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Theorem tgpconncomp 21856
Description: The identity component, the connected component containing the identity element, is a closed (conncompcld 21177) normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
Hypotheses
Ref Expression
tgpconncomp.x 𝑋 = (Base‘𝐺)
tgpconncomp.z 0 = (0g𝐺)
tgpconncomp.j 𝐽 = (TopOpen‘𝐺)
tgpconncomp.s 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
Assertion
Ref Expression
tgpconncomp (𝐺 ∈ TopGrp → 𝑆 ∈ (NrmSGrp‘𝐺))
Distinct variable groups:   𝑥, 0   𝑥,𝐽   𝑥,𝐺   𝑥,𝑋
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem tgpconncomp
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgpconncomp.s . . . . 5 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
2 ssrab2 3672 . . . . . 6 {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋
3 sspwuni 4584 . . . . . 6 ({𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋 {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝑋)
42, 3mpbi 220 . . . . 5 {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝑋
51, 4eqsstri 3620 . . . 4 𝑆𝑋
65a1i 11 . . 3 (𝐺 ∈ TopGrp → 𝑆𝑋)
7 tgpconncomp.j . . . . . 6 𝐽 = (TopOpen‘𝐺)
8 tgpconncomp.x . . . . . 6 𝑋 = (Base‘𝐺)
97, 8tgptopon 21826 . . . . 5 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
10 tgpgrp 21822 . . . . . 6 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
11 tgpconncomp.z . . . . . . 7 0 = (0g𝐺)
128, 11grpidcl 17390 . . . . . 6 (𝐺 ∈ Grp → 0𝑋)
1310, 12syl 17 . . . . 5 (𝐺 ∈ TopGrp → 0𝑋)
141conncompid 21174 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 0𝑋) → 0𝑆)
159, 13, 14syl2anc 692 . . . 4 (𝐺 ∈ TopGrp → 0𝑆)
16 ne0i 3903 . . . 4 ( 0𝑆𝑆 ≠ ∅)
1715, 16syl 17 . . 3 (𝐺 ∈ TopGrp → 𝑆 ≠ ∅)
18 df-ima 5097 . . . . . . . 8 ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) = ran ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) ↾ 𝑆)
19 resmpt 5418 . . . . . . . . . 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 5322 . . . . . . . 8 ran ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) ↾ 𝑆) = ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧))
2218, 21eqtri 2643 . . . . . . 7 ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) = ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧))
23 imassrn 5446 . . . . . . . . 9 ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) ⊆ ran (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧))
2410adantr 481 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 𝐺 ∈ Grp)
2524adantr 481 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑦𝑆) ∧ 𝑧𝑋) → 𝐺 ∈ Grp)
266sselda 3588 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 𝑦𝑋)
2726adantr 481 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑦𝑆) ∧ 𝑧𝑋) → 𝑦𝑋)
28 simpr 477 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑦𝑆) ∧ 𝑧𝑋) → 𝑧𝑋)
29 eqid 2621 . . . . . . . . . . . . 13 (-g𝐺) = (-g𝐺)
308, 29grpsubcl 17435 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑦𝑋𝑧𝑋) → (𝑦(-g𝐺)𝑧) ∈ 𝑋)
3125, 27, 28, 30syl3anc 1323 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑦𝑆) ∧ 𝑧𝑋) → (𝑦(-g𝐺)𝑧) ∈ 𝑋)
32 eqid 2621 . . . . . . . . . . 11 (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) = (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧))
3331, 32fmptd 6351 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)):𝑋𝑋)
34 frn 6020 . . . . . . . . . 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 3599 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) ⊆ 𝑋)
378, 11, 29grpsubid 17439 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → (𝑦(-g𝐺)𝑦) = 0 )
3824, 26, 37syl2anc 692 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑦(-g𝐺)𝑦) = 0 )
39 simpr 477 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 𝑦𝑆)
40 ovex 6643 . . . . . . . . . . 11 (𝑦(-g𝐺)𝑦) ∈ V
41 eqid 2621 . . . . . . . . . . . 12 (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)) = (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧))
42 oveq2 6623 . . . . . . . . . . . 12 (𝑧 = 𝑦 → (𝑦(-g𝐺)𝑧) = (𝑦(-g𝐺)𝑦))
4341, 42elrnmpt1s 5343 . . . . . . . . . . 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 2699 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 0 ∈ ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)))
4645, 22syl6eleqr 2709 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 0 ∈ ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆))
47 eqid 2621 . . . . . . . . 9 𝐽 = 𝐽
48 eqid 2621 . . . . . . . . . . . . . . 15 (+g𝐺) = (+g𝐺)
49 eqid 2621 . . . . . . . . . . . . . . 15 (invg𝐺) = (invg𝐺)
508, 48, 49, 29grpsubval 17405 . . . . . . . . . . . . . 14 ((𝑦𝑋𝑧𝑋) → (𝑦(-g𝐺)𝑧) = (𝑦(+g𝐺)((invg𝐺)‘𝑧)))
5126, 50sylan 488 . . . . . . . . . . . . 13 (((𝐺 ∈ TopGrp ∧ 𝑦𝑆) ∧ 𝑧𝑋) → (𝑦(-g𝐺)𝑧) = (𝑦(+g𝐺)((invg𝐺)‘𝑧)))
5251mpteq2dva 4714 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) = (𝑧𝑋 ↦ (𝑦(+g𝐺)((invg𝐺)‘𝑧))))
538, 49grpinvcl 17407 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → ((invg𝐺)‘𝑧) ∈ 𝑋)
5424, 53sylan 488 . . . . . . . . . . . . 13 (((𝐺 ∈ TopGrp ∧ 𝑦𝑆) ∧ 𝑧𝑋) → ((invg𝐺)‘𝑧) ∈ 𝑋)
558, 49grpinvf 17406 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Grp → (invg𝐺):𝑋𝑋)
5610, 55syl 17 . . . . . . . . . . . . . . 15 (𝐺 ∈ TopGrp → (invg𝐺):𝑋𝑋)
5756adantr 481 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (invg𝐺):𝑋𝑋)
5857feqmptd 6216 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (invg𝐺) = (𝑧𝑋 ↦ ((invg𝐺)‘𝑧)))
59 eqidd 2622 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) = (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)))
60 oveq2 6623 . . . . . . . . . . . . 13 (𝑤 = ((invg𝐺)‘𝑧) → (𝑦(+g𝐺)𝑤) = (𝑦(+g𝐺)((invg𝐺)‘𝑧)))
6154, 58, 59, 60fmptco 6362 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ((𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∘ (invg𝐺)) = (𝑧𝑋 ↦ (𝑦(+g𝐺)((invg𝐺)‘𝑧))))
6252, 61eqtr4d 2658 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) = ((𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∘ (invg𝐺)))
637, 49grpinvhmeo 21830 . . . . . . . . . . . . 13 (𝐺 ∈ TopGrp → (invg𝐺) ∈ (𝐽Homeo𝐽))
6463adantr 481 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (invg𝐺) ∈ (𝐽Homeo𝐽))
65 eqid 2621 . . . . . . . . . . . . . 14 (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) = (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤))
6665, 8, 48, 7tgplacthmeo 21847 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑦𝑋) → (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∈ (𝐽Homeo𝐽))
6726, 66syldan 487 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∈ (𝐽Homeo𝐽))
68 hmeoco 21515 . . . . . . . . . . . 12 (((invg𝐺) ∈ (𝐽Homeo𝐽) ∧ (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∈ (𝐽Homeo𝐽)) → ((𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∘ (invg𝐺)) ∈ (𝐽Homeo𝐽))
6964, 67, 68syl2anc 692 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ((𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∘ (invg𝐺)) ∈ (𝐽Homeo𝐽))
7062, 69eqeltrd 2698 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
71 hmeocn 21503 . . . . . . . . . 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 20659 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
749, 73syl 17 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝑋 = 𝐽)
7574adantr 481 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 𝑋 = 𝐽)
765, 75syl5sseq 3638 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 𝑆 𝐽)
771conncompconn 21175 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 0𝑋) → (𝐽t 𝑆) ∈ Conn)
789, 13, 77syl2anc 692 . . . . . . . . . 10 (𝐺 ∈ TopGrp → (𝐽t 𝑆) ∈ Conn)
7978adantr 481 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝐽t 𝑆) ∈ Conn)
8047, 72, 76, 79connima 21168 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝐽t ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆)) ∈ Conn)
811conncompss 21176 . . . . . . . 8 ((((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) ⊆ 𝑋0 ∈ ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) ∧ (𝐽t ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆)) ∈ Conn) → ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) ⊆ 𝑆)
8236, 46, 80, 81syl3anc 1323 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) ⊆ 𝑆)
8322, 82syl5eqssr 3635 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)) ⊆ 𝑆)
84 ovex 6643 . . . . . . . 8 (𝑦(-g𝐺)𝑧) ∈ V
8584, 41fnmpti 5989 . . . . . . 7 (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)) Fn 𝑆
86 df-f 5861 . . . . . . 7 ((𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)):𝑆𝑆 ↔ ((𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)) Fn 𝑆 ∧ ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)) ⊆ 𝑆))
8785, 86mpbiran 952 . . . . . 6 ((𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)):𝑆𝑆 ↔ ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)) ⊆ 𝑆)
8883, 87sylibr 224 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)):𝑆𝑆)
8941fmpt 6347 . . . . 5 (∀𝑧𝑆 (𝑦(-g𝐺)𝑧) ∈ 𝑆 ↔ (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)):𝑆𝑆)
9088, 89sylibr 224 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ∀𝑧𝑆 (𝑦(-g𝐺)𝑧) ∈ 𝑆)
9190ralrimiva 2962 . . 3 (𝐺 ∈ TopGrp → ∀𝑦𝑆𝑧𝑆 (𝑦(-g𝐺)𝑧) ∈ 𝑆)
928, 29issubg4 17553 . . . 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 1243 . 2 (𝐺 ∈ TopGrp → 𝑆 ∈ (SubGrp‘𝐺))
9510adantr 481 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → 𝐺 ∈ Grp)
96 eqid 2621 . . . . . . . . . . 11 (oppg𝐺) = (oppg𝐺)
9796, 49oppginv 17729 . . . . . . . . . 10 (𝐺 ∈ Grp → (invg𝐺) = (invg‘(oppg𝐺)))
9895, 97syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (invg𝐺) = (invg‘(oppg𝐺)))
9998fveq1d 6160 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → ((invg𝐺)‘((invg𝐺)‘𝑦)) = ((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦)))
100 simprll 801 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → 𝑦𝑋)
1018, 49grpinvinv 17422 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → ((invg𝐺)‘((invg𝐺)‘𝑦)) = 𝑦)
10295, 100, 101syl2anc 692 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → ((invg𝐺)‘((invg𝐺)‘𝑦)) = 𝑦)
10399, 102eqtr3d 2657 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → ((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦)) = 𝑦)
104103oveq1d 6630 . . . . . 6 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦))(+g‘(oppg𝐺))𝑧) = (𝑦(+g‘(oppg𝐺))𝑧))
105 eqid 2621 . . . . . . 7 (+g‘(oppg𝐺)) = (+g‘(oppg𝐺))
10648, 96, 105oppgplus 17719 . . . . . 6 (𝑦(+g‘(oppg𝐺))𝑧) = (𝑧(+g𝐺)𝑦)
107104, 106syl6eq 2671 . . . . 5 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦))(+g‘(oppg𝐺))𝑧) = (𝑧(+g𝐺)𝑦))
1088, 49grpinvcl 17407 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → ((invg𝐺)‘𝑦) ∈ 𝑋)
10995, 100, 108syl2anc 692 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → ((invg𝐺)‘𝑦) ∈ 𝑋)
110 simprlr 802 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → 𝑧𝑋)
111102oveq1d 6630 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg𝐺)‘((invg𝐺)‘𝑦))(+g𝐺)𝑧) = (𝑦(+g𝐺)𝑧))
112 simprr 795 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (𝑦(+g𝐺)𝑧) ∈ 𝑆)
113111, 112eqeltrd 2698 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg𝐺)‘((invg𝐺)‘𝑦))(+g𝐺)𝑧) ∈ 𝑆)
114 eqid 2621 . . . . . . . . . . 11 (𝐺 ~QG 𝑆) = (𝐺 ~QG 𝑆)
1158, 49, 48, 114eqgval 17583 . . . . . . . . . 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 1243 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → ((invg𝐺)‘𝑦)(𝐺 ~QG 𝑆)𝑧)
1188, 11, 7, 1, 114tgpconncompeqg 21855 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ ((invg𝐺)‘𝑦) ∈ 𝑋) → [((invg𝐺)‘𝑦)](𝐺 ~QG 𝑆) = {𝑥 ∈ 𝒫 𝑋 ∣ (((invg𝐺)‘𝑦) ∈ 𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
119109, 118syldan 487 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → [((invg𝐺)‘𝑦)](𝐺 ~QG 𝑆) = {𝑥 ∈ 𝒫 𝑋 ∣ (((invg𝐺)‘𝑦) ∈ 𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
12096oppgtgp 21842 . . . . . . . . . . . . 13 (𝐺 ∈ TopGrp → (oppg𝐺) ∈ TopGrp)
121120adantr 481 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (oppg𝐺) ∈ TopGrp)
12296, 8oppgbas 17721 . . . . . . . . . . . . 13 𝑋 = (Base‘(oppg𝐺))
12396, 11oppgid 17726 . . . . . . . . . . . . 13 0 = (0g‘(oppg𝐺))
12496, 7oppgtopn 17723 . . . . . . . . . . . . 13 𝐽 = (TopOpen‘(oppg𝐺))
125 eqid 2621 . . . . . . . . . . . . 13 ((oppg𝐺) ~QG 𝑆) = ((oppg𝐺) ~QG 𝑆)
126122, 123, 124, 1, 125tgpconncompeqg 21855 . . . . . . . . . . . 12 (((oppg𝐺) ∈ TopGrp ∧ ((invg𝐺)‘𝑦) ∈ 𝑋) → [((invg𝐺)‘𝑦)]((oppg𝐺) ~QG 𝑆) = {𝑥 ∈ 𝒫 𝑋 ∣ (((invg𝐺)‘𝑦) ∈ 𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
127121, 109, 126syl2anc 692 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → [((invg𝐺)‘𝑦)]((oppg𝐺) ~QG 𝑆) = {𝑥 ∈ 𝒫 𝑋 ∣ (((invg𝐺)‘𝑦) ∈ 𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
128119, 127eqtr4d 2658 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → [((invg𝐺)‘𝑦)](𝐺 ~QG 𝑆) = [((invg𝐺)‘𝑦)]((oppg𝐺) ~QG 𝑆))
129128eleq2d 2684 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (𝑧 ∈ [((invg𝐺)‘𝑦)](𝐺 ~QG 𝑆) ↔ 𝑧 ∈ [((invg𝐺)‘𝑦)]((oppg𝐺) ~QG 𝑆)))
130 vex 3193 . . . . . . . . . 10 𝑧 ∈ V
131 fvex 6168 . . . . . . . . . 10 ((invg𝐺)‘𝑦) ∈ V
132130, 131elec 7746 . . . . . . . . 9 (𝑧 ∈ [((invg𝐺)‘𝑦)](𝐺 ~QG 𝑆) ↔ ((invg𝐺)‘𝑦)(𝐺 ~QG 𝑆)𝑧)
133130, 131elec 7746 . . . . . . . . 9 (𝑧 ∈ [((invg𝐺)‘𝑦)]((oppg𝐺) ~QG 𝑆) ↔ ((invg𝐺)‘𝑦)((oppg𝐺) ~QG 𝑆)𝑧)
134129, 132, 1333bitr3g 302 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg𝐺)‘𝑦)(𝐺 ~QG 𝑆)𝑧 ↔ ((invg𝐺)‘𝑦)((oppg𝐺) ~QG 𝑆)𝑧))
135117, 134mpbid 222 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → ((invg𝐺)‘𝑦)((oppg𝐺) ~QG 𝑆)𝑧)
136 eqid 2621 . . . . . . . . 9 (invg‘(oppg𝐺)) = (invg‘(oppg𝐺))
137122, 136, 105, 125eqgval 17583 . . . . . . . 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 222 . . . . . 6 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg𝐺)‘𝑦) ∈ 𝑋𝑧𝑋 ∧ (((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦))(+g‘(oppg𝐺))𝑧) ∈ 𝑆))
140139simp3d 1073 . . . . 5 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦))(+g‘(oppg𝐺))𝑧) ∈ 𝑆)
141107, 140eqeltrrd 2699 . . . 4 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (𝑧(+g𝐺)𝑦) ∈ 𝑆)
142141expr 642 . . 3 ((𝐺 ∈ TopGrp ∧ (𝑦𝑋𝑧𝑋)) → ((𝑦(+g𝐺)𝑧) ∈ 𝑆 → (𝑧(+g𝐺)𝑦) ∈ 𝑆))
143142ralrimivva 2967 . 2 (𝐺 ∈ TopGrp → ∀𝑦𝑋𝑧𝑋 ((𝑦(+g𝐺)𝑧) ∈ 𝑆 → (𝑧(+g𝐺)𝑦) ∈ 𝑆))
1448, 48isnsg2 17564 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦(+g𝐺)𝑧) ∈ 𝑆 → (𝑧(+g𝐺)𝑦) ∈ 𝑆)))
14594, 143, 144sylanbrc 697 1 (𝐺 ∈ TopGrp → 𝑆 ∈ (NrmSGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2908  {crab 2912  Vcvv 3190  wss 3560  c0 3897  𝒫 cpw 4136   cuni 4409   class class class wbr 4623  cmpt 4683  ran crn 5085  cres 5086  cima 5087  ccom 5088   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  [cec 7700  Basecbs 15800  +gcplusg 15881  t crest 16021  TopOpenctopn 16022  0gc0g 16040  Grpcgrp 17362  invgcminusg 17363  -gcsg 17364  SubGrpcsubg 17528  NrmSGrpcnsg 17529   ~QG cqg 17530  oppgcoppg 17715  TopOnctopon 20655   Cn ccn 20968  Conncconn 21154  Homeochmeo 21496  TopGrpctgp 21815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-tpos 7312  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-oadd 7524  df-er 7702  df-ec 7704  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fi 8277  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-tset 15900  df-rest 16023  df-topn 16024  df-0g 16042  df-topgen 16044  df-plusf 17181  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365  df-minusg 17366  df-sbg 17367  df-subg 17531  df-nsg 17532  df-eqg 17533  df-oppg 17716  df-top 20639  df-topon 20656  df-topsp 20677  df-bases 20690  df-cld 20763  df-cn 20971  df-cnp 20972  df-conn 21155  df-tx 21305  df-hmeo 21498  df-tmd 21816  df-tgp 21817
This theorem is referenced by: (None)
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