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| Mirrors > Home > MPE Home > Th. List > tgpconncompss | Structured version Visualization version GIF version | ||
| Description: The identity component is a subset of any open subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.) | 
| Ref | Expression | 
|---|---|
| tgpconncomp.x | ⊢ 𝑋 = (Base‘𝐺) | 
| tgpconncomp.z | ⊢ 0 = (0g‘𝐺) | 
| tgpconncomp.j | ⊢ 𝐽 = (TopOpen‘𝐺) | 
| tgpconncomp.s | ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} | 
| Ref | Expression | 
|---|---|
| tgpconncompss | ⊢ ((𝐺 ∈ TopGrp ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ 𝐽) → 𝑆 ⊆ 𝑇) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | tgpconncomp.j | . . . 4 ⊢ 𝐽 = (TopOpen‘𝐺) | |
| 2 | tgpconncomp.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
| 3 | 1, 2 | tgptopon 24091 | . . 3 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋)) | 
| 4 | 3 | 3ad2ant1 1133 | . 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋)) | 
| 5 | simp3 1138 | . . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ 𝐽) → 𝑇 ∈ 𝐽) | |
| 6 | 1 | opnsubg 24117 | . . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ 𝐽) → 𝑇 ∈ (Clsd‘𝐽)) | 
| 7 | 5, 6 | elind 4199 | . 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ 𝐽) → 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽))) | 
| 8 | tgpconncomp.z | . . . 4 ⊢ 0 = (0g‘𝐺) | |
| 9 | 8 | subg0cl 19153 | . . 3 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑇) | 
| 10 | 9 | 3ad2ant2 1134 | . 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ 𝐽) → 0 ∈ 𝑇) | 
| 11 | tgpconncomp.s | . . 3 ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} | |
| 12 | 11 | conncompclo 23444 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 0 ∈ 𝑇) → 𝑆 ⊆ 𝑇) | 
| 13 | 4, 7, 10, 12 | syl3anc 1372 | 1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ 𝐽) → 𝑆 ⊆ 𝑇) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 {crab 3435 ∩ cin 3949 ⊆ wss 3950 𝒫 cpw 4599 ∪ cuni 4906 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 ↾t crest 17466 TopOpenctopn 17467 0gc0g 17485 SubGrpcsubg 19139 TopOnctopon 22917 Clsdccld 23025 Conncconn 23420 TopGrpctgp 24080 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fi 9452 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-rest 17468 df-0g 17487 df-topgen 17489 df-plusf 18653 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-minusg 18956 df-sbg 18957 df-subg 19142 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-cn 23236 df-cnp 23237 df-conn 23421 df-tx 23571 df-hmeo 23764 df-tmd 24081 df-tgp 24082 | 
| This theorem is referenced by: (None) | 
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