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Theorem tgpconncomp 21897
Description: The identity component, the connected component containing the identity element, is a closed (conncompcld 21218) 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 3679 . . . . . 6 {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋
3 sspwuni 4602 . . . . . 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 3627 . . . 4 𝑆𝑋
65a1i 11 . . 3 (𝐺 ∈ TopGrp → 𝑆𝑋)
7 tgpconncomp.j . . . . . 6 𝐽 = (TopOpen‘𝐺)
8 tgpconncomp.x . . . . . 6 𝑋 = (Base‘𝐺)
97, 8tgptopon 21867 . . . . 5 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
10 tgpgrp 21863 . . . . . 6 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
11 tgpconncomp.z . . . . . . 7 0 = (0g𝐺)
128, 11grpidcl 17431 . . . . . 6 (𝐺 ∈ Grp → 0𝑋)
1310, 12syl 17 . . . . 5 (𝐺 ∈ TopGrp → 0𝑋)
141conncompid 21215 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 0𝑋) → 0𝑆)
159, 13, 14syl2anc 692 . . . 4 (𝐺 ∈ TopGrp → 0𝑆)
16 ne0i 3913 . . . 4 ( 0𝑆𝑆 ≠ ∅)
1715, 16syl 17 . . 3 (𝐺 ∈ TopGrp → 𝑆 ≠ ∅)
18 df-ima 5117 . . . . . . . 8 ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) = ran ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) ↾ 𝑆)
19 resmpt 5437 . . . . . . . . . 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 5341 . . . . . . . 8 ran ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) ↾ 𝑆) = ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧))
2218, 21eqtri 2642 . . . . . . 7 ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) = ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧))
23 imassrn 5465 . . . . . . . . 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 3595 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 𝑦𝑋)
2726adantr 481 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑦𝑆) ∧ 𝑧𝑋) → 𝑦𝑋)
28 simpr 477 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑦𝑆) ∧ 𝑧𝑋) → 𝑧𝑋)
29 eqid 2620 . . . . . . . . . . . . 13 (-g𝐺) = (-g𝐺)
308, 29grpsubcl 17476 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑦𝑋𝑧𝑋) → (𝑦(-g𝐺)𝑧) ∈ 𝑋)
3125, 27, 28, 30syl3anc 1324 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑦𝑆) ∧ 𝑧𝑋) → (𝑦(-g𝐺)𝑧) ∈ 𝑋)
32 eqid 2620 . . . . . . . . . . 11 (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) = (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧))
3331, 32fmptd 6371 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)):𝑋𝑋)
34 frn 6040 . . . . . . . . . 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 3606 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) ⊆ 𝑋)
378, 11, 29grpsubid 17480 . . . . . . . . . . 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 6663 . . . . . . . . . . 11 (𝑦(-g𝐺)𝑦) ∈ V
41 eqid 2620 . . . . . . . . . . . 12 (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)) = (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧))
42 oveq2 6643 . . . . . . . . . . . 12 (𝑧 = 𝑦 → (𝑦(-g𝐺)𝑧) = (𝑦(-g𝐺)𝑦))
4341, 42elrnmpt1s 5362 . . . . . . . . . . 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 2700 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 0 ∈ ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)))
4645, 22syl6eleqr 2710 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 0 ∈ ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆))
47 eqid 2620 . . . . . . . . 9 𝐽 = 𝐽
48 eqid 2620 . . . . . . . . . . . . . . 15 (+g𝐺) = (+g𝐺)
49 eqid 2620 . . . . . . . . . . . . . . 15 (invg𝐺) = (invg𝐺)
508, 48, 49, 29grpsubval 17446 . . . . . . . . . . . . . 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 4735 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) = (𝑧𝑋 ↦ (𝑦(+g𝐺)((invg𝐺)‘𝑧))))
538, 49grpinvcl 17448 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → ((invg𝐺)‘𝑧) ∈ 𝑋)
5424, 53sylan 488 . . . . . . . . . . . . 13 (((𝐺 ∈ TopGrp ∧ 𝑦𝑆) ∧ 𝑧𝑋) → ((invg𝐺)‘𝑧) ∈ 𝑋)
558, 49grpinvf 17447 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Grp → (invg𝐺):𝑋𝑋)
5610, 55syl 17 . . . . . . . . . . . . . . 15 (𝐺 ∈ TopGrp → (invg𝐺):𝑋𝑋)
5756adantr 481 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (invg𝐺):𝑋𝑋)
5857feqmptd 6236 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (invg𝐺) = (𝑧𝑋 ↦ ((invg𝐺)‘𝑧)))
59 eqidd 2621 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) = (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)))
60 oveq2 6643 . . . . . . . . . . . . 13 (𝑤 = ((invg𝐺)‘𝑧) → (𝑦(+g𝐺)𝑤) = (𝑦(+g𝐺)((invg𝐺)‘𝑧)))
6154, 58, 59, 60fmptco 6382 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ((𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∘ (invg𝐺)) = (𝑧𝑋 ↦ (𝑦(+g𝐺)((invg𝐺)‘𝑧))))
6252, 61eqtr4d 2657 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) = ((𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∘ (invg𝐺)))
637, 49grpinvhmeo 21871 . . . . . . . . . . . . 13 (𝐺 ∈ TopGrp → (invg𝐺) ∈ (𝐽Homeo𝐽))
6463adantr 481 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (invg𝐺) ∈ (𝐽Homeo𝐽))
65 eqid 2620 . . . . . . . . . . . . . 14 (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) = (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤))
6665, 8, 48, 7tgplacthmeo 21888 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑦𝑋) → (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∈ (𝐽Homeo𝐽))
6726, 66syldan 487 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑤𝑋 ↦ (𝑦(+g𝐺)𝑤)) ∈ (𝐽Homeo𝐽))
68 hmeoco 21556 . . . . . . . . . . . 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 2699 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
71 hmeocn 21544 . . . . . . . . . 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 20700 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
749, 73syl 17 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝑋 = 𝐽)
7574adantr 481 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 𝑋 = 𝐽)
765, 75syl5sseq 3645 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → 𝑆 𝐽)
771conncompconn 21216 . . . . . . . . . . 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 21209 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → (𝐽t ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆)) ∈ Conn)
811conncompss 21217 . . . . . . . 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 1324 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ((𝑧𝑋 ↦ (𝑦(-g𝐺)𝑧)) “ 𝑆) ⊆ 𝑆)
8322, 82syl5eqssr 3642 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ran (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)) ⊆ 𝑆)
84 ovex 6663 . . . . . . . 8 (𝑦(-g𝐺)𝑧) ∈ V
8584, 41fnmpti 6009 . . . . . . 7 (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)) Fn 𝑆
86 df-f 5880 . . . . . . 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 6367 . . . . 5 (∀𝑧𝑆 (𝑦(-g𝐺)𝑧) ∈ 𝑆 ↔ (𝑧𝑆 ↦ (𝑦(-g𝐺)𝑧)):𝑆𝑆)
9088, 89sylibr 224 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑦𝑆) → ∀𝑧𝑆 (𝑦(-g𝐺)𝑧) ∈ 𝑆)
9190ralrimiva 2963 . . 3 (𝐺 ∈ TopGrp → ∀𝑦𝑆𝑧𝑆 (𝑦(-g𝐺)𝑧) ∈ 𝑆)
928, 29issubg4 17594 . . . 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 2620 . . . . . . . . . . 11 (oppg𝐺) = (oppg𝐺)
9796, 49oppginv 17770 . . . . . . . . . 10 (𝐺 ∈ Grp → (invg𝐺) = (invg‘(oppg𝐺)))
9895, 97syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (invg𝐺) = (invg‘(oppg𝐺)))
9998fveq1d 6180 . . . . . . . 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 17463 . . . . . . . . 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 2656 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → ((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦)) = 𝑦)
104103oveq1d 6650 . . . . . 6 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦))(+g‘(oppg𝐺))𝑧) = (𝑦(+g‘(oppg𝐺))𝑧))
105 eqid 2620 . . . . . . 7 (+g‘(oppg𝐺)) = (+g‘(oppg𝐺))
10648, 96, 105oppgplus 17760 . . . . . 6 (𝑦(+g‘(oppg𝐺))𝑧) = (𝑧(+g𝐺)𝑦)
107104, 106syl6eq 2670 . . . . 5 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg‘(oppg𝐺))‘((invg𝐺)‘𝑦))(+g‘(oppg𝐺))𝑧) = (𝑧(+g𝐺)𝑦))
1088, 49grpinvcl 17448 . . . . . . . . . 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 6650 . . . . . . . . . 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 2699 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (((invg𝐺)‘((invg𝐺)‘𝑦))(+g𝐺)𝑧) ∈ 𝑆)
114 eqid 2620 . . . . . . . . . . 11 (𝐺 ~QG 𝑆) = (𝐺 ~QG 𝑆)
1158, 49, 48, 114eqgval 17624 . . . . . . . . . 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 21896 . . . . . . . . . . . 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 21883 . . . . . . . . . . . . 13 (𝐺 ∈ TopGrp → (oppg𝐺) ∈ TopGrp)
121120adantr 481 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (oppg𝐺) ∈ TopGrp)
12296, 8oppgbas 17762 . . . . . . . . . . . . 13 𝑋 = (Base‘(oppg𝐺))
12396, 11oppgid 17767 . . . . . . . . . . . . 13 0 = (0g‘(oppg𝐺))
12496, 7oppgtopn 17764 . . . . . . . . . . . . 13 𝐽 = (TopOpen‘(oppg𝐺))
125 eqid 2620 . . . . . . . . . . . . 13 ((oppg𝐺) ~QG 𝑆) = ((oppg𝐺) ~QG 𝑆)
126122, 123, 124, 1, 125tgpconncompeqg 21896 . . . . . . . . . . . 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 2657 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → [((invg𝐺)‘𝑦)](𝐺 ~QG 𝑆) = [((invg𝐺)‘𝑦)]((oppg𝐺) ~QG 𝑆))
129128eleq2d 2685 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ ((𝑦𝑋𝑧𝑋) ∧ (𝑦(+g𝐺)𝑧) ∈ 𝑆)) → (𝑧 ∈ [((invg𝐺)‘𝑦)](𝐺 ~QG 𝑆) ↔ 𝑧 ∈ [((invg𝐺)‘𝑦)]((oppg𝐺) ~QG 𝑆)))
130 vex 3198 . . . . . . . . . 10 𝑧 ∈ V
131 fvex 6188 . . . . . . . . . 10 ((invg𝐺)‘𝑦) ∈ V
132130, 131elec 7771 . . . . . . . . 9 (𝑧 ∈ [((invg𝐺)‘𝑦)](𝐺 ~QG 𝑆) ↔ ((invg𝐺)‘𝑦)(𝐺 ~QG 𝑆)𝑧)
133130, 131elec 7771 . . . . . . . . 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 2620 . . . . . . . . 9 (invg‘(oppg𝐺)) = (invg‘(oppg𝐺))
137122, 136, 105, 125eqgval 17624 . . . . . . . 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 2700 . . . 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 2968 . 2 (𝐺 ∈ TopGrp → ∀𝑦𝑋𝑧𝑋 ((𝑦(+g𝐺)𝑧) ∈ 𝑆 → (𝑧(+g𝐺)𝑦) ∈ 𝑆))
1448, 48isnsg2 17605 . 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 1481  wcel 1988  wne 2791  wral 2909  {crab 2913  Vcvv 3195  wss 3567  c0 3907  𝒫 cpw 4149   cuni 4427   class class class wbr 4644  cmpt 4720  ran crn 5105  cres 5106  cima 5107  ccom 5108   Fn wfn 5871  wf 5872  cfv 5876  (class class class)co 6635  [cec 7725  Basecbs 15838  +gcplusg 15922  t crest 16062  TopOpenctopn 16063  0gc0g 16081  Grpcgrp 17403  invgcminusg 17404  -gcsg 17405  SubGrpcsubg 17569  NrmSGrpcnsg 17570   ~QG cqg 17571  oppgcoppg 17756  TopOnctopon 20696   Cn ccn 21009  Conncconn 21195  Homeochmeo 21537  TopGrpctgp 21856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-tpos 7337  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-oadd 7549  df-er 7727  df-ec 7729  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fi 8302  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-plusg 15935  df-tset 15941  df-rest 16064  df-topn 16065  df-0g 16083  df-topgen 16085  df-plusf 17222  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-minusg 17407  df-sbg 17408  df-subg 17572  df-nsg 17573  df-eqg 17574  df-oppg 17757  df-top 20680  df-topon 20697  df-topsp 20718  df-bases 20731  df-cld 20804  df-cn 21012  df-cnp 21013  df-conn 21196  df-tx 21346  df-hmeo 21539  df-tmd 21857  df-tgp 21858
This theorem is referenced by: (None)
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