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Theorem submtmd 20476
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 15859 . . 3  |-  ( S  e.  (SubMnd `  G
)  ->  H  e.  Mnd )
32adantl 466 . 2  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  H  e.  Mnd )
4 tmdtps 20448 . . . 4  |-  ( G  e. TopMnd  ->  G  e.  TopSp )
5 resstps 19561 . . . 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 2535 . 2  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  H  e.  TopSp
)
81submbas 15860 . . . . . . 7  |-  ( S  e.  (SubMnd `  G
)  ->  S  =  ( Base `  H )
)
98adantl 466 . . . . . 6  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  S  =  ( Base `  H )
)
10 eqid 2443 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
111, 10ressplusg 14616 . . . . . . . 8  |-  ( S  e.  (SubMnd `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
1211adantl 466 . . . . . . 7  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
1312oveqd 6298 . . . . . 6  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( x
( +g  `  G ) y )  =  ( x ( +g  `  H
) y ) )
149, 9, 13mpt2eq123dv 6344 . . . . 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 2443 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
16 eqid 2443 . . . . . 6  |-  ( +g  `  H )  =  ( +g  `  H )
17 eqid 2443 . . . . . 6  |-  ( +f `  H )  =  ( +f `  H )
1815, 16, 17plusffval 15751 . . . . 5  |-  ( +f `  H )  =  ( x  e.  ( Base `  H
) ,  y  e.  ( Base `  H
)  |->  ( x ( +g  `  H ) y ) )
1914, 18syl6reqr 2503 . . . 4  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( +f `  H )  =  ( x  e.  S ,  y  e.  S  |->  ( x ( +g  `  G ) y ) ) )
20 eqid 2443 . . . . 5  |-  ( (
TopOpen `  G )t  S )  =  ( ( TopOpen `  G )t  S )
21 eqid 2443 . . . . . . 7  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
22 eqid 2443 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
2321, 22tmdtopon 20453 . . . . . 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 15855 . . . . . 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 2443 . . . . . . . 8  |-  ( +f `  G )  =  ( +f `  G )
2822, 10, 27plusffval 15751 . . . . . . 7  |-  ( +f `  G )  =  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  ( x ( +g  `  G ) y ) )
2921, 27tmdcn 20455 . . . . . . 7  |-  ( G  e. TopMnd  ->  ( +f `  G )  e.  ( ( ( TopOpen `  G
)  tX  ( TopOpen `  G ) )  Cn  ( TopOpen `  G )
) )
3028, 29syl5eqelr 2536 . . . . . 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 20051 . . . 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 2531 . . 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 15813 . . . . . 6  |-  ( H  e.  Mnd  ->  ( +f `  H
) : ( (
Base `  H )  X.  ( Base `  H
) ) --> ( Base `  H ) )
35 frn 5727 . . . . . 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 3526 . . . 4  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ran  ( +f `  H ) 
C_  S )
38 cnrest2 19660 . . . 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 1229 . . 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 19560 . . 3  |-  ( (
TopOpen `  G )t  S )  =  ( TopOpen `  H
)
4217, 41istmd 20446 . 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 1181 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 1383    e. wcel 1804    C_ wss 3461    X. cxp 4987   ran crn 4990   -->wf 5574   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   Basecbs 14509   ↾s cress 14510   +g cplusg 14574   ↾t crest 14695   TopOpenctopn 14696   +fcplusf 15743   Mndcmnd 15793  SubMndcsubmnd 15839  TopOnctopon 19268   TopSpctps 19270    Cn ccn 19598    tX ctx 19934  TopMndctmd 20442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fi 7873  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-tset 14593  df-rest 14697  df-topn 14698  df-0g 14716  df-topgen 14718  df-plusf 15745  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cn 19601  df-tx 19936  df-tmd 20444
This theorem is referenced by:  subgtgp  20477  nrgtdrg  21074  iistmd  27757
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