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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 | ⊢ 𝑂 = (oppg‘𝐺) |
Ref | Expression |
---|---|
oppgtmd | ⊢ (𝐺 ∈ TopMnd → 𝑂 ∈ TopMnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tmdmnd 22680 | . . 3 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd) | |
2 | oppgtmd.1 | . . . 4 ⊢ 𝑂 = (oppg‘𝐺) | |
3 | 2 | oppgmnd 18474 | . . 3 ⊢ (𝐺 ∈ Mnd → 𝑂 ∈ Mnd) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝐺 ∈ TopMnd → 𝑂 ∈ Mnd) |
5 | eqid 2798 | . . . 4 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
6 | eqid 2798 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
7 | 5, 6 | tmdtopon 22686 | . . 3 ⊢ (𝐺 ∈ TopMnd → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺))) |
8 | 2, 6 | oppgbas 18471 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝑂) |
9 | 2, 5 | oppgtopn 18473 | . . . 4 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝑂) |
10 | 8, 9 | istps 21539 | . . 3 ⊢ (𝑂 ∈ TopSp ↔ (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺))) |
11 | 7, 10 | sylibr 237 | . 2 ⊢ (𝐺 ∈ TopMnd → 𝑂 ∈ TopSp) |
12 | eqid 2798 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
13 | id 22 | . . 3 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopMnd) | |
14 | 7, 7 | cnmpt2nd 22274 | . . 3 ⊢ (𝐺 ∈ TopMnd → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))) |
15 | 7, 7 | cnmpt1st 22273 | . . 3 ⊢ (𝐺 ∈ TopMnd → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))) |
16 | 5, 12, 13, 7, 7, 14, 15 | cnmpt2plusg 22693 | . 2 ⊢ (𝐺 ∈ TopMnd → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑦(+g‘𝐺)𝑥)) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))) |
17 | eqid 2798 | . . . . 5 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
18 | eqid 2798 | . . . . 5 ⊢ (+𝑓‘𝑂) = (+𝑓‘𝑂) | |
19 | 8, 17, 18 | plusffval 17850 | . . . 4 ⊢ (+𝑓‘𝑂) = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝑂)𝑦)) |
20 | 12, 2, 17 | oppgplus 18469 | . . . . 5 ⊢ (𝑥(+g‘𝑂)𝑦) = (𝑦(+g‘𝐺)𝑥) |
21 | 6, 6, 20 | mpoeq123i 7209 | . . . 4 ⊢ (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝑂)𝑦)) = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑦(+g‘𝐺)𝑥)) |
22 | 19, 21 | eqtr2i 2822 | . . 3 ⊢ (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑦(+g‘𝐺)𝑥)) = (+𝑓‘𝑂) |
23 | 22, 9 | istmd 22679 | . 2 ⊢ (𝑂 ∈ TopMnd ↔ (𝑂 ∈ Mnd ∧ 𝑂 ∈ TopSp ∧ (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑦(+g‘𝐺)𝑥)) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺)))) |
24 | 4, 11, 16, 23 | syl3anbrc 1340 | 1 ⊢ (𝐺 ∈ TopMnd → 𝑂 ∈ TopMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 Basecbs 16475 +gcplusg 16557 TopOpenctopn 16687 +𝑓cplusf 17841 Mndcmnd 17903 oppgcoppg 18465 TopOnctopon 21515 TopSpctps 21537 Cn ccn 21829 ×t ctx 22165 TopMndctmd 22675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-tset 16576 df-rest 16688 df-topn 16689 df-0g 16707 df-topgen 16709 df-plusf 17843 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-oppg 18466 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cn 21832 df-tx 22167 df-tmd 22677 |
This theorem is referenced by: oppgtgp 22703 |
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