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Theorem mhmhmeotmd 26491
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 15570 . . 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 15569 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  ->  S  e.  Mnd )
71, 6ax-mp 5 . . . 4  |-  S  e. 
Mnd
8 eqid 2451 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
9 eqid 2451 . . . . 5  |-  ( +f `  S )  =  ( +f `  S )
108, 9mndplusf 15533 . . . 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 2451 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
13 eqid 2451 . . . . 5  |-  ( +f `  T )  =  ( +f `  T )
1412, 13mndplusf 15533 . . . 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 2451 . . . . 5  |-  ( TopOpen `  S )  =  (
TopOpen `  S )
1817, 8tmdtopon 19768 . . . 4  |-  ( S  e. TopMnd  ->  ( TopOpen `  S
)  e.  (TopOn `  ( Base `  S )
) )
1916, 18ax-mp 5 . . 3  |-  ( TopOpen `  S )  e.  (TopOn `  ( Base `  S
) )
20 eqid 2451 . . . . 5  |-  ( TopOpen `  T )  =  (
TopOpen `  T )
2112, 20istps 18657 . . . 4  |-  ( T  e.  TopSp 
<->  ( TopOpen `  T )  e.  (TopOn `  ( Base `  T ) ) )
224, 21mpbi 208 . . 3  |-  ( TopOpen `  T )  e.  (TopOn `  ( Base `  T
) )
23 eqid 2451 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
24 eqid 2451 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
258, 23, 24mhmlin 15573 . . . . 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 1302 . . . 4  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  ( F `  ( x
( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T ) ( F `
 y ) ) )
278, 23, 9plusfval 15530 . . . . 5  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  (
x ( +f `  S ) y )  =  ( x ( +g  `  S ) y ) )
2827fveq2d 5793 . . . 4  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  ( F `  ( x
( +f `  S ) y ) )  =  ( F `
 ( x ( +g  `  S ) y ) ) )
298, 12mhmf 15571 . . . . . . 7  |-  ( F  e.  ( S MndHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
301, 29ax-mp 5 . . . . . 6  |-  F :
( Base `  S ) --> ( Base `  T )
3130ffvelrni 5941 . . . . 5  |-  ( x  e.  ( Base `  S
)  ->  ( F `  x )  e.  (
Base `  T )
)
3230ffvelrni 5941 . . . . 5  |-  ( y  e.  ( Base `  S
)  ->  ( F `  y )  e.  (
Base `  T )
)
3312, 24, 13plusfval 15530 . . . . 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 2502 . . 3  |-  ( ( x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) )  ->  ( F `  ( x
( +f `  S ) y ) )  =  ( ( F `  x ) ( +f `  T ) ( F `
 y ) ) )
3617, 9tmdcn 19770 . . . 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 26490 . 2  |-  ( +f `  T )  e.  ( ( (
TopOpen `  T )  tX  ( TopOpen `  T )
)  Cn  ( TopOpen `  T ) )
3913, 20istmd 19761 . 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 1170 1  |-  T  e. TopMnd
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370    e. wcel 1758    X. cxp 4936   -->wf 5512   ` cfv 5516  (class class class)co 6190   Basecbs 14276   +g cplusg 14340   TopOpenctopn 14462   Mndcmnd 15511   +fcplusf 15514   MndHom cmhm 15564  TopOnctopon 18615   TopSpctps 18617    Cn ccn 18944    tX ctx 19249   Homeochmeo 19442  TopMndctmd 19757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-map 7316  df-topgen 14484  df-mnd 15517  df-plusf 15518  df-mhm 15566  df-top 18619  df-bases 18621  df-topon 18622  df-topsp 18623  df-cn 18947  df-tx 19251  df-hmeo 19444  df-tmd 19759
This theorem is referenced by:  xrge0tmd  26510
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