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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 482 | . . 3 β’ ((πΊ β TopMnd β§ π΄ β π) β πΊ β TopMnd) | |
5 | tgplacthmeo.2 | . . . . 5 β’ π = (BaseβπΊ) | |
6 | 2, 5 | tmdtopon 23940 | . . . 4 β’ (πΊ β TopMnd β π½ β (TopOnβπ)) |
7 | 6 | adantr 480 | . . 3 β’ ((πΊ β TopMnd β§ π΄ β π) β π½ β (TopOnβπ)) |
8 | simpr 484 | . . . 4 β’ ((πΊ β TopMnd β§ π΄ β π) β π΄ β π) | |
9 | 7, 7, 8 | cnmptc 23521 | . . 3 β’ ((πΊ β TopMnd β§ π΄ β π) β (π₯ β π β¦ π΄) β (π½ Cn π½)) |
10 | 7 | cnmptid 23520 | . . 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 23946 | . 2 β’ ((πΊ β TopMnd β§ π΄ β π) β (π₯ β π β¦ (π΄ + π₯)) β (π½ Cn π½)) |
12 | 1, 11 | eqeltrid 2831 | 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 395 = wceq 1533 β wcel 2098 β¦ cmpt 5224 βcfv 6537 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 TopOpenctopn 17376 TopOnctopon 22767 Cn ccn 23083 TopMndctmd 23929 |
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-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-ral 3056 df-rex 3065 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-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-id 5567 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-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-map 8824 df-topgen 17398 df-plusf 18572 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cn 23086 df-cnp 23087 df-tx 23421 df-tmd 23931 |
This theorem is referenced by: tgplacthmeo 23962 ghmcnp 23974 |
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