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Mirrors > Home > MPE Home > Th. List > oppgtmd | Structured version Visualization version GIF version |
Description: The opposite of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.) |
Ref | Expression |
---|---|
oppgtmd.1 | โข ๐ = (oppgโ๐บ) |
Ref | Expression |
---|---|
oppgtmd | โข (๐บ โ TopMnd โ ๐ โ TopMnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tmdmnd 23929 | . . 3 โข (๐บ โ TopMnd โ ๐บ โ Mnd) | |
2 | oppgtmd.1 | . . . 4 โข ๐ = (oppgโ๐บ) | |
3 | 2 | oppgmnd 19270 | . . 3 โข (๐บ โ Mnd โ ๐ โ Mnd) |
4 | 1, 3 | syl 17 | . 2 โข (๐บ โ TopMnd โ ๐ โ Mnd) |
5 | eqid 2726 | . . . 4 โข (TopOpenโ๐บ) = (TopOpenโ๐บ) | |
6 | eqid 2726 | . . . 4 โข (Baseโ๐บ) = (Baseโ๐บ) | |
7 | 5, 6 | tmdtopon 23935 | . . 3 โข (๐บ โ TopMnd โ (TopOpenโ๐บ) โ (TopOnโ(Baseโ๐บ))) |
8 | 2, 6 | oppgbas 19265 | . . . 4 โข (Baseโ๐บ) = (Baseโ๐) |
9 | 2, 5 | oppgtopn 19269 | . . . 4 โข (TopOpenโ๐บ) = (TopOpenโ๐) |
10 | 8, 9 | istps 22786 | . . 3 โข (๐ โ TopSp โ (TopOpenโ๐บ) โ (TopOnโ(Baseโ๐บ))) |
11 | 7, 10 | sylibr 233 | . 2 โข (๐บ โ TopMnd โ ๐ โ TopSp) |
12 | eqid 2726 | . . 3 โข (+gโ๐บ) = (+gโ๐บ) | |
13 | id 22 | . . 3 โข (๐บ โ TopMnd โ ๐บ โ TopMnd) | |
14 | 7, 7 | cnmpt2nd 23523 | . . 3 โข (๐บ โ TopMnd โ (๐ฅ โ (Baseโ๐บ), ๐ฆ โ (Baseโ๐บ) โฆ ๐ฆ) โ (((TopOpenโ๐บ) รt (TopOpenโ๐บ)) Cn (TopOpenโ๐บ))) |
15 | 7, 7 | cnmpt1st 23522 | . . 3 โข (๐บ โ TopMnd โ (๐ฅ โ (Baseโ๐บ), ๐ฆ โ (Baseโ๐บ) โฆ ๐ฅ) โ (((TopOpenโ๐บ) รt (TopOpenโ๐บ)) Cn (TopOpenโ๐บ))) |
16 | 5, 12, 13, 7, 7, 14, 15 | cnmpt2plusg 23942 | . 2 โข (๐บ โ TopMnd โ (๐ฅ โ (Baseโ๐บ), ๐ฆ โ (Baseโ๐บ) โฆ (๐ฆ(+gโ๐บ)๐ฅ)) โ (((TopOpenโ๐บ) รt (TopOpenโ๐บ)) Cn (TopOpenโ๐บ))) |
17 | eqid 2726 | . . . . 5 โข (+gโ๐) = (+gโ๐) | |
18 | eqid 2726 | . . . . 5 โข (+๐โ๐) = (+๐โ๐) | |
19 | 8, 17, 18 | plusffval 18576 | . . . 4 โข (+๐โ๐) = (๐ฅ โ (Baseโ๐บ), ๐ฆ โ (Baseโ๐บ) โฆ (๐ฅ(+gโ๐)๐ฆ)) |
20 | 12, 2, 17 | oppgplus 19262 | . . . . 5 โข (๐ฅ(+gโ๐)๐ฆ) = (๐ฆ(+gโ๐บ)๐ฅ) |
21 | 6, 6, 20 | mpoeq123i 7480 | . . . 4 โข (๐ฅ โ (Baseโ๐บ), ๐ฆ โ (Baseโ๐บ) โฆ (๐ฅ(+gโ๐)๐ฆ)) = (๐ฅ โ (Baseโ๐บ), ๐ฆ โ (Baseโ๐บ) โฆ (๐ฆ(+gโ๐บ)๐ฅ)) |
22 | 19, 21 | eqtr2i 2755 | . . 3 โข (๐ฅ โ (Baseโ๐บ), ๐ฆ โ (Baseโ๐บ) โฆ (๐ฆ(+gโ๐บ)๐ฅ)) = (+๐โ๐) |
23 | 22, 9 | istmd 23928 | . 2 โข (๐ โ TopMnd โ (๐ โ Mnd โง ๐ โ TopSp โง (๐ฅ โ (Baseโ๐บ), ๐ฆ โ (Baseโ๐บ) โฆ (๐ฆ(+gโ๐บ)๐ฅ)) โ (((TopOpenโ๐บ) รt (TopOpenโ๐บ)) Cn (TopOpenโ๐บ)))) |
24 | 4, 11, 16, 23 | syl3anbrc 1340 | 1 โข (๐บ โ TopMnd โ ๐ โ TopMnd) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โcfv 6536 (class class class)co 7404 โ cmpo 7406 Basecbs 17150 +gcplusg 17203 TopOpenctopn 17373 +๐cplusf 18567 Mndcmnd 18664 oppgcoppg 19258 TopOnctopon 22762 TopSpctps 22784 Cn ccn 23078 รt ctx 23414 TopMndctmd 23924 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-tset 17222 df-rest 17374 df-topn 17375 df-0g 17393 df-topgen 17395 df-plusf 18569 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-oppg 19259 df-top 22746 df-topon 22763 df-topsp 22785 df-bases 22799 df-cn 23081 df-tx 23416 df-tmd 23926 |
This theorem is referenced by: oppgtgp 23952 |
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