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Theorem mhmhmeotmd 28807
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 16664 . . 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 16663 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  ->  S  e.  Mnd )
71, 6ax-mp 5 . . . 4  |-  S  e. 
Mnd
8 eqid 2471 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
9 eqid 2471 . . . . 5  |-  ( +f `  S )  =  ( +f `  S )
108, 9mndplusf 16633 . . . 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 2471 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
13 eqid 2471 . . . . 5  |-  ( +f `  T )  =  ( +f `  T )
1412, 13mndplusf 16633 . . . 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 2471 . . . . 5  |-  ( TopOpen `  S )  =  (
TopOpen `  S )
1817, 8tmdtopon 21174 . . . 4  |-  ( S  e. TopMnd  ->  ( TopOpen `  S
)  e.  (TopOn `  ( Base `  S )
) )
1916, 18ax-mp 5 . . 3  |-  ( TopOpen `  S )  e.  (TopOn `  ( Base `  S
) )
20 eqid 2471 . . . . 5  |-  ( TopOpen `  T )  =  (
TopOpen `  T )
2112, 20istps 20028 . . . 4  |-  ( T  e.  TopSp 
<->  ( TopOpen `  T )  e.  (TopOn `  ( Base `  T ) ) )
224, 21mpbi 213 . . 3  |-  ( TopOpen `  T )  e.  (TopOn `  ( Base `  T
) )
23 eqid 2471 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
24 eqid 2471 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
258, 23, 24mhmlin 16667 . . . . 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 1377 . . . 4  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  ( F `  ( x
( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T ) ( F `
 y ) ) )
278, 23, 9plusfval 16572 . . . . 5  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  (
x ( +f `  S ) y )  =  ( x ( +g  `  S ) y ) )
2827fveq2d 5883 . . . 4  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  ( F `  ( x
( +f `  S ) y ) )  =  ( F `
 ( x ( +g  `  S ) y ) ) )
298, 12mhmf 16665 . . . . . . 7  |-  ( F  e.  ( S MndHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
301, 29ax-mp 5 . . . . . 6  |-  F :
( Base `  S ) --> ( Base `  T )
3130ffvelrni 6036 . . . . 5  |-  ( x  e.  ( Base `  S
)  ->  ( F `  x )  e.  (
Base `  T )
)
3230ffvelrni 6036 . . . . 5  |-  ( y  e.  ( Base `  S
)  ->  ( F `  y )  e.  (
Base `  T )
)
3312, 24, 13plusfval 16572 . . . . 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 485 . . . 4  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  (
( F `  x
) ( +f `  T ) ( F `
 y ) )  =  ( ( F `
 x ) ( +g  `  T ) ( F `  y
) ) )
3526, 28, 343eqtr4d 2515 . . 3  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  ( F `  ( x
( +f `  S ) y ) )  =  ( ( F `  x ) ( +f `  T ) ( F `
 y ) ) )
3617, 9tmdcn 21176 . . . 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 28806 . 2  |-  ( +f `  T )  e.  ( ( (
TopOpen `  T )  tX  ( TopOpen `  T )
)  Cn  ( TopOpen `  T ) )
3913, 20istmd 21167 . 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 1212 1  |-  T  e. TopMnd
Colors of variables: wff setvar class
Syntax hints:    /\ wa 376    = wceq 1452    e. wcel 1904    X. cxp 4837   -->wf 5585   ` cfv 5589  (class class class)co 6308   Basecbs 15199   +g cplusg 15268   TopOpenctopn 15398   +fcplusf 16563   Mndcmnd 16613   MndHom cmhm 16658  TopOnctopon 19995   TopSpctps 19996    Cn ccn 20317    tX ctx 20652   Homeochmeo 20845  TopMndctmd 21163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-map 7492  df-topgen 15420  df-plusf 16565  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cn 20320  df-tx 20654  df-hmeo 20847  df-tmd 21165
This theorem is referenced by:  xrge0tmd  28826
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