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Theorem clsnsg 23605
Description: The closure of a normal subgroup is a normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
Hypothesis
Ref Expression
subgntr.h 𝐽 = (TopOpenβ€˜πΊ)
Assertion
Ref Expression
clsnsg ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) β†’ ((clsβ€˜π½)β€˜π‘†) ∈ (NrmSGrpβ€˜πΊ))

Proof of Theorem clsnsg
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nsgsubg 19032 . . 3 (𝑆 ∈ (NrmSGrpβ€˜πΊ) β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))
2 subgntr.h . . . 4 𝐽 = (TopOpenβ€˜πΊ)
32clssubg 23604 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ ((clsβ€˜π½)β€˜π‘†) ∈ (SubGrpβ€˜πΊ))
41, 3sylan2 593 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) β†’ ((clsβ€˜π½)β€˜π‘†) ∈ (SubGrpβ€˜πΊ))
5 df-ima 5688 . . . . . . 7 ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β€œ ((clsβ€˜π½)β€˜π‘†)) = ran ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β†Ύ ((clsβ€˜π½)β€˜π‘†))
6 eqid 2732 . . . . . . . . . . . . . 14 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
72, 6tgptopon 23577 . . . . . . . . . . . . 13 (𝐺 ∈ TopGrp β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
87ad2antrr 724 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
9 topontop 22406 . . . . . . . . . . . 12 (𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)) β†’ 𝐽 ∈ Top)
108, 9syl 17 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ 𝐽 ∈ Top)
111ad2antlr 725 . . . . . . . . . . . . 13 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))
126subgss 19001 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ 𝑆 βŠ† (Baseβ€˜πΊ))
1311, 12syl 17 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ 𝑆 βŠ† (Baseβ€˜πΊ))
14 toponuni 22407 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)) β†’ (Baseβ€˜πΊ) = βˆͺ 𝐽)
158, 14syl 17 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ (Baseβ€˜πΊ) = βˆͺ 𝐽)
1613, 15sseqtrd 4021 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ 𝑆 βŠ† βˆͺ 𝐽)
17 eqid 2732 . . . . . . . . . . . 12 βˆͺ 𝐽 = βˆͺ 𝐽
1817clsss3 22554 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† βˆͺ 𝐽)
1910, 16, 18syl2anc 584 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† βˆͺ 𝐽)
2019, 15sseqtrrd 4022 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† (Baseβ€˜πΊ))
2120resmptd 6038 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β†Ύ ((clsβ€˜π½)β€˜π‘†)) = (𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)))
2221rneqd 5935 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ran ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β†Ύ ((clsβ€˜π½)β€˜π‘†)) = ran (𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)))
235, 22eqtrid 2784 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β€œ ((clsβ€˜π½)β€˜π‘†)) = ran (𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)))
24 eqid 2732 . . . . . . . . . 10 (+gβ€˜πΊ) = (+gβ€˜πΊ)
25 tgptmd 23574 . . . . . . . . . . 11 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ TopMnd)
2625ad2antrr 724 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ 𝐺 ∈ TopMnd)
27 simpr 485 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ π‘₯ ∈ (Baseβ€˜πΊ))
288, 8, 27cnmptc 23157 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ (𝑦 ∈ (Baseβ€˜πΊ) ↦ π‘₯) ∈ (𝐽 Cn 𝐽))
298cnmptid 23156 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ (𝑦 ∈ (Baseβ€˜πΊ) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
302, 24, 26, 8, 28, 29cnmpt1plusg 23582 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ (𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)𝑦)) ∈ (𝐽 Cn 𝐽))
31 eqid 2732 . . . . . . . . . . 11 (-gβ€˜πΊ) = (-gβ€˜πΊ)
322, 31tgpsubcn 23585 . . . . . . . . . 10 (𝐺 ∈ TopGrp β†’ (-gβ€˜πΊ) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
3332ad2antrr 724 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ (-gβ€˜πΊ) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
348, 30, 28, 33cnmpt12f 23161 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ (𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) ∈ (𝐽 Cn 𝐽))
3517cnclsi 22767 . . . . . . . 8 (((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) ∈ (𝐽 Cn 𝐽) ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β€œ ((clsβ€˜π½)β€˜π‘†)) βŠ† ((clsβ€˜π½)β€˜((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β€œ 𝑆)))
3634, 16, 35syl2anc 584 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β€œ ((clsβ€˜π½)β€˜π‘†)) βŠ† ((clsβ€˜π½)β€˜((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β€œ 𝑆)))
37 df-ima 5688 . . . . . . . . . 10 ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β€œ 𝑆) = ran ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β†Ύ 𝑆)
3813resmptd 6038 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β†Ύ 𝑆) = (𝑦 ∈ 𝑆 ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)))
3938rneqd 5935 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ran ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β†Ύ 𝑆) = ran (𝑦 ∈ 𝑆 ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)))
4037, 39eqtrid 2784 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β€œ 𝑆) = ran (𝑦 ∈ 𝑆 ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)))
416, 24, 31nsgconj 19033 . . . . . . . . . . . 12 ((𝑆 ∈ (NrmSGrpβ€˜πΊ) ∧ π‘₯ ∈ (Baseβ€˜πΊ) ∧ 𝑦 ∈ 𝑆) β†’ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯) ∈ 𝑆)
4241ad4ant234 1175 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) ∧ 𝑦 ∈ 𝑆) β†’ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯) ∈ 𝑆)
4342fmpttd 7111 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ (𝑦 ∈ 𝑆 ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)):π‘†βŸΆπ‘†)
4443frnd 6722 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ran (𝑦 ∈ 𝑆 ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) βŠ† 𝑆)
4540, 44eqsstrd 4019 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β€œ 𝑆) βŠ† 𝑆)
4617clsss 22549 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽 ∧ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β€œ 𝑆) βŠ† 𝑆) β†’ ((clsβ€˜π½)β€˜((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β€œ 𝑆)) βŠ† ((clsβ€˜π½)β€˜π‘†))
4710, 16, 45, 46syl3anc 1371 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((clsβ€˜π½)β€˜((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β€œ 𝑆)) βŠ† ((clsβ€˜π½)β€˜π‘†))
4836, 47sstrd 3991 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((𝑦 ∈ (Baseβ€˜πΊ) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) β€œ ((clsβ€˜π½)β€˜π‘†)) βŠ† ((clsβ€˜π½)β€˜π‘†))
4923, 48eqsstrrd 4020 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ran (𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) βŠ† ((clsβ€˜π½)β€˜π‘†))
50 ovex 7438 . . . . . . 7 ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯) ∈ V
51 eqid 2732 . . . . . . 7 (𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) = (𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯))
5250, 51fnmpti 6690 . . . . . 6 (𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) Fn ((clsβ€˜π½)β€˜π‘†)
53 df-f 6544 . . . . . 6 ((𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)):((clsβ€˜π½)β€˜π‘†)⟢((clsβ€˜π½)β€˜π‘†) ↔ ((𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) Fn ((clsβ€˜π½)β€˜π‘†) ∧ ran (𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) βŠ† ((clsβ€˜π½)β€˜π‘†)))
5452, 53mpbiran 707 . . . . 5 ((𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)):((clsβ€˜π½)β€˜π‘†)⟢((clsβ€˜π½)β€˜π‘†) ↔ ran (𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)) βŠ† ((clsβ€˜π½)β€˜π‘†))
5549, 54sylibr 233 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ (𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)):((clsβ€˜π½)β€˜π‘†)⟢((clsβ€˜π½)β€˜π‘†))
5651fmpt 7106 . . . 4 (βˆ€π‘¦ ∈ ((clsβ€˜π½)β€˜π‘†)((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯) ∈ ((clsβ€˜π½)β€˜π‘†) ↔ (𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ↦ ((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯)):((clsβ€˜π½)β€˜π‘†)⟢((clsβ€˜π½)β€˜π‘†))
5755, 56sylibr 233 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ βˆ€π‘¦ ∈ ((clsβ€˜π½)β€˜π‘†)((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯) ∈ ((clsβ€˜π½)β€˜π‘†))
5857ralrimiva 3146 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) β†’ βˆ€π‘₯ ∈ (Baseβ€˜πΊ)βˆ€π‘¦ ∈ ((clsβ€˜π½)β€˜π‘†)((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯) ∈ ((clsβ€˜π½)β€˜π‘†))
596, 24, 31isnsg3 19034 . 2 (((clsβ€˜π½)β€˜π‘†) ∈ (NrmSGrpβ€˜πΊ) ↔ (((clsβ€˜π½)β€˜π‘†) ∈ (SubGrpβ€˜πΊ) ∧ βˆ€π‘₯ ∈ (Baseβ€˜πΊ)βˆ€π‘¦ ∈ ((clsβ€˜π½)β€˜π‘†)((π‘₯(+gβ€˜πΊ)𝑦)(-gβ€˜πΊ)π‘₯) ∈ ((clsβ€˜π½)β€˜π‘†)))
604, 58, 59sylanbrc 583 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrpβ€˜πΊ)) β†’ ((clsβ€˜π½)β€˜π‘†) ∈ (NrmSGrpβ€˜πΊ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βŠ† wss 3947  βˆͺ cuni 4907   ↦ cmpt 5230  ran crn 5676   β†Ύ cres 5677   β€œ cima 5678   Fn wfn 6535  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  +gcplusg 17193  TopOpenctopn 17363  -gcsg 18817  SubGrpcsubg 18994  NrmSGrpcnsg 18995  Topctop 22386  TopOnctopon 22403  clsccl 22513   Cn ccn 22719   Γ—t ctx 23055  TopMndctmd 23565  TopGrpctgp 23566
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
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 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-int 4950  df-iun 4998  df-iin 4999  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 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-0g 17383  df-topgen 17385  df-plusf 18556  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-sbg 18820  df-subg 18997  df-nsg 18998  df-top 22387  df-topon 22404  df-topsp 22426  df-bases 22440  df-cld 22514  df-ntr 22515  df-cls 22516  df-cn 22722  df-cnp 22723  df-tx 23057  df-tmd 23567  df-tgp 23568
This theorem is referenced by: (None)
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