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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 | β’ π = (BaseβπΊ) |
| tgpconncomp.z | β’ 0 = (0gβπΊ) |
| tgpconncomp.j | β’ π½ = (TopOpenβπΊ) |
| tgpconncomp.s | β’ π = βͺ {π₯ β π« π β£ ( 0 β π₯ β§ (π½ βΎt π₯) β Conn)} |
| Ref | Expression |
|---|---|
| tgpconncompss | β’ ((πΊ β TopGrp β§ π β (SubGrpβπΊ) β§ π β π½) β π β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tgpconncomp.j | . . . 4 β’ π½ = (TopOpenβπΊ) | |
| 2 | tgpconncomp.x | . . . 4 β’ π = (BaseβπΊ) | |
| 3 | 1, 2 | tgptopon 23908 | . . 3 β’ (πΊ β TopGrp β π½ β (TopOnβπ)) |
| 4 | 3 | 3ad2ant1 1130 | . 2 β’ ((πΊ β TopGrp β§ π β (SubGrpβπΊ) β§ π β π½) β π½ β (TopOnβπ)) |
| 5 | simp3 1135 | . . 3 β’ ((πΊ β TopGrp β§ π β (SubGrpβπΊ) β§ π β π½) β π β π½) | |
| 6 | 1 | opnsubg 23934 | . . 3 β’ ((πΊ β TopGrp β§ π β (SubGrpβπΊ) β§ π β π½) β π β (Clsdβπ½)) |
| 7 | 5, 6 | elind 4186 | . 2 β’ ((πΊ β TopGrp β§ π β (SubGrpβπΊ) β§ π β π½) β π β (π½ β© (Clsdβπ½))) |
| 8 | tgpconncomp.z | . . . 4 β’ 0 = (0gβπΊ) | |
| 9 | 8 | subg0cl 19051 | . . 3 β’ (π β (SubGrpβπΊ) β 0 β π) |
| 10 | 9 | 3ad2ant2 1131 | . 2 β’ ((πΊ β TopGrp β§ π β (SubGrpβπΊ) β§ π β π½) β 0 β π) |
| 11 | tgpconncomp.s | . . 3 β’ π = βͺ {π₯ β π« π β£ ( 0 β π₯ β§ (π½ βΎt π₯) β Conn)} | |
| 12 | 11 | conncompclo 23261 | . 2 β’ ((π½ β (TopOnβπ) β§ π β (π½ β© (Clsdβπ½)) β§ 0 β π) β π β π) |
| 13 | 4, 7, 10, 12 | syl3anc 1368 | 1 β’ ((πΊ β TopGrp β§ π β (SubGrpβπΊ) β§ π β π½) β π β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 {crab 3424 β© cin 3939 β wss 3940 π« cpw 4594 βͺ cuni 4899 βcfv 6533 (class class class)co 7401 Basecbs 17143 βΎt crest 17365 TopOpenctopn 17366 0gc0g 17384 SubGrpcsubg 19037 TopOnctopon 22734 Clsdccld 22842 Conncconn 23237 TopGrpctgp 23897 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 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 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-rest 17367 df-0g 17386 df-topgen 17388 df-plusf 18562 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-minusg 18857 df-sbg 18858 df-subg 19040 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-cld 22845 df-ntr 22846 df-cls 22847 df-cn 23053 df-cnp 23054 df-conn 23238 df-tx 23388 df-hmeo 23581 df-tmd 23898 df-tgp 23899 |
| This theorem is referenced by: (None) |
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