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Theorem submtmd 17803
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 14447 . . 3  |-  ( S  e.  (SubMnd `  G
)  ->  H  e.  Mnd )
32adantl 452 . 2  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  H  e.  Mnd )
4 tmdtps 17775 . . . 4  |-  ( G  e. TopMnd  ->  G  e.  TopSp )
5 resstps 16933 . . . 4  |-  ( ( G  e.  TopSp  /\  S  e.  (SubMnd `  G )
)  ->  ( Gs  S
)  e.  TopSp )
64, 5sylan 457 . . 3  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( Gs  S
)  e.  TopSp )
71, 6syl5eqel 2380 . 2  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  H  e.  TopSp
)
81submbas 14448 . . . . . . 7  |-  ( S  e.  (SubMnd `  G
)  ->  S  =  ( Base `  H )
)
98adantl 452 . . . . . 6  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  S  =  ( Base `  H )
)
10 eqid 2296 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
111, 10ressplusg 13266 . . . . . . . 8  |-  ( S  e.  (SubMnd `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
1211adantl 452 . . . . . . 7  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
1312oveqd 5891 . . . . . 6  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( x
( +g  `  G ) y )  =  ( x ( +g  `  H
) y ) )
149, 9, 13mpt2eq123dv 5926 . . . . 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 2296 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
16 eqid 2296 . . . . . 6  |-  ( +g  `  H )  =  ( +g  `  H )
17 eqid 2296 . . . . . 6  |-  ( + f `  H )  =  ( + f `  H )
1815, 16, 17plusffval 14395 . . . . 5  |-  ( + f `  H )  =  ( x  e.  ( Base `  H
) ,  y  e.  ( Base `  H
)  |->  ( x ( +g  `  H ) y ) )
1914, 18syl6reqr 2347 . . . 4  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( + f `  H )  =  ( x  e.  S ,  y  e.  S  |->  ( x ( +g  `  G ) y ) ) )
20 eqid 2296 . . . . 5  |-  ( (
TopOpen `  G )t  S )  =  ( ( TopOpen `  G )t  S )
21 eqid 2296 . . . . . . 7  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
22 eqid 2296 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
2321, 22tmdtopon 17780 . . . . . 6  |-  ( G  e. TopMnd  ->  ( TopOpen `  G
)  e.  (TopOn `  ( Base `  G )
) )
2423adantr 451 . . . . 5  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( TopOpen `  G )  e.  (TopOn `  ( Base `  G
) ) )
2522submss 14443 . . . . . 6  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  ( Base `  G ) )
2625adantl 452 . . . . 5  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  S  C_  ( Base `  G ) )
27 eqid 2296 . . . . . . . 8  |-  ( + f `  G )  =  ( + f `  G )
2822, 10, 27plusffval 14395 . . . . . . 7  |-  ( + f `  G )  =  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  ( x ( +g  `  G ) y ) )
2921, 27tmdcn 17782 . . . . . . 7  |-  ( G  e. TopMnd  ->  ( + f `  G )  e.  ( ( ( TopOpen `  G
)  tX  ( TopOpen `  G ) )  Cn  ( TopOpen `  G )
) )
3028, 29syl5eqelr 2381 . . . . . 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 451 . . . . 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 17387 . . . 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 2370 . . 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 14399 . . . . . 6  |-  ( H  e.  Mnd  ->  ( + f `  H ) : ( ( Base `  H )  X.  ( Base `  H ) ) --> ( Base `  H
) )
35 frn 5411 . . . . . 6  |-  ( ( + f `  H
) : ( (
Base `  H )  X.  ( Base `  H
) ) --> ( Base `  H )  ->  ran  ( + f `  H
)  C_  ( Base `  H ) )
363, 34, 353syl 18 . . . . 5  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ran  ( + f `  H ) 
C_  ( Base `  H
) )
3736, 9sseqtr4d 3228 . . . 4  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ran  ( + f `  H ) 
C_  S )
38 cnrest2 17030 . . . 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 1182 . . 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 201 . 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 16932 . . 3  |-  ( (
TopOpen `  G )t  S )  =  ( TopOpen `  H
)
4217, 41istmd 17773 . 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 1136 1  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  H  e. TopMnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165    X. cxp 4703   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   Basecbs 13164   ↾s cress 13165   +g cplusg 13224   ↾t crest 13341   TopOpenctopn 13342   Mndcmnd 14377   + fcplusf 14380  SubMndcsubmnd 14430  TopOnctopon 16648   TopSpctps 16650    Cn ccn 16970    tX ctx 17271  TopMndctmd 17769
This theorem is referenced by:  subgtgp  17804  nrgtdrg  18219  iistmd  23301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-tset 13243  df-rest 13343  df-topn 13344  df-topgen 13360  df-0g 13420  df-mnd 14383  df-plusf 14384  df-submnd 14432  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cn 16973  df-tx 17273  df-tmd 17771
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