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Theorem subgntr 23962
Description: A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 23964, 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 5682 . . . . . 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 2726 . . . . . . . . . . . 12 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
42, 3tgptopon 23937 . . . . . . . . . . 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 480 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
7 topontop 22766 . . . . . . . . . . . 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 480 . . . . . . . . . 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 19052 . . . . . . . . . . . 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 22767 . . . . . . . . . . . 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 4017 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝑆 βŠ† βˆͺ 𝐽)
16 eqid 2726 . . . . . . . . . . 11 βˆͺ 𝐽 = βˆͺ 𝐽
1716ntropn 22904 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜π‘†) ∈ 𝐽)
189, 15, 17syl2anc 583 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ ((intβ€˜π½)β€˜π‘†) ∈ 𝐽)
19 toponss 22780 . . . . . . . . 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 583 . . . . . . . 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 6033 . . . . . . 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 5930 . . . . . 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 2778 . . . . 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 484 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ 𝑆)
2616ntrss2 22912 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† 𝑆)
279, 15, 26syl2anc 583 . . . . . . . . . 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 3978 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐴 ∈ 𝑆)
30 eqid 2726 . . . . . . . . . 10 (-gβ€˜πΊ) = (-gβ€˜πΊ)
3130subgsubcl 19062 . . . . . . . . 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 3978 . . . . . . 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 2726 . . . . . . . 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 2726 . . . . . . . 8 (+gβ€˜πΊ) = (+gβ€˜πΊ)
3634, 3, 35, 2tgplacthmeo 23958 . . . . . . 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 583 . . . . . 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 23620 . . . . . 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 583 . . . . 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 2828 . . . 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 23933 . . . . . . 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 3977 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ (Baseβ€˜πΊ))
4520, 28sseldd 3978 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐴 ∈ (Baseβ€˜πΊ))
463, 35, 30grpnpcan 18958 . . . . . 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 7437 . . . . . 6 ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝐴) ∈ V
49 eqid 2726 . . . . . . 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 7412 . . . . . . 7 (𝑦 = 𝐴 β†’ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦) = ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝐴))
5149, 50elrnmpt1s 5949 . . . . . 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 585 . . . . 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 2828 . . . 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 480 . . . . . . 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 480 . . . . . . 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 3977 . . . . . . 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 19061 . . . . . . 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 7109 . . . . 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 6718 . . . 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 2816 . . . . . 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 4002 . . . . . 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 3606 . . . 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 834 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ 𝑒 βŠ† 𝑆))
6665ralrimiva 3140 . 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 22829 . . 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 257 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064  Vcvv 3468   βŠ† wss 3943  βˆͺ cuni 4902   ↦ cmpt 5224  ran crn 5670   β†Ύ cres 5671   β€œ cima 5672  β€˜cfv 6536  (class class class)co 7404  Basecbs 17151  +gcplusg 17204  TopOpenctopn 17374  Grpcgrp 18861  -gcsg 18863  SubGrpcsubg 19045  Topctop 22746  TopOnctopon 22763  intcnt 22872  Homeochmeo 23608  TopGrpctgp 23926
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 7721  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 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-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 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-0g 17394  df-topgen 17396  df-plusf 18570  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-grp 18864  df-minusg 18865  df-sbg 18866  df-subg 19048  df-top 22747  df-topon 22764  df-topsp 22786  df-bases 22800  df-ntr 22875  df-cn 23082  df-cnp 23083  df-tx 23417  df-hmeo 23610  df-tmd 23927  df-tgp 23928
This theorem is referenced by: (None)
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