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Mirrors > Home > MPE Home > Th. List > tmdlactcn | Structured version Visualization version GIF version |
Description: The left group action of element π΄ in a topological monoid πΊ is a continuous function. (Contributed by FL, 18-Mar-2008.) (Revised by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
tgplacthmeo.1 | β’ πΉ = (π₯ β π β¦ (π΄ + π₯)) |
tgplacthmeo.2 | β’ π = (BaseβπΊ) |
tgplacthmeo.3 | β’ + = (+gβπΊ) |
tgplacthmeo.4 | β’ π½ = (TopOpenβπΊ) |
Ref | Expression |
---|---|
tmdlactcn | β’ ((πΊ β TopMnd β§ π΄ β π) β πΉ β (π½ Cn π½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgplacthmeo.1 | . 2 β’ πΉ = (π₯ β π β¦ (π΄ + π₯)) | |
2 | tgplacthmeo.4 | . . 3 β’ π½ = (TopOpenβπΊ) | |
3 | tgplacthmeo.3 | . . 3 β’ + = (+gβπΊ) | |
4 | simpl 484 | . . 3 β’ ((πΊ β TopMnd β§ π΄ β π) β πΊ β TopMnd) | |
5 | tgplacthmeo.2 | . . . . 5 β’ π = (BaseβπΊ) | |
6 | 2, 5 | tmdtopon 23455 | . . . 4 β’ (πΊ β TopMnd β π½ β (TopOnβπ)) |
7 | 6 | adantr 482 | . . 3 β’ ((πΊ β TopMnd β§ π΄ β π) β π½ β (TopOnβπ)) |
8 | simpr 486 | . . . 4 β’ ((πΊ β TopMnd β§ π΄ β π) β π΄ β π) | |
9 | 7, 7, 8 | cnmptc 23036 | . . 3 β’ ((πΊ β TopMnd β§ π΄ β π) β (π₯ β π β¦ π΄) β (π½ Cn π½)) |
10 | 7 | cnmptid 23035 | . . 3 β’ ((πΊ β TopMnd β§ π΄ β π) β (π₯ β π β¦ π₯) β (π½ Cn π½)) |
11 | 2, 3, 4, 7, 9, 10 | cnmpt1plusg 23461 | . 2 β’ ((πΊ β TopMnd β§ π΄ β π) β (π₯ β π β¦ (π΄ + π₯)) β (π½ Cn π½)) |
12 | 1, 11 | eqeltrid 2838 | 1 β’ ((πΊ β TopMnd β§ π΄ β π) β πΉ β (π½ Cn π½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β¦ cmpt 5192 βcfv 6500 (class class class)co 7361 Basecbs 17091 +gcplusg 17141 TopOpenctopn 17311 TopOnctopon 22282 Cn ccn 22598 TopMndctmd 23444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-map 8773 df-topgen 17333 df-plusf 18504 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-cn 22601 df-cnp 22602 df-tx 22936 df-tmd 23446 |
This theorem is referenced by: tgplacthmeo 23477 ghmcnp 23489 |
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