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Theorem mhmhmeotmd 27545
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 15781 . . 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 15780 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  ->  S  e.  Mnd )
71, 6ax-mp 5 . . . 4  |-  S  e. 
Mnd
8 eqid 2467 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
9 eqid 2467 . . . . 5  |-  ( +f `  S )  =  ( +f `  S )
108, 9mndplusf 15744 . . . 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 2467 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
13 eqid 2467 . . . . 5  |-  ( +f `  T )  =  ( +f `  T )
1412, 13mndplusf 15744 . . . 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 2467 . . . . 5  |-  ( TopOpen `  S )  =  (
TopOpen `  S )
1817, 8tmdtopon 20315 . . . 4  |-  ( S  e. TopMnd  ->  ( TopOpen `  S
)  e.  (TopOn `  ( Base `  S )
) )
1916, 18ax-mp 5 . . 3  |-  ( TopOpen `  S )  e.  (TopOn `  ( Base `  S
) )
20 eqid 2467 . . . . 5  |-  ( TopOpen `  T )  =  (
TopOpen `  T )
2112, 20istps 19204 . . . 4  |-  ( T  e.  TopSp 
<->  ( TopOpen `  T )  e.  (TopOn `  ( Base `  T ) ) )
224, 21mpbi 208 . . 3  |-  ( TopOpen `  T )  e.  (TopOn `  ( Base `  T
) )
23 eqid 2467 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
24 eqid 2467 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
258, 23, 24mhmlin 15784 . . . . 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 1311 . . . 4  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  ( F `  ( x
( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T ) ( F `
 y ) ) )
278, 23, 9plusfval 15741 . . . . 5  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  (
x ( +f `  S ) y )  =  ( x ( +g  `  S ) y ) )
2827fveq2d 5868 . . . 4  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  ( F `  ( x
( +f `  S ) y ) )  =  ( F `
 ( x ( +g  `  S ) y ) ) )
298, 12mhmf 15782 . . . . . . 7  |-  ( F  e.  ( S MndHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
301, 29ax-mp 5 . . . . . 6  |-  F :
( Base `  S ) --> ( Base `  T )
3130ffvelrni 6018 . . . . 5  |-  ( x  e.  ( Base `  S
)  ->  ( F `  x )  e.  (
Base `  T )
)
3230ffvelrni 6018 . . . . 5  |-  ( y  e.  ( Base `  S
)  ->  ( F `  y )  e.  (
Base `  T )
)
3312, 24, 13plusfval 15741 . . . . 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 2518 . . 3  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  ( F `  ( x
( +f `  S ) y ) )  =  ( ( F `  x ) ( +f `  T ) ( F `
 y ) ) )
3617, 9tmdcn 20317 . . . 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 27544 . 2  |-  ( +f `  T )  e.  ( ( (
TopOpen `  T )  tX  ( TopOpen `  T )
)  Cn  ( TopOpen `  T ) )
3913, 20istmd 20308 . 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 1178 1  |-  T  e. TopMnd
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767    X. cxp 4997   -->wf 5582   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   TopOpenctopn 14673   Mndcmnd 15722   +fcplusf 15725   MndHom cmhm 15775  TopOnctopon 19162   TopSpctps 19164    Cn ccn 19491    tX ctx 19796   Homeochmeo 19989  TopMndctmd 20304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-map 7419  df-topgen 14695  df-mnd 15728  df-plusf 15729  df-mhm 15777  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cn 19494  df-tx 19798  df-hmeo 19991  df-tmd 20306
This theorem is referenced by:  xrge0tmd  27564
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