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Theorem subgntr 23481
Description: A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 23483, 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 5650 . . . . . 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 2733 . . . . . . . . . . . 12 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
42, 3tgptopon 23456 . . . . . . . . . . 11 (𝐺 ∈ TopGrp β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
543ad2ant1 1134 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
65adantr 482 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
7 topontop 22285 . . . . . . . . . . . 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 482 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐽 ∈ Top)
10 simpl2 1193 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))
113subgss 18937 . . . . . . . . . . . 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 22286 . . . . . . . . . . . 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 3988 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝑆 βŠ† βˆͺ 𝐽)
16 eqid 2733 . . . . . . . . . . 11 βˆͺ 𝐽 = βˆͺ 𝐽
1716ntropn 22423 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜π‘†) ∈ 𝐽)
189, 15, 17syl2anc 585 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ ((intβ€˜π½)β€˜π‘†) ∈ 𝐽)
19 toponss 22299 . . . . . . . . 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 585 . . . . . . . 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 5998 . . . . . . 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 5897 . . . . . 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 2785 . . . . 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 1192 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐺 ∈ TopGrp)
25 simpr 486 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ 𝑆)
2616ntrss2 22431 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† 𝑆)
279, 15, 26syl2anc 585 . . . . . . . . . 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 1194 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†))
2927, 28sseldd 3949 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐴 ∈ 𝑆)
30 eqid 2733 . . . . . . . . . 10 (-gβ€˜πΊ) = (-gβ€˜πΊ)
3130subgsubcl 18947 . . . . . . . . 9 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) β†’ (π‘₯(-gβ€˜πΊ)𝐴) ∈ 𝑆)
3210, 25, 29, 31syl3anc 1372 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ (π‘₯(-gβ€˜πΊ)𝐴) ∈ 𝑆)
3312, 32sseldd 3949 . . . . . . 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 2733 . . . . . . . 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 2733 . . . . . . . 8 (+gβ€˜πΊ) = (+gβ€˜πΊ)
3634, 3, 35, 2tgplacthmeo 23477 . . . . . . 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 585 . . . . . 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 23139 . . . . . 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 585 . . . . 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 2835 . . . 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 23452 . . . . . . 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 1135 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† (Baseβ€˜πΊ))
4443sselda 3948 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ (Baseβ€˜πΊ))
4520, 28sseldd 3949 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝐴 ∈ ((intβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝑆) β†’ 𝐴 ∈ (Baseβ€˜πΊ))
463, 35, 30grpnpcan 18847 . . . . . 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 1372 . . . . 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 7394 . . . . . 6 ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝐴) ∈ V
49 eqid 2733 . . . . . . 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 7369 . . . . . . 7 (𝑦 = 𝐴 β†’ ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝑦) = ((π‘₯(-gβ€˜πΊ)𝐴)(+gβ€˜πΊ)𝐴))
5149, 50elrnmpt1s 5916 . . . . . 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 587 . . . . 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 2835 . . . 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 482 . . . . . . 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 482 . . . . . . 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 3948 . . . . . . 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 18946 . . . . . . 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 1372 . . . . . 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 7067 . . . . 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 6680 . . . 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 2823 . . . . . 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 3973 . . . . . 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 632 . . . . 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 3583 . . . 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 836 . . 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 22348 . . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3447   βŠ† wss 3914  βˆͺ cuni 4869   ↦ cmpt 5192  ran crn 5638   β†Ύ cres 5639   β€œ cima 5640  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  +gcplusg 17141  TopOpenctopn 17311  Grpcgrp 18756  -gcsg 18758  SubGrpcsubg 18930  Topctop 22265  TopOnctopon 22282  intcnt 22391  Homeochmeo 23127  TopGrpctgp 23445
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-0g 17331  df-topgen 17333  df-plusf 18504  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-grp 18759  df-minusg 18760  df-sbg 18761  df-subg 18933  df-top 22266  df-topon 22283  df-topsp 22305  df-bases 22319  df-ntr 22394  df-cn 22601  df-cnp 22602  df-tx 22936  df-hmeo 23129  df-tmd 23446  df-tgp 23447
This theorem is referenced by: (None)
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