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Theorem submtmd 19793
Description: A submonoid of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 6-Oct-2015.)
Hypothesis
Ref Expression
subgtgp.h  |-  H  =  ( Gs  S )
Assertion
Ref Expression
submtmd  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  H  e. TopMnd )

Proof of Theorem submtmd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subgtgp.h . . . 4  |-  H  =  ( Gs  S )
21submmnd 15586 . . 3  |-  ( S  e.  (SubMnd `  G
)  ->  H  e.  Mnd )
32adantl 466 . 2  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  H  e.  Mnd )
4 tmdtps 19765 . . . 4  |-  ( G  e. TopMnd  ->  G  e.  TopSp )
5 resstps 18909 . . . 4  |-  ( ( G  e.  TopSp  /\  S  e.  (SubMnd `  G )
)  ->  ( Gs  S
)  e.  TopSp )
64, 5sylan 471 . . 3  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( Gs  S
)  e.  TopSp )
71, 6syl5eqel 2543 . 2  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  H  e.  TopSp
)
81submbas 15587 . . . . . . 7  |-  ( S  e.  (SubMnd `  G
)  ->  S  =  ( Base `  H )
)
98adantl 466 . . . . . 6  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  S  =  ( Base `  H )
)
10 eqid 2451 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
111, 10ressplusg 14384 . . . . . . . 8  |-  ( S  e.  (SubMnd `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
1211adantl 466 . . . . . . 7  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
1312oveqd 6209 . . . . . 6  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( x
( +g  `  G ) y )  =  ( x ( +g  `  H
) y ) )
149, 9, 13mpt2eq123dv 6249 . . . . 5  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( x  e.  S ,  y  e.  S  |->  ( x ( +g  `  G ) y ) )  =  ( x  e.  (
Base `  H ) ,  y  e.  ( Base `  H )  |->  ( x ( +g  `  H
) y ) ) )
15 eqid 2451 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
16 eqid 2451 . . . . . 6  |-  ( +g  `  H )  =  ( +g  `  H )
17 eqid 2451 . . . . . 6  |-  ( +f `  H )  =  ( +f `  H )
1815, 16, 17plusffval 15531 . . . . 5  |-  ( +f `  H )  =  ( x  e.  ( Base `  H
) ,  y  e.  ( Base `  H
)  |->  ( x ( +g  `  H ) y ) )
1914, 18syl6reqr 2511 . . . 4  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( +f `  H )  =  ( x  e.  S ,  y  e.  S  |->  ( x ( +g  `  G ) y ) ) )
20 eqid 2451 . . . . 5  |-  ( (
TopOpen `  G )t  S )  =  ( ( TopOpen `  G )t  S )
21 eqid 2451 . . . . . . 7  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
22 eqid 2451 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
2321, 22tmdtopon 19770 . . . . . 6  |-  ( G  e. TopMnd  ->  ( TopOpen `  G
)  e.  (TopOn `  ( Base `  G )
) )
2423adantr 465 . . . . 5  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( TopOpen `  G )  e.  (TopOn `  ( Base `  G
) ) )
2522submss 15582 . . . . . 6  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  ( Base `  G ) )
2625adantl 466 . . . . 5  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  S  C_  ( Base `  G ) )
27 eqid 2451 . . . . . . . 8  |-  ( +f `  G )  =  ( +f `  G )
2822, 10, 27plusffval 15531 . . . . . . 7  |-  ( +f `  G )  =  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  ( x ( +g  `  G ) y ) )
2921, 27tmdcn 19772 . . . . . . 7  |-  ( G  e. TopMnd  ->  ( +f `  G )  e.  ( ( ( TopOpen `  G
)  tX  ( TopOpen `  G ) )  Cn  ( TopOpen `  G )
) )
3028, 29syl5eqelr 2544 . . . . . 6  |-  ( G  e. TopMnd  ->  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  ( x ( +g  `  G ) y ) )  e.  ( ( ( TopOpen `  G )  tX  ( TopOpen
`  G ) )  Cn  ( TopOpen `  G
) ) )
3130adantr 465 . . . . 5  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  ( x ( +g  `  G ) y ) )  e.  ( ( ( TopOpen `  G )  tX  ( TopOpen
`  G ) )  Cn  ( TopOpen `  G
) ) )
3220, 24, 26, 20, 24, 26, 31cnmpt2res 19368 . . . 4  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( x  e.  S ,  y  e.  S  |->  ( x ( +g  `  G ) y ) )  e.  ( ( ( (
TopOpen `  G )t  S ) 
tX  ( ( TopOpen `  G )t  S ) )  Cn  ( TopOpen `  G )
) )
3319, 32eqeltrd 2539 . . 3  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( +f `  H )  e.  ( ( ( (
TopOpen `  G )t  S ) 
tX  ( ( TopOpen `  G )t  S ) )  Cn  ( TopOpen `  G )
) )
3415, 17mndplusf 15535 . . . . . 6  |-  ( H  e.  Mnd  ->  ( +f `  H
) : ( (
Base `  H )  X.  ( Base `  H
) ) --> ( Base `  H ) )
35 frn 5665 . . . . . 6  |-  ( ( +f `  H
) : ( (
Base `  H )  X.  ( Base `  H
) ) --> ( Base `  H )  ->  ran  ( +f `  H
)  C_  ( Base `  H ) )
363, 34, 353syl 20 . . . . 5  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ran  ( +f `  H ) 
C_  ( Base `  H
) )
3736, 9sseqtr4d 3493 . . . 4  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ran  ( +f `  H ) 
C_  S )
38 cnrest2 19008 . . . 4  |-  ( ( ( TopOpen `  G )  e.  (TopOn `  ( Base `  G ) )  /\  ran  ( +f `  H )  C_  S  /\  S  C_  ( Base `  G ) )  -> 
( ( +f `  H )  e.  ( ( ( ( TopOpen `  G )t  S )  tX  (
( TopOpen `  G )t  S
) )  Cn  ( TopOpen
`  G ) )  <-> 
( +f `  H )  e.  ( ( ( ( TopOpen `  G )t  S )  tX  (
( TopOpen `  G )t  S
) )  Cn  (
( TopOpen `  G )t  S
) ) ) )
3924, 37, 26, 38syl3anc 1219 . . 3  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( ( +f `  H
)  e.  ( ( ( ( TopOpen `  G
)t 
S )  tX  (
( TopOpen `  G )t  S
) )  Cn  ( TopOpen
`  G ) )  <-> 
( +f `  H )  e.  ( ( ( ( TopOpen `  G )t  S )  tX  (
( TopOpen `  G )t  S
) )  Cn  (
( TopOpen `  G )t  S
) ) ) )
4033, 39mpbid 210 . 2  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( +f `  H )  e.  ( ( ( (
TopOpen `  G )t  S ) 
tX  ( ( TopOpen `  G )t  S ) )  Cn  ( ( TopOpen `  G
)t 
S ) ) )
411, 21resstopn 18908 . . 3  |-  ( (
TopOpen `  G )t  S )  =  ( TopOpen `  H
)
4217, 41istmd 19763 . 2  |-  ( H  e. TopMnd 
<->  ( H  e.  Mnd  /\  H  e.  TopSp  /\  ( +f `  H
)  e.  ( ( ( ( TopOpen `  G
)t 
S )  tX  (
( TopOpen `  G )t  S
) )  Cn  (
( TopOpen `  G )t  S
) ) ) )
433, 7, 40, 42syl3anbrc 1172 1  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  H  e. TopMnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3428    X. cxp 4938   ran crn 4941   -->wf 5514   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194   Basecbs 14278   ↾s cress 14279   +g cplusg 14342   ↾t crest 14463   TopOpenctopn 14464   Mndcmnd 15513   +fcplusf 15516  SubMndcsubmnd 15567  TopOnctopon 18617   TopSpctps 18619    Cn ccn 18946    tX ctx 19251  TopMndctmd 19759
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-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-oadd 7026  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-fi 7764  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-tset 14361  df-rest 14465  df-topn 14466  df-0g 14484  df-topgen 14486  df-mnd 15519  df-plusf 15520  df-submnd 15569  df-top 18621  df-bases 18623  df-topon 18624  df-topsp 18625  df-cn 18949  df-tx 19253  df-tmd 19761
This theorem is referenced by:  subgtgp  19794  nrgtdrg  20391  iistmd  26468
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