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Theorem mhmhmeotmd 28375
Description: Deduce a Topological Monoid using mapping that is both a homeomorphism and a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)
Hypotheses
Ref Expression
mhmhmeotmd.m  |-  F  e.  ( S MndHom  T )
mhmhmeotmd.h  |-  F  e.  ( ( TopOpen `  S
) Homeo ( TopOpen `  T
) )
mhmhmeotmd.t  |-  S  e. TopMnd
mhmhmeotmd.s  |-  T  e. 
TopSp
Assertion
Ref Expression
mhmhmeotmd  |-  T  e. TopMnd

Proof of Theorem mhmhmeotmd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhmhmeotmd.m . . 3  |-  F  e.  ( S MndHom  T )
2 mhmrcl2 16296 . . 3  |-  ( F  e.  ( S MndHom  T
)  ->  T  e.  Mnd )
31, 2ax-mp 5 . 2  |-  T  e. 
Mnd
4 mhmhmeotmd.s . 2  |-  T  e. 
TopSp
5 mhmhmeotmd.h . . 3  |-  F  e.  ( ( TopOpen `  S
) Homeo ( TopOpen `  T
) )
6 mhmrcl1 16295 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  ->  S  e.  Mnd )
71, 6ax-mp 5 . . . 4  |-  S  e. 
Mnd
8 eqid 2404 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
9 eqid 2404 . . . . 5  |-  ( +f `  S )  =  ( +f `  S )
108, 9mndplusf 16265 . . . 4  |-  ( S  e.  Mnd  ->  ( +f `  S
) : ( (
Base `  S )  X.  ( Base `  S
) ) --> ( Base `  S ) )
117, 10ax-mp 5 . . 3  |-  ( +f `  S ) : ( ( Base `  S )  X.  ( Base `  S ) ) --> ( Base `  S
)
12 eqid 2404 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
13 eqid 2404 . . . . 5  |-  ( +f `  T )  =  ( +f `  T )
1412, 13mndplusf 16265 . . . 4  |-  ( T  e.  Mnd  ->  ( +f `  T
) : ( (
Base `  T )  X.  ( Base `  T
) ) --> ( Base `  T ) )
153, 14ax-mp 5 . . 3  |-  ( +f `  T ) : ( ( Base `  T )  X.  ( Base `  T ) ) --> ( Base `  T
)
16 mhmhmeotmd.t . . . 4  |-  S  e. TopMnd
17 eqid 2404 . . . . 5  |-  ( TopOpen `  S )  =  (
TopOpen `  S )
1817, 8tmdtopon 20874 . . . 4  |-  ( S  e. TopMnd  ->  ( TopOpen `  S
)  e.  (TopOn `  ( Base `  S )
) )
1916, 18ax-mp 5 . . 3  |-  ( TopOpen `  S )  e.  (TopOn `  ( Base `  S
) )
20 eqid 2404 . . . . 5  |-  ( TopOpen `  T )  =  (
TopOpen `  T )
2112, 20istps 19731 . . . 4  |-  ( T  e.  TopSp 
<->  ( TopOpen `  T )  e.  (TopOn `  ( Base `  T ) ) )
224, 21mpbi 210 . . 3  |-  ( TopOpen `  T )  e.  (TopOn `  ( Base `  T
) )
23 eqid 2404 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
24 eqid 2404 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
258, 23, 24mhmlin 16299 . . . . 5  |-  ( ( F  e.  ( S MndHom  T )  /\  x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )
)  ->  ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T
) ( F `  y ) ) )
261, 25mp3an1 1315 . . . 4  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  ( F `  ( x
( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T ) ( F `
 y ) ) )
278, 23, 9plusfval 16204 . . . . 5  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  (
x ( +f `  S ) y )  =  ( x ( +g  `  S ) y ) )
2827fveq2d 5855 . . . 4  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  ( F `  ( x
( +f `  S ) y ) )  =  ( F `
 ( x ( +g  `  S ) y ) ) )
298, 12mhmf 16297 . . . . . . 7  |-  ( F  e.  ( S MndHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
301, 29ax-mp 5 . . . . . 6  |-  F :
( Base `  S ) --> ( Base `  T )
3130ffvelrni 6010 . . . . 5  |-  ( x  e.  ( Base `  S
)  ->  ( F `  x )  e.  (
Base `  T )
)
3230ffvelrni 6010 . . . . 5  |-  ( y  e.  ( Base `  S
)  ->  ( F `  y )  e.  (
Base `  T )
)
3312, 24, 13plusfval 16204 . . . . 5  |-  ( ( ( F `  x
)  e.  ( Base `  T )  /\  ( F `  y )  e.  ( Base `  T
) )  ->  (
( F `  x
) ( +f `  T ) ( F `
 y ) )  =  ( ( F `
 x ) ( +g  `  T ) ( F `  y
) ) )
3431, 32, 33syl2an 477 . . . 4  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  (
( F `  x
) ( +f `  T ) ( F `
 y ) )  =  ( ( F `
 x ) ( +g  `  T ) ( F `  y
) ) )
3526, 28, 343eqtr4d 2455 . . 3  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  ( F `  ( x
( +f `  S ) y ) )  =  ( ( F `  x ) ( +f `  T ) ( F `
 y ) ) )
3617, 9tmdcn 20876 . . . 4  |-  ( S  e. TopMnd  ->  ( +f `  S )  e.  ( ( ( TopOpen `  S
)  tX  ( TopOpen `  S ) )  Cn  ( TopOpen `  S )
) )
3716, 36ax-mp 5 . . 3  |-  ( +f `  S )  e.  ( ( (
TopOpen `  S )  tX  ( TopOpen `  S )
)  Cn  ( TopOpen `  S ) )
385, 11, 15, 19, 22, 35, 37mndpluscn 28374 . 2  |-  ( +f `  T )  e.  ( ( (
TopOpen `  T )  tX  ( TopOpen `  T )
)  Cn  ( TopOpen `  T ) )
3913, 20istmd 20867 . 2  |-  ( T  e. TopMnd 
<->  ( T  e.  Mnd  /\  T  e.  TopSp  /\  ( +f `  T
)  e.  ( ( ( TopOpen `  T )  tX  ( TopOpen `  T )
)  Cn  ( TopOpen `  T ) ) ) )
403, 4, 38, 39mpbir3an 1181 1  |-  T  e. TopMnd
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1407    e. wcel 1844    X. cxp 4823   -->wf 5567   ` cfv 5571  (class class class)co 6280   Basecbs 14843   +g cplusg 14911   TopOpenctopn 15038   +fcplusf 16195   Mndcmnd 16245   MndHom cmhm 16290  TopOnctopon 19689   TopSpctps 19691    Cn ccn 20020    tX ctx 20355   Homeochmeo 20548  TopMndctmd 20863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-map 7461  df-topgen 15060  df-plusf 16197  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-mhm 16292  df-top 19693  df-bases 19695  df-topon 19696  df-topsp 19697  df-cn 20023  df-tx 20357  df-hmeo 20550  df-tmd 20865
This theorem is referenced by:  xrge0tmd  28394
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