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Theorem submtmd 18134
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 14754 . . 3  |-  ( S  e.  (SubMnd `  G
)  ->  H  e.  Mnd )
32adantl 453 . 2  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  H  e.  Mnd )
4 tmdtps 18106 . . . 4  |-  ( G  e. TopMnd  ->  G  e.  TopSp )
5 resstps 17251 . . . 4  |-  ( ( G  e.  TopSp  /\  S  e.  (SubMnd `  G )
)  ->  ( Gs  S
)  e.  TopSp )
64, 5sylan 458 . . 3  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( Gs  S
)  e.  TopSp )
71, 6syl5eqel 2520 . 2  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  H  e.  TopSp
)
81submbas 14755 . . . . . . 7  |-  ( S  e.  (SubMnd `  G
)  ->  S  =  ( Base `  H )
)
98adantl 453 . . . . . 6  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  S  =  ( Base `  H )
)
10 eqid 2436 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
111, 10ressplusg 13571 . . . . . . . 8  |-  ( S  e.  (SubMnd `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
1211adantl 453 . . . . . . 7  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
1312oveqd 6098 . . . . . 6  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( x
( +g  `  G ) y )  =  ( x ( +g  `  H
) y ) )
149, 9, 13mpt2eq123dv 6136 . . . . 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 2436 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
16 eqid 2436 . . . . . 6  |-  ( +g  `  H )  =  ( +g  `  H )
17 eqid 2436 . . . . . 6  |-  ( + f `  H )  =  ( + f `  H )
1815, 16, 17plusffval 14702 . . . . 5  |-  ( + f `  H )  =  ( x  e.  ( Base `  H
) ,  y  e.  ( Base `  H
)  |->  ( x ( +g  `  H ) y ) )
1914, 18syl6reqr 2487 . . . 4  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( + f `  H )  =  ( x  e.  S ,  y  e.  S  |->  ( x ( +g  `  G ) y ) ) )
20 eqid 2436 . . . . 5  |-  ( (
TopOpen `  G )t  S )  =  ( ( TopOpen `  G )t  S )
21 eqid 2436 . . . . . . 7  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
22 eqid 2436 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
2321, 22tmdtopon 18111 . . . . . 6  |-  ( G  e. TopMnd  ->  ( TopOpen `  G
)  e.  (TopOn `  ( Base `  G )
) )
2423adantr 452 . . . . 5  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ( TopOpen `  G )  e.  (TopOn `  ( Base `  G
) ) )
2522submss 14750 . . . . . 6  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  ( Base `  G ) )
2625adantl 453 . . . . 5  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  S  C_  ( Base `  G ) )
27 eqid 2436 . . . . . . . 8  |-  ( + f `  G )  =  ( + f `  G )
2822, 10, 27plusffval 14702 . . . . . . 7  |-  ( + f `  G )  =  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  ( x ( +g  `  G ) y ) )
2921, 27tmdcn 18113 . . . . . . 7  |-  ( G  e. TopMnd  ->  ( + f `  G )  e.  ( ( ( TopOpen `  G
)  tX  ( TopOpen `  G ) )  Cn  ( TopOpen `  G )
) )
3028, 29syl5eqelr 2521 . . . . . 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 452 . . . . 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 17709 . . . 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 2510 . . 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 14706 . . . . . 6  |-  ( H  e.  Mnd  ->  ( + f `  H ) : ( ( Base `  H )  X.  ( Base `  H ) ) --> ( Base `  H
) )
35 frn 5597 . . . . . 6  |-  ( ( + f `  H
) : ( (
Base `  H )  X.  ( Base `  H
) ) --> ( Base `  H )  ->  ran  ( + f `  H
)  C_  ( Base `  H ) )
363, 34, 353syl 19 . . . . 5  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ran  ( + f `  H ) 
C_  ( Base `  H
) )
3736, 9sseqtr4d 3385 . . . 4  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  ran  ( + f `  H ) 
C_  S )
38 cnrest2 17350 . . . 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 1184 . . 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 202 . 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 17250 . . 3  |-  ( (
TopOpen `  G )t  S )  =  ( TopOpen `  H
)
4217, 41istmd 18104 . 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 1138 1  |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G )
)  ->  H  e. TopMnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3320    X. cxp 4876   ran crn 4879   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   Basecbs 13469   ↾s cress 13470   +g cplusg 13529   ↾t crest 13648   TopOpenctopn 13649   Mndcmnd 14684   + fcplusf 14687  SubMndcsubmnd 14737  TopOnctopon 16959   TopSpctps 16961    Cn ccn 17288    tX ctx 17592  TopMndctmd 18100
This theorem is referenced by:  subgtgp  18135  nrgtdrg  18728  iistmd  24300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-tset 13548  df-rest 13650  df-topn 13651  df-topgen 13667  df-0g 13727  df-mnd 14690  df-plusf 14691  df-submnd 14739  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cn 17291  df-tx 17594  df-tmd 18102
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