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Theorem tgpconcomp 21726
 Description: The identity component, the connected component containing the identity element, is a closed (concompcld 21047) normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
Hypotheses
Ref Expression
tgpconcomp.x 𝑋 = (Base‘𝐺)
tgpconcomp.z 0 = (0g𝐺)
tgpconcomp.j 𝐽 = (TopOpen‘𝐺)
tgpconcomp.s 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Con)}
Assertion
Ref Expression
tgpconcomp (𝐺 ∈ TopGrp → 𝑆 ∈ (NrmSGrp‘𝐺))
Distinct variable groups:   𝑥, 0   𝑥,𝐽   𝑥,𝐺   𝑥,𝑋
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem tgpconcomp
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgpconcomp.s . . . . 5 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Con)}
2 ssrab2 3650 . . . . . 6 {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Con)} ⊆ 𝒫 𝑋
3 sspwuni 4547 . . . . . 6 ({𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Con)} ⊆ 𝒫 𝑋 {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Con)} ⊆ 𝑋)
42, 3mpbi 219 . . . . 5 {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Con)} ⊆ 𝑋
51, 4eqsstri 3598 . . . 4 𝑆𝑋
65a1i 11 . . 3 (𝐺 ∈ TopGrp → 𝑆𝑋)
7 tgpconcomp.j . . . . . 6 𝐽 = (TopOpen‘𝐺)
8 tgpconcomp.x . . . . . 6 𝑋 = (Base‘𝐺)
97, 8tgptopon 21696 . . . . 5 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
10 tgpgrp 21692 . . . . . 6 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
11 tgpconcomp.z . . . . . . 7 0 = (0g𝐺)
128, 11grpidcl 17273 . . . . . 6 (𝐺 ∈ Grp → 0𝑋)
1310, 12syl 17 . . . . 5 (𝐺 ∈ TopGrp → 0𝑋)
141concompid 21044 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 0𝑋) → 0𝑆)
159, 13, 14syl2anc 691 . . . 4 (𝐺 ∈ TopGrp → 0𝑆)
16 ne0i 3880 . . . 4 ( 0𝑆𝑆 ≠ ∅)
1715, 16syl 17 . . 3 (𝐺 ∈ TopGrp → 𝑆 ≠ ∅)
18 df-ima 5051 . . . . . . . 8 ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) = ran ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) ↾ 𝑆)
19 resmpt 5369 . . . . . . . . . 10 (𝑆𝑋 → ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) ↾ 𝑆) = (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)))
205, 19ax-mp 5 . . . . . . . . 9 ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) ↾ 𝑆) = (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧))
2120rneqi 5273 . . . . . . . 8 ran ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) ↾ 𝑆) = ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧))
2218, 21eqtri 2632 . . . . . . 7 ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) = ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧))
23 imassrn 5396 . . . . . . . . 9 ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) ⊆ ran (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧))
2410adantr 480 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 𝐺 ∈ Grp)
2524adantr 480 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑦𝑆) ∧ 𝑧𝑋) → 𝐺 ∈ Grp)
266sselda 3568 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 𝑦𝑋)
2726adantr 480 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑦𝑆) ∧ 𝑧𝑋) → 𝑦𝑋)
28 simpr 476 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑦𝑆) ∧ 𝑧𝑋) → 𝑧𝑋)
29 eqid 2610 . . . . . . . . . . . . 13 (-g𝐺) = (-g𝐺)
308, 29grpsubcl 17318 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑦𝑋𝑧𝑋) → (𝑦(-g𝐺)𝑧) ∈ 𝑋)
3125, 27, 28, 30syl3anc 1318 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑦𝑆) ∧ 𝑧𝑋) → (𝑦(-g𝐺)𝑧) ∈ 𝑋)
32 eqid 2610 . . . . . . . . . . 11 (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) = (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧))
3331, 32fmptd 6292 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)):𝑋𝑋)
34 frn 5966 . . . . . . . . . 10 ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)):𝑋𝑋 → ran (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) ⊆ 𝑋)
3533, 34syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ran (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) ⊆ 𝑋)
3623, 35syl5ss 3579 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) ⊆ 𝑋)
378, 11, 29grpsubid 17322 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → (𝑦(-g𝐺)𝑦) = 0 )
3824, 26, 37syl2anc 691 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑦(-g𝐺)𝑦) = 0 )
39 simpr 476 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 𝑦𝑆)
40 ovex 6577 . . . . . . . . . . 11 (𝑦(-g𝐺)𝑦) ∈ V
41 eqid 2610 . . . . . . . . . . . 12 (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)) = (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧))
42 oveq2 6557 . . . . . . . . . . . 12 (𝑧 = 𝑦 → (𝑦(-g𝐺)𝑧) = (𝑦(-g𝐺)𝑦))
4341, 42elrnmpt1s 5294 . . . . . . . . . . 11 ((𝑦𝑆 ∧ (𝑦(-g𝐺)𝑦) ∈ V) → (𝑦(-g𝐺)𝑦) ∈ ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)))
4439, 40, 43sylancl 693 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑦(-g𝐺)𝑦) ∈ ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)))
4538, 44eqeltrrd 2689 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 0 ∈ ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)))
4645, 22syl6eleqr 2699 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 0 ∈ ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆))
47 eqid 2610 . . . . . . . . 9 𝐽 = 𝐽
48 eqid 2610 . . . . . . . . . . . . . . 15 (+g𝐺) = (+g𝐺)
49 eqid 2610 . . . . . . . . . . . . . . 15 (invg𝐺) = (invg𝐺)
508, 48, 49, 29grpsubval 17288 . . . . . . . . . . . . . 14 ((𝑦𝑋𝑧𝑋) → (𝑦(-g𝐺)𝑧) = (𝑦(+g𝐺)((invg𝐺)‘𝑧)))
5126, 50sylan 487 . . . . . . . . . . . . 13 (((𝐺 ∈ TopGrp ∧ 𝑦𝑆) ∧ 𝑧𝑋) → (𝑦(-g𝐺)𝑧) = (𝑦(+g𝐺)((invg𝐺)‘𝑧)))
5251mpteq2dva 4672 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) = (𝑧𝑋 ↦ (𝑦(+g𝐺)((invg𝐺)‘𝑧))))
538, 49grpinvcl 17290 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → ((invg𝐺)‘𝑧) ∈ 𝑋)
5424, 53sylan 487 . . . . . . . . . . . . 13 (((𝐺 ∈ TopGrp ∧ 𝑦𝑆) ∧ 𝑧𝑋) → ((invg𝐺)‘𝑧) ∈ 𝑋)
558, 49grpinvf 17289 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Grp → (invg𝐺):𝑋𝑋)
5610, 55syl 17 . . . . . . . . . . . . . . 15 (𝐺 ∈ TopGrp → (invg𝐺):𝑋𝑋)
5756adantr 480 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (invg𝐺):𝑋𝑋)
5857feqmptd 6159 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (invg𝐺) = (𝑧𝑋 ↦ ((invg𝐺)‘𝑧)))
59 eqidd 2611 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) = (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)))
60 oveq2 6557 . . . . . . . . . . . . 13 (𝑤 = ((invg𝐺)‘𝑧) → (𝑦(+g𝐺)𝑤) = (𝑦(+g𝐺)((invg𝐺)‘𝑧)))
6154, 58, 59, 60fmptco 6303 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ((𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∘ (invg𝐺)) = (𝑧𝑋 ↦ (𝑦(+g𝐺)((invg𝐺)‘𝑧))))
6252, 61eqtr4d 2647 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) = ((𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∘ (invg𝐺)))
637, 49grpinvhmeo 21700 . . . . . . . . . . . . 13 (𝐺 ∈ TopGrp → (invg𝐺) ∈ (𝐽Homeo𝐽))
6463adantr 480 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (invg𝐺) ∈ (𝐽Homeo𝐽))
65 eqid 2610 . . . . . . . . . . . . . 14 (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) = (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤))
6665, 8, 48, 7tgplacthmeo 21717 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑦𝑋) → (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∈ (𝐽Homeo𝐽))
6726, 66syldan 486 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∈ (𝐽Homeo𝐽))
68 hmeoco 21385 . . . . . . . . . . . 12 (((invg𝐺) ∈ (𝐽Homeo𝐽) ∧ (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∈ (𝐽Homeo𝐽)) → ((𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∘ (invg𝐺)) ∈ (𝐽Homeo𝐽))
6964, 67, 68syl2anc 691 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ((𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∘ (invg𝐺)) ∈ (𝐽Homeo𝐽))
7062, 69eqeltrd 2688 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
71 hmeocn 21373 . . . . . . . . . 10 ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) ∈ (𝐽Homeo𝐽) → (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
7270, 71syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
73 toponuni 20542 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
749, 73syl 17 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝑋 = 𝐽)
7574adantr 480 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 𝑋 = 𝐽)
765, 75syl5sseq 3616 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 𝑆 𝐽)
771concompcon 21045 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 0𝑋) → (𝐽t 𝑆) ∈ Con)
789, 13, 77syl2anc 691 . . . . . . . . . 10 (𝐺 ∈ TopGrp → (𝐽t 𝑆) ∈ Con)
7978adantr 480 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝐽t 𝑆) ∈ Con)
8047, 72, 76, 79conima 21038 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝐽t ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆)) ∈ Con)
811concompss 21046 . . . . . . . 8 ((((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) ⊆ 𝑋0 ∈ ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) ∧ (𝐽t ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆)) ∈ Con) → ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) ⊆ 𝑆)
8236, 46, 80, 81syl3anc 1318 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) ⊆ 𝑆)
8322, 82syl5eqssr 3613 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)) ⊆ 𝑆)
84 ovex 6577 . . . . . . . 8 (𝑦(-g𝐺)𝑧) ∈ V
8584, 41fnmpti 5935 . . . . . . 7 (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)) Fn 𝑆
86 df-f 5808 . . . . . . 7 ((𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)):𝑆𝑆 ↔ ((𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)) Fn 𝑆 ∧ ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)) ⊆ 𝑆))
8785, 86mpbiran 955 . . . . . 6 ((𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)):𝑆𝑆 ↔ ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)) ⊆ 𝑆)
8883, 87sylibr 223 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)):𝑆𝑆)
8941fmpt 6289 . . . . 5 (∀𝑧𝑆 (𝑦(-g𝐺)𝑧) ∈ 𝑆 ↔ (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)):𝑆𝑆)
9088, 89sylibr 223 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ∀𝑧𝑆 (𝑦(-g𝐺)𝑧) ∈ 𝑆)
9190ralrimiva 2949 . . 3 (𝐺 ∈ TopGrp → ∀𝑦𝑆𝑧𝑆 (𝑦(-g𝐺)𝑧) ∈ 𝑆)
928, 29issubg4 17436 . . . 4 (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆𝑋𝑆 ≠ ∅ ∧ ∀𝑦𝑆𝑧𝑆 (𝑦(-g𝐺)𝑧) ∈ 𝑆)))
9310, 92syl 17 . . 3 (𝐺 ∈ TopGrp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆𝑋𝑆 ≠ ∅ ∧ ∀𝑦𝑆𝑧𝑆 (𝑦(-g𝐺)𝑧) ∈ 𝑆)))
946, 17, 91, 93mpbir3and 1238 . 2 (𝐺 ∈ TopGrp → 𝑆 ∈ (SubGrp‘𝐺))
9510adantr 480 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → 𝐺 ∈ Grp)
96 eqid 2610 . . . . . . . . . . 11 (oppg𝐺) = (oppg𝐺)
9796, 49oppginv 17612 . . . . . . . . . 10 (𝐺 ∈ Grp → (invg𝐺) = (invg‘(oppg𝐺)))
9895, 97syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (invg𝐺) = (invg‘(oppg𝐺)))
9998fveq1d 6105 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → ((invg𝐺)‘((invg𝐺)‘𝑦)) = ((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦)))
100 simprll 798 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → 𝑦𝑋)
1018, 49grpinvinv 17305 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → ((invg𝐺)‘((invg𝐺)‘𝑦)) = 𝑦)
10295, 100, 101syl2anc 691 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → ((invg𝐺)‘((invg𝐺)‘𝑦)) = 𝑦)
10399, 102eqtr3d 2646 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → ((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦)) = 𝑦)
104103oveq1d 6564 . . . . . 6 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦))(+g‘(oppg𝐺))𝑧) = (𝑦(+g‘(oppg𝐺))𝑧))
105 eqid 2610 . . . . . . 7 (+g‘(oppg𝐺)) = (+g‘(oppg𝐺))
10648, 96, 105oppgplus 17602 . . . . . 6 (𝑦(+g‘(oppg𝐺))𝑧) = (𝑧(+g𝐺)𝑦)
107104, 106syl6eq 2660 . . . . 5 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦))(+g‘(oppg𝐺))𝑧) = (𝑧(+g𝐺)𝑦))
1088, 49grpinvcl 17290 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → ((invg𝐺)‘𝑦) ∈ 𝑋)
10995, 100, 108syl2anc 691 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → ((invg𝐺)‘𝑦) ∈ 𝑋)
110 simprlr 799 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → 𝑧𝑋)
111102oveq1d 6564 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg𝐺)‘((invg𝐺)‘𝑦))(+g𝐺)𝑧) = (𝑦(+g𝐺)𝑧))
112 simprr 792 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (𝑦(+g𝐺)𝑧) ∈ 𝑆)
113111, 112eqeltrd 2688 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg𝐺)‘((invg𝐺)‘𝑦))(+g𝐺)𝑧) ∈ 𝑆)
114 eqid 2610 . . . . . . . . . . 11 (𝐺 ~QG 𝑆) = (𝐺 ~QG 𝑆)
1158, 49, 48, 114eqgval 17466 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑆𝑋) → (((invg𝐺)‘𝑦)(𝐺 ~QG 𝑆)𝑧 ↔ (((invg𝐺)‘𝑦) ∈ 𝑋𝑧𝑋 ∧ (((invg𝐺)‘((invg𝐺)‘𝑦))(+g𝐺)𝑧) ∈ 𝑆)))
11695, 5, 115sylancl 693 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg𝐺)‘𝑦)(𝐺 ~QG 𝑆)𝑧 ↔ (((invg𝐺)‘𝑦) ∈ 𝑋𝑧𝑋 ∧ (((invg𝐺)‘((invg𝐺)‘𝑦))(+g𝐺)𝑧) ∈ 𝑆)))
117109, 110, 113, 116mpbir3and 1238 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → ((invg𝐺)‘𝑦)(𝐺 ~QG 𝑆)𝑧)
1188, 11, 7, 1, 114tgpconcompeqg 21725 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ ((invg𝐺)‘𝑦) ∈ 𝑋) → [((invg𝐺)‘𝑦)](𝐺 ~QG 𝑆) = {𝑥 ∈ 𝒫 𝑋 ∣ (((invg𝐺)‘𝑦) ∈ 𝑥 ∧ (𝐽t 𝑥) ∈ Con)})
119109, 118syldan 486 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → [((invg𝐺)‘𝑦)](𝐺 ~QG 𝑆) = {𝑥 ∈ 𝒫 𝑋 ∣ (((invg𝐺)‘𝑦) ∈ 𝑥 ∧ (𝐽t 𝑥) ∈ Con)})
12096oppgtgp 21712 . . . . . . . . . . . . 13 (𝐺 ∈ TopGrp → (oppg𝐺) ∈ TopGrp)
121120adantr 480 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (oppg𝐺) ∈ TopGrp)
12296, 8oppgbas 17604 . . . . . . . . . . . . 13 𝑋 = (Base‘(oppg𝐺))
12396, 11oppgid 17609 . . . . . . . . . . . . 13 0 = (0g‘(oppg𝐺))
12496, 7oppgtopn 17606 . . . . . . . . . . . . 13 𝐽 = (TopOpen‘(oppg𝐺))
125 eqid 2610 . . . . . . . . . . . . 13 ((oppg𝐺) ~QG 𝑆) = ((oppg𝐺) ~QG 𝑆)
126122, 123, 124, 1, 125tgpconcompeqg 21725 . . . . . . . . . . . 12 (((oppg𝐺) ∈ TopGrp ∧ ((invg𝐺)‘𝑦) ∈ 𝑋) → [((invg𝐺)‘𝑦)]((oppg𝐺) ~QG 𝑆) = {𝑥 ∈ 𝒫 𝑋 ∣ (((invg𝐺)‘𝑦) ∈ 𝑥 ∧ (𝐽t 𝑥) ∈ Con)})
127121, 109, 126syl2anc 691 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → [((invg𝐺)‘𝑦)]((oppg𝐺) ~QG 𝑆) = {𝑥 ∈ 𝒫 𝑋 ∣ (((invg𝐺)‘𝑦) ∈ 𝑥 ∧ (𝐽t 𝑥) ∈ Con)})
128119, 127eqtr4d 2647 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → [((invg𝐺)‘𝑦)](𝐺 ~QG 𝑆) = [((invg𝐺)‘𝑦)]((oppg𝐺) ~QG 𝑆))
129128eleq2d 2673 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (𝑧 ∈ [((invg𝐺)‘𝑦)](𝐺 ~QG 𝑆) ↔ 𝑧 ∈ [((invg𝐺)‘𝑦)]((oppg𝐺) ~QG 𝑆)))
130 vex 3176 . . . . . . . . . 10 𝑧 ∈ V
131 fvex 6113 . . . . . . . . . 10 ((invg𝐺)‘𝑦) ∈ V
132130, 131elec 7673 . . . . . . . . 9 (𝑧 ∈ [((invg𝐺)‘𝑦)](𝐺 ~QG 𝑆) ↔ ((invg𝐺)‘𝑦)(𝐺 ~QG 𝑆)𝑧)
133130, 131elec 7673 . . . . . . . . 9 (𝑧 ∈ [((invg𝐺)‘𝑦)]((oppg𝐺) ~QG 𝑆) ↔ ((invg𝐺)‘𝑦)((oppg𝐺) ~QG 𝑆)𝑧)
134129, 132, 1333bitr3g 301 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg𝐺)‘𝑦)(𝐺 ~QG 𝑆)𝑧 ↔ ((invg𝐺)‘𝑦)((oppg𝐺) ~QG 𝑆)𝑧))
135117, 134mpbid 221 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → ((invg𝐺)‘𝑦)((oppg𝐺) ~QG 𝑆)𝑧)
136 eqid 2610 . . . . . . . . 9 (invg‘(oppg𝐺)) = (invg‘(oppg𝐺))
137122, 136, 105, 125eqgval 17466 . . . . . . . 8 (((oppg𝐺) ∈ TopGrp ∧ 𝑆𝑋) → (((invg𝐺)‘𝑦)((oppg𝐺) ~QG 𝑆)𝑧 ↔ (((invg𝐺)‘𝑦) ∈ 𝑋𝑧𝑋 ∧ (((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦))(+g‘(oppg𝐺))𝑧) ∈ 𝑆)))
138121, 5, 137sylancl 693 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg𝐺)‘𝑦)((oppg𝐺) ~QG 𝑆)𝑧 ↔ (((invg𝐺)‘𝑦) ∈ 𝑋𝑧𝑋 ∧ (((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦))(+g‘(oppg𝐺))𝑧) ∈ 𝑆)))
139135, 138mpbid 221 . . . . . 6 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg𝐺)‘𝑦) ∈ 𝑋𝑧𝑋 ∧ (((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦))(+g‘(oppg𝐺))𝑧) ∈ 𝑆))
140139simp3d 1068 . . . . 5 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦))(+g‘(oppg𝐺))𝑧) ∈ 𝑆)
141107, 140eqeltrrd 2689 . . . 4 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (𝑧(+g𝐺)𝑦) ∈ 𝑆)
142141expr 641 . . 3 ((𝐺 ∈ TopGrp ∧ (𝑦𝑋𝑧𝑋)) → ((𝑦(+g𝐺)𝑧) ∈ 𝑆 → (𝑧(+g𝐺)𝑦) ∈ 𝑆))
143142ralrimivva 2954 . 2 (𝐺 ∈ TopGrp → ∀𝑦𝑋𝑧𝑋 ((𝑦(+g𝐺)𝑧) ∈ 𝑆 → (𝑧(+g𝐺)𝑦) ∈ 𝑆))
1448, 48isnsg2 17447 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦(+g𝐺)𝑧) ∈ 𝑆 → (𝑧(+g𝐺)𝑦) ∈ 𝑆)))
14594, 143, 144sylanbrc 695 1 (𝐺 ∈ TopGrp → 𝑆 ∈ (NrmSGrp‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039   ↾ cres 5040   “ cima 5041   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  [cec 7627  Basecbs 15695  +gcplusg 15768   ↾t crest 15904  TopOpenctopn 15905  0gc0g 15923  Grpcgrp 17245  invgcminusg 17246  -gcsg 17247  SubGrpcsubg 17411  NrmSGrpcnsg 17412   ~QG cqg 17413  oppgcoppg 17598  TopOnctopon 20518   Cn ccn 20838  Conccon 21024  Homeochmeo 21366  TopGrpctgp 21685 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-ec 7631  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-tset 15787  df-rest 15906  df-topn 15907  df-0g 15925  df-topgen 15927  df-plusf 17064  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-nsg 17415  df-eqg 17416  df-oppg 17599  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-cn 20841  df-cnp 20842  df-con 21025  df-tx 21175  df-hmeo 21368  df-tmd 21686  df-tgp 21687 This theorem is referenced by: (None)
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