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Theorem subgntr 23610
Description: A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 23612, 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 5689 . . . . . 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 2732 . . . . . . . . . . . 12 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
42, 3tgptopon 23585 . . . . . . . . . . 11 (𝐺 ∈ TopGrp β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
543ad2ant1 1133 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
65adantr 481 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
7 topontop 22414 . . . . . . . . . . . 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 481 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐽 ∈ Top)
10 simpl2 1192 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))
113subgss 19006 . . . . . . . . . . . 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 22415 . . . . . . . . . . . 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 4022 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝑆 βŠ† βˆͺ 𝐽)
16 eqid 2732 . . . . . . . . . . 11 βˆͺ 𝐽 = βˆͺ 𝐽
1716ntropn 22552 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜π‘†) ∈ 𝐽)
189, 15, 17syl2anc 584 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ ((intβ€˜π½)β€˜π‘†) ∈ 𝐽)
19 toponss 22428 . . . . . . . . 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 584 . . . . . . . 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 6040 . . . . . . 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 5937 . . . . . 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 2784 . . . . 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 1191 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐺 ∈ TopGrp)
25 simpr 485 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ 𝑆)
2616ntrss2 22560 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† 𝑆)
279, 15, 26syl2anc 584 . . . . . . . . . 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 1193 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†))
2927, 28sseldd 3983 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐴 ∈ 𝑆)
30 eqid 2732 . . . . . . . . . 10 (-gβ€˜πΊ) = (-gβ€˜πΊ)
3130subgsubcl 19016 . . . . . . . . 9 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) β†’ (π‘₯(-gβ€˜πΊ)𝐴) ∈ 𝑆)
3210, 25, 29, 31syl3anc 1371 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ (π‘₯(-gβ€˜πΊ)𝐴) ∈ 𝑆)
3312, 32sseldd 3983 . . . . . . 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 2732 . . . . . . . 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 2732 . . . . . . . 8 (+gβ€˜πΊ) = (+gβ€˜πΊ)
3634, 3, 35, 2tgplacthmeo 23606 . . . . . . 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 584 . . . . . 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 23268 . . . . . 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 584 . . . . 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 2834 . . . 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 23581 . . . . . . 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 1134 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† (Baseβ€˜πΊ))
4443sselda 3982 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ (Baseβ€˜πΊ))
4520, 28sseldd 3983 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐴 ∈ (Baseβ€˜πΊ))
463, 35, 30grpnpcan 18914 . . . . . 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 1371 . . . . 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 7441 . . . . . 6 ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝐴) ∈ V
49 eqid 2732 . . . . . . 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 7416 . . . . . . 7 (𝑦 = 𝐴 β†’ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦) = ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝐴))
5149, 50elrnmpt1s 5956 . . . . . 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 586 . . . . 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 2834 . . . 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 481 . . . . . . 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 481 . . . . . . 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 3982 . . . . . . 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 19015 . . . . . . 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 1371 . . . . . 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 7114 . . . . 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 6725 . . . 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 2822 . . . . . 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 4007 . . . . . 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 631 . . . . 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 3612 . . . 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 3146 . 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 22477 . . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βŠ† wss 3948  βˆͺ cuni 4908   ↦ cmpt 5231  ran crn 5677   β†Ύ cres 5678   β€œ cima 5679  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  +gcplusg 17196  TopOpenctopn 17366  Grpcgrp 18818  -gcsg 18820  SubGrpcsubg 18999  Topctop 22394  TopOnctopon 22411  intcnt 22520  Homeochmeo 23256  TopGrpctgp 23574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-0g 17386  df-topgen 17388  df-plusf 18559  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821  df-minusg 18822  df-sbg 18823  df-subg 19002  df-top 22395  df-topon 22412  df-topsp 22434  df-bases 22448  df-ntr 22523  df-cn 22730  df-cnp 22731  df-tx 23065  df-hmeo 23258  df-tmd 23575  df-tgp 23576
This theorem is referenced by: (None)
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