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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 23999 | . . 3 โข (๐บ โ TopMnd โ ๐บ โ Mnd) | |
2 | oppgtmd.1 | . . . 4 โข ๐ = (oppgโ๐บ) | |
3 | 2 | oppgmnd 19315 | . . 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 2728 | . . . 4 โข (TopOpenโ๐บ) = (TopOpenโ๐บ) | |
6 | eqid 2728 | . . . 4 โข (Baseโ๐บ) = (Baseโ๐บ) | |
7 | 5, 6 | tmdtopon 24005 | . . 3 โข (๐บ โ TopMnd โ (TopOpenโ๐บ) โ (TopOnโ(Baseโ๐บ))) |
8 | 2, 6 | oppgbas 19310 | . . . 4 โข (Baseโ๐บ) = (Baseโ๐) |
9 | 2, 5 | oppgtopn 19314 | . . . 4 โข (TopOpenโ๐บ) = (TopOpenโ๐) |
10 | 8, 9 | istps 22856 | . . 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 2728 | . . 3 โข (+gโ๐บ) = (+gโ๐บ) | |
13 | id 22 | . . 3 โข (๐บ โ TopMnd โ ๐บ โ TopMnd) | |
14 | 7, 7 | cnmpt2nd 23593 | . . 3 โข (๐บ โ TopMnd โ (๐ฅ โ (Baseโ๐บ), ๐ฆ โ (Baseโ๐บ) โฆ ๐ฆ) โ (((TopOpenโ๐บ) รt (TopOpenโ๐บ)) Cn (TopOpenโ๐บ))) |
15 | 7, 7 | cnmpt1st 23592 | . . 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 24012 | . 2 โข (๐บ โ TopMnd โ (๐ฅ โ (Baseโ๐บ), ๐ฆ โ (Baseโ๐บ) โฆ (๐ฆ(+gโ๐บ)๐ฅ)) โ (((TopOpenโ๐บ) รt (TopOpenโ๐บ)) Cn (TopOpenโ๐บ))) |
17 | eqid 2728 | . . . . 5 โข (+gโ๐) = (+gโ๐) | |
18 | eqid 2728 | . . . . 5 โข (+๐โ๐) = (+๐โ๐) | |
19 | 8, 17, 18 | plusffval 18613 | . . . 4 โข (+๐โ๐) = (๐ฅ โ (Baseโ๐บ), ๐ฆ โ (Baseโ๐บ) โฆ (๐ฅ(+gโ๐)๐ฆ)) |
20 | 12, 2, 17 | oppgplus 19307 | . . . . 5 โข (๐ฅ(+gโ๐)๐ฆ) = (๐ฆ(+gโ๐บ)๐ฅ) |
21 | 6, 6, 20 | mpoeq123i 7502 | . . . 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 2757 | . . 3 โข (๐ฅ โ (Baseโ๐บ), ๐ฆ โ (Baseโ๐บ) โฆ (๐ฆ(+gโ๐บ)๐ฅ)) = (+๐โ๐) |
23 | 22, 9 | istmd 23998 | . 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 6553 (class class class)co 7426 โ cmpo 7428 Basecbs 17187 +gcplusg 17240 TopOpenctopn 17410 +๐cplusf 18604 Mndcmnd 18701 oppgcoppg 19303 TopOnctopon 22832 TopSpctps 22854 Cn ccn 23148 รt ctx 23484 TopMndctmd 23994 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-tset 17259 df-rest 17411 df-topn 17412 df-0g 17430 df-topgen 17432 df-plusf 18606 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-oppg 19304 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cn 23151 df-tx 23486 df-tmd 23996 |
This theorem is referenced by: oppgtgp 24022 |
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