . . . . . 9 (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)) = (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦))
37 oveq2 7419 . . . . . . . . 9 (𝑦 = (0gβ€˜πΊ) β†’ (π‘₯(+gβ€˜πΊ)𝑦) = (π‘₯(+gβ€˜πΊ)(0gβ€˜πΊ)))
3836, 37elrnmpt1s 5955 . . . . . . . 8 (((0gβ€˜πΊ) ∈ 𝑆 ∧ (π‘₯(+gβ€˜πΊ)(0gβ€˜πΊ)) ∈ V) β†’ (π‘₯(+gβ€˜πΊ)(0gβ€˜πΊ)) ∈ ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)))
3934, 35, 38sylancl 584 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ (π‘₯(+gβ€˜πΊ)(0gβ€˜πΊ)) ∈ ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)))
4031, 39eqeltrrd 2832 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ π‘₯ ∈ ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)))
4128adantr 479 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) ∧ 𝑦 ∈ 𝑆) β†’ 𝐺 ∈ Grp)
4218adantr 479 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) ∧ 𝑦 ∈ 𝑆) β†’ π‘₯ ∈ (Baseβ€˜πΊ))
4312sselda 3981 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) ∧ 𝑦 ∈ 𝑆) β†’ 𝑦 ∈ (Baseβ€˜πΊ))
441, 20grpcl 18863 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ π‘₯ ∈ (Baseβ€˜πΊ) ∧ 𝑦 ∈ (Baseβ€˜πΊ)) β†’ (π‘₯(+gβ€˜πΊ)𝑦) ∈ (Baseβ€˜πΊ))
4541, 42, 43, 44syl3anc 1369 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) ∧ 𝑦 ∈ 𝑆) β†’ (π‘₯(+gβ€˜πΊ)𝑦) ∈ (Baseβ€˜πΊ))
46 eldifn 4126 . . . . . . . . . . 11 (π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆) β†’ Β¬ π‘₯ ∈ 𝑆)
4746ad2antlr 723 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) ∧ 𝑦 ∈ 𝑆) β†’ Β¬ π‘₯ ∈ 𝑆)
48 eqid 2730 . . . . . . . . . . . . . . 15 (-gβ€˜πΊ) = (-gβ€˜πΊ)
4948subgsubcl 19053 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (π‘₯(+gβ€˜πΊ)𝑦) ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) β†’ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)𝑦) ∈ 𝑆)
50493com23 1124 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑦 ∈ 𝑆 ∧ (π‘₯(+gβ€˜πΊ)𝑦) ∈ 𝑆) β†’ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)𝑦) ∈ 𝑆)
51503expia 1119 . . . . . . . . . . . 12 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑦 ∈ 𝑆) β†’ ((π‘₯(+gβ€˜πΊ)𝑦) ∈ 𝑆 β†’ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)𝑦) ∈ 𝑆))
5232, 51sylan 578 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) ∧ 𝑦 ∈ 𝑆) β†’ ((π‘₯(+gβ€˜πΊ)𝑦) ∈ 𝑆 β†’ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)𝑦) ∈ 𝑆))
531, 20, 48grppncan 18950 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ π‘₯ ∈ (Baseβ€˜πΊ) ∧ 𝑦 ∈ (Baseβ€˜πΊ)) β†’ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)𝑦) = π‘₯)
5441, 42, 43, 53syl3anc 1369 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) ∧ 𝑦 ∈ 𝑆) β†’ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)𝑦) = π‘₯)
5554eleq1d 2816 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) ∧ 𝑦 ∈ 𝑆) β†’ (((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)𝑦) ∈ 𝑆 ↔ π‘₯ ∈ 𝑆))
5652, 55sylibd 238 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) ∧ 𝑦 ∈ 𝑆) β†’ ((π‘₯(+gβ€˜πΊ)𝑦) ∈ 𝑆 β†’ π‘₯ ∈ 𝑆))
5747, 56mtod 197 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) ∧ 𝑦 ∈ 𝑆) β†’ Β¬ (π‘₯(+gβ€˜πΊ)𝑦) ∈ 𝑆)
5845, 57eldifd 3958 . . . . . . . 8 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) ∧ 𝑦 ∈ 𝑆) β†’ (π‘₯(+gβ€˜πΊ)𝑦) ∈ ((Baseβ€˜πΊ) βˆ– 𝑆))
5958fmpttd 7115 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)):π‘†βŸΆ((Baseβ€˜πΊ) βˆ– 𝑆))
6059frnd 6724 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)) βŠ† ((Baseβ€˜πΊ) βˆ– 𝑆))
61 eleq2 2820 . . . . . . . 8 (𝑒 = ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)) β†’ (π‘₯ ∈ 𝑒 ↔ π‘₯ ∈ ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦))))
62 sseq1 4006 . . . . . . . 8 (𝑒 = ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)) β†’ (𝑒 βŠ† ((Baseβ€˜πΊ) βˆ– 𝑆) ↔ ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)) βŠ† ((Baseβ€˜πΊ) βˆ– 𝑆)))
6361, 62anbi12d 629 . . . . . . 7 (𝑒 = ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)) β†’ ((π‘₯ ∈ 𝑒 ∧ 𝑒 βŠ† ((Baseβ€˜πΊ) βˆ– 𝑆)) ↔ (π‘₯ ∈ ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)) ∧ ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)) βŠ† ((Baseβ€˜πΊ) βˆ– 𝑆))))
6463rspcev 3611 . . . . . 6 ((ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)) ∈ 𝐽 ∧ (π‘₯ ∈ ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)) ∧ ran (𝑦 ∈ 𝑆 ↦ (π‘₯(+gβ€˜πΊ)𝑦)) βŠ† ((Baseβ€˜πΊ) βˆ– 𝑆))) β†’ βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ 𝑒 βŠ† ((Baseβ€˜πΊ) βˆ– 𝑆)))
6526, 40, 60, 64syl12anc 833 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) ∧ π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)) β†’ βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ 𝑒 βŠ† ((Baseβ€˜πΊ) βˆ– 𝑆)))
6665ralrimiva 3144 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) β†’ βˆ€π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ 𝑒 βŠ† ((Baseβ€˜πΊ) βˆ– 𝑆)))
67 topontop 22635 . . . . . 6 (𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)) β†’ 𝐽 ∈ Top)
686, 67syl 17 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) β†’ 𝐽 ∈ Top)
69 eltop2 22698 . . . . 5 (𝐽 ∈ Top β†’ (((Baseβ€˜πΊ) βˆ– 𝑆) ∈ 𝐽 ↔ βˆ€π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ 𝑒 βŠ† ((Baseβ€˜πΊ) βˆ– 𝑆))))
7068, 69syl 17 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) β†’ (((Baseβ€˜πΊ) βˆ– 𝑆) ∈ 𝐽 ↔ βˆ€π‘₯ ∈ ((Baseβ€˜πΊ) βˆ– 𝑆)βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ 𝑒 βŠ† ((Baseβ€˜πΊ) βˆ– 𝑆))))
7166, 70mpbird 256 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) β†’ ((Baseβ€˜πΊ) βˆ– 𝑆) ∈ 𝐽)
7210, 71eqeltrrd 2832 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) β†’ (βˆͺ 𝐽 βˆ– 𝑆) ∈ 𝐽)
73 eqid 2730 . . . 4 βˆͺ 𝐽 = βˆͺ 𝐽
7473iscld 22751 . . 3 (𝐽 ∈ Top β†’ (𝑆 ∈ (Clsdβ€˜π½) ↔ (𝑆 βŠ† βˆͺ 𝐽 ∧ (βˆͺ 𝐽 βˆ– 𝑆) ∈ 𝐽)))
7568, 74syl 17 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) β†’ (𝑆 ∈ (Clsdβ€˜π½) ↔ (𝑆 βŠ† βˆͺ 𝐽 ∧ (βˆͺ 𝐽 βˆ– 𝑆) ∈ 𝐽)))
769, 72, 75mpbir2and 709 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 ∈ 𝐽) β†’ 𝑆 ∈ (Clsdβ€˜π½))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068  Vcvv 3472   βˆ– cdif 3944   βŠ† wss 3947  βˆͺ cuni 4907   ↦ cmpt 5230  ran crn 5676   β†Ύ cres 5677   β€œ cima 5678  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  TopOpenctopn 17371  0gc0g 17389  Grpcgrp 18855  -gcsg 18857  SubGrpcsubg 19036  Topctop 22615  TopOnctopon 22632  Clsdccld 22740  Homeochmeo 23477  TopGrpctgp 23795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  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 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  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-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-0g 17391  df-topgen 17393  df-plusf 18564  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-sbg 18860  df-subg 19039  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-cld 22743  df-cn 22951  df-cnp 22952  df-tx 23286  df-hmeo 23479  df-tmd 23796  df-tgp 23797
This theorem is referenced by:  cldsubg  23835  tgpconncompss  23838
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