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Theorem subgntr 24029
Description: A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 24031, the interior of a subgroup is either a subgroup, or empty. (Contributed by Mario Carneiro, 19-Sep-2015.)
Hypothesis
Ref Expression
subgntr.h 𝐽 = (TopOpenβ€˜πΊ)
Assertion
Ref Expression
subgntr ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) β†’ 𝑆 ∈ 𝐽)
Proof of Theorem subgntr
Dummy variables π‘₯ 𝑒 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ima 5693 . . . . . 6 ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) β€œ ((intβ€˜π½)β€˜π‘†)) = ran ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) β†Ύ ((intβ€˜π½)β€˜π‘†))
2 subgntr.h . . . . . . . . . . . 12 𝐽 = (TopOpenβ€˜πΊ)
3 eqid 2727 . . . . . . . . . . . 12 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
42, 3tgptopon 24004 . . . . . . . . . . 11 (𝐺 ∈ TopGrp β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
543ad2ant1 1130 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
65adantr 479 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
7 topontop 22833 . . . . . . . . . . . 12 (𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)) β†’ 𝐽 ∈ Top)
85, 7syl 17 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) β†’ 𝐽 ∈ Top)
98adantr 479 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐽 ∈ Top)
10 simpl2 1189 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))
113subgss 19087 . . . . . . . . . . . 12 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ 𝑆 βŠ† (Baseβ€˜πΊ))
1210, 11syl 17 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝑆 βŠ† (Baseβ€˜πΊ))
13 toponuni 22834 . . . . . . . . . . . 12 (𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)) β†’ (Baseβ€˜πΊ) = βˆͺ 𝐽)
146, 13syl 17 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ (Baseβ€˜πΊ) = βˆͺ 𝐽)
1512, 14sseqtrd 4020 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝑆 βŠ† βˆͺ 𝐽)
16 eqid 2727 . . . . . . . . . . 11 βˆͺ 𝐽 = βˆͺ 𝐽
1716ntropn 22971 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜π‘†) ∈ 𝐽)
189, 15, 17syl2anc 582 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ ((intβ€˜π½)β€˜π‘†) ∈ 𝐽)
19 toponss 22847 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)) ∧ ((intβ€˜π½)β€˜π‘†) ∈ 𝐽) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† (Baseβ€˜πΊ))
206, 18, 19syl2anc 582 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† (Baseβ€˜πΊ))
2120resmptd 6047 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) β†Ύ ((intβ€˜π½)β€˜π‘†)) = (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)))
2221rneqd 5942 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ ran ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) β†Ύ ((intβ€˜π½)β€˜π‘†)) = ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)))
231, 22eqtrid 2779 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) β€œ ((intβ€˜π½)β€˜π‘†)) = ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)))
24 simpl1 1188 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐺 ∈ TopGrp)
25 simpr 483 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ 𝑆)
2616ntrss2 22979 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† 𝑆)
279, 15, 26syl2anc 582 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† 𝑆)
28 simpl3 1190 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†))
2927, 28sseldd 3981 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐴 ∈ 𝑆)
30 eqid 2727 . . . . . . . . . 10 (-gβ€˜πΊ) = (-gβ€˜πΊ)
3130subgsubcl 19097 . . . . . . . . 9 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) β†’ (π‘₯(-gβ€˜πΊ)𝐴) ∈ 𝑆)
3210, 25, 29, 31syl3anc 1368 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ (π‘₯(-gβ€˜πΊ)𝐴) ∈ 𝑆)
3312, 32sseldd 3981 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ (π‘₯(-gβ€˜πΊ)𝐴) ∈ (Baseβ€˜πΊ))
34 eqid 2727 . . . . . . . 8 (𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) = (𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦))
35 eqid 2727 . . . . . . . 8 (+gβ€˜πΊ) = (+gβ€˜πΊ)
3634, 3, 35, 2tgplacthmeo 24025 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ (π‘₯(-gβ€˜πΊ)𝐴) ∈ (Baseβ€˜πΊ)) β†’ (𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) ∈ (𝐽Homeo𝐽))
3724, 33, 36syl2anc 582 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ (𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) ∈ (𝐽Homeo𝐽))
38 hmeoima 23687 . . . . . 6 (((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) ∈ (𝐽Homeo𝐽) ∧ ((intβ€˜π½)β€˜π‘†) ∈ 𝐽) β†’ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) β€œ ((intβ€˜π½)β€˜π‘†)) ∈ 𝐽)
3937, 18, 38syl2anc 582 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) β€œ ((intβ€˜π½)β€˜π‘†)) ∈ 𝐽)
4023, 39eqeltrrd 2829 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) ∈ 𝐽)
41 tgpgrp 24000 . . . . . . 7 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ Grp)
4224, 41syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐺 ∈ Grp)
43113ad2ant2 1131 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† (Baseβ€˜πΊ))
4443sselda 3980 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ (Baseβ€˜πΊ))
4520, 28sseldd 3981 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐴 ∈ (Baseβ€˜πΊ))
463, 35, 30grpnpcan 18993 . . . . . 6 ((𝐺 ∈ Grp ∧ π‘₯ ∈ (Baseβ€˜πΊ) ∧ 𝐴 ∈ (Baseβ€˜πΊ)) β†’ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝐴) = π‘₯)
4742, 44, 45, 46syl3anc 1368 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝐴) = π‘₯)
48 ovex 7457 . . . . . 6 ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝐴) ∈ V
49 eqid 2727 . . . . . . 7 (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) = (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦))
50 oveq2 7432 . . . . . . 7 (𝑦 = 𝐴 β†’ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦) = ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝐴))
5149, 50elrnmpt1s 5961 . . . . . 6 ((𝐴 ∈ ((intβ€˜π½)β€˜π‘†) ∧ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝐴) ∈ V) β†’ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝐴) ∈ ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)))
5228, 48, 51sylancl 584 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝐴) ∈ ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)))
5347, 52eqeltrrd 2829 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)))
5410adantr 479 . . . . . . 7 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) ∧ 𝑦 ∈ ((intβ€˜π½)β€˜π‘†)) β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))
5532adantr 479 . . . . . . 7 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) ∧ 𝑦 ∈ ((intβ€˜π½)β€˜π‘†)) β†’ (π‘₯(-gβ€˜πΊ)𝐴) ∈ 𝑆)
5627sselda 3980 . . . . . . 7 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) ∧ 𝑦 ∈ ((intβ€˜π½)β€˜π‘†)) β†’ 𝑦 ∈ 𝑆)
5735subgcl 19096 . . . . . . 7 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (π‘₯(-gβ€˜πΊ)𝐴) ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) β†’ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦) ∈ 𝑆)
5854, 55, 56, 57syl3anc 1368 . . . . . 6 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) ∧ 𝑦 ∈ ((intβ€˜π½)β€˜π‘†)) β†’ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦) ∈ 𝑆)
5958fmpttd 7128 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)):((intβ€˜π½)β€˜π‘†)βŸΆπ‘†)
6059frnd 6733 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) βŠ† 𝑆)
61 eleq2 2817 . . . . . 6 (𝑒 = ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) β†’ (π‘₯ ∈ 𝑒 ↔ π‘₯ ∈ ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦))))
62 sseq1 4005 . . . . . 6 (𝑒 = ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) β†’ (𝑒 βŠ† 𝑆 ↔ ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) βŠ† 𝑆))
6361, 62anbi12d 630 . . . . 5 (𝑒 = ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) β†’ ((π‘₯ ∈ 𝑒 ∧ 𝑒 βŠ† 𝑆) ↔ (π‘₯ ∈ ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) ∧ ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) βŠ† 𝑆)))
6463rspcev 3609 . . . 4 ((ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) ∈ 𝐽 ∧ (π‘₯ ∈ ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) ∧ ran (𝑦 ∈ ((intβ€˜π½)β€˜π‘†) ↦ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦)) βŠ† 𝑆)) β†’ βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ 𝑒 βŠ† 𝑆))
6540, 53, 60, 64syl12anc 835 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ 𝑒 βŠ† 𝑆))
6665ralrimiva 3142 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) β†’ βˆ€π‘₯ ∈ 𝑆 βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ 𝑒 βŠ† 𝑆))
67 eltop2 22896 . . 3 (𝐽 ∈ Top β†’ (𝑆 ∈ 𝐽 ↔ βˆ€π‘₯ ∈ 𝑆 βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ 𝑒 βŠ† 𝑆)))
688, 67syl 17 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) β†’ (𝑆 ∈ 𝐽 ↔ βˆ€π‘₯ ∈ 𝑆 βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ 𝑒 βŠ† 𝑆)))
6966, 68mpbird 256 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) β†’ 𝑆 ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3057  βˆƒwrex 3066  Vcvv 3471   βŠ† wss 3947  βˆͺ cuni 4910   ↦ cmpt 5233  ran crn 5681   β†Ύ cres 5682   β€œ cima 5683  β€˜cfv 6551  (class class class)co 7424  Basecbs 17185  +gcplusg 17238  TopOpenctopn 17408  Grpcgrp 18895  -gcsg 18897  SubGrpcsubg 19080  Topctop 22813  TopOnctopon 22830  intcnt 22939  Homeochmeo 23675  TopGrpctgp 23993
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 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221
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 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-er 8729  df-map 8851  df-en 8969  df-dom 8970  df-sdom 8971  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-nn 12249  df-2 12311  df-sets 17138  df-slot 17156  df-ndx 17168  df-base 17186  df-ress 17215  df-plusg 17251  df-0g 17428  df-topgen 17430  df-plusf 18604  df-mgm 18605  df-sgrp 18684  df-mnd 18700  df-grp 18898  df-minusg 18899  df-sbg 18900  df-subg 19083  df-top 22814  df-topon 22831  df-topsp 22853  df-bases 22867  df-ntr 22942  df-cn 23149  df-cnp 23150  df-tx 23484  df-hmeo 23677  df-tmd 23994  df-tgp 23995
This theorem is referenced by: (None)
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