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Theorem submnd0 16521
Description: The zero of a submonoid is the same as the zero in the parent monoid. (Note that we must add the condition that the zero of the parent monoid is actually contained in the submonoid, because it is possible to have "subsets that are monoids" which are not submonoids because they have a different identity element.) (Contributed by Mario Carneiro, 10-Jan-2015.)
Hypotheses
Ref Expression
submnd0.b  |-  B  =  ( Base `  G
)
submnd0.z  |-  .0.  =  ( 0g `  G )
submnd0.h  |-  H  =  ( Gs  S )
Assertion
Ref Expression
submnd0  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  ->  .0.  =  ( 0g `  H ) )

Proof of Theorem submnd0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2429 . 2  |-  ( Base `  H )  =  (
Base `  H )
2 eqid 2429 . 2  |-  ( 0g
`  H )  =  ( 0g `  H
)
3 eqid 2429 . 2  |-  ( +g  `  H )  =  ( +g  `  H )
4 simprr 764 . . 3  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  ->  .0.  e.  S )
5 submnd0.h . . . . 5  |-  H  =  ( Gs  S )
6 submnd0.b . . . . 5  |-  B  =  ( Base `  G
)
75, 6ressbas2 15143 . . . 4  |-  ( S 
C_  B  ->  S  =  ( Base `  H
) )
87ad2antrl 732 . . 3  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  ->  S  =  ( Base `  H )
)
94, 8eleqtrd 2519 . 2  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  ->  .0.  e.  ( Base `  H )
)
10 fvex 5891 . . . . . . 7  |-  ( Base `  H )  e.  _V
118, 10syl6eqel 2525 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  ->  S  e.  _V )
1211adantr 466 . . . . 5  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  /\  x  e.  ( Base `  H ) )  ->  S  e.  _V )
13 eqid 2429 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
145, 13ressplusg 15202 . . . . 5  |-  ( S  e.  _V  ->  ( +g  `  G )  =  ( +g  `  H
) )
1512, 14syl 17 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  /\  x  e.  ( Base `  H ) )  -> 
( +g  `  G )  =  ( +g  `  H
) )
1615oveqd 6322 . . 3  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  /\  x  e.  ( Base `  H ) )  -> 
(  .0.  ( +g  `  G ) x )  =  (  .0.  ( +g  `  H ) x ) )
17 simpll 758 . . . 4  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  ->  G  e.  Mnd )
185, 6ressbasss 15144 . . . . 5  |-  ( Base `  H )  C_  B
1918sseli 3466 . . . 4  |-  ( x  e.  ( Base `  H
)  ->  x  e.  B )
20 submnd0.z . . . . 5  |-  .0.  =  ( 0g `  G )
216, 13, 20mndlid 16512 . . . 4  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  (  .0.  ( +g  `  G ) x )  =  x )
2217, 19, 21syl2an 479 . . 3  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  /\  x  e.  ( Base `  H ) )  -> 
(  .0.  ( +g  `  G ) x )  =  x )
2316, 22eqtr3d 2472 . 2  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  /\  x  e.  ( Base `  H ) )  -> 
(  .0.  ( +g  `  H ) x )  =  x )
2415oveqd 6322 . . 3  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  /\  x  e.  ( Base `  H ) )  -> 
( x ( +g  `  G )  .0.  )  =  ( x ( +g  `  H )  .0.  ) )
256, 13, 20mndrid 16513 . . . 4  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  ( x ( +g  `  G )  .0.  )  =  x )
2617, 19, 25syl2an 479 . . 3  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  /\  x  e.  ( Base `  H ) )  -> 
( x ( +g  `  G )  .0.  )  =  x )
2724, 26eqtr3d 2472 . 2  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  /\  x  e.  ( Base `  H ) )  -> 
( x ( +g  `  H )  .0.  )  =  x )
281, 2, 3, 9, 23, 27ismgmid2 16465 1  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  ->  .0.  =  ( 0g `  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087    C_ wss 3442   ` cfv 5601  (class class class)co 6305   Basecbs 15084   ↾s cress 15085   +g cplusg 15153   0gc0g 15301   Mndcmnd 16490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15166  df-0g 15303  df-mgm 16443  df-sgrp 16482  df-mnd 16492
This theorem is referenced by:  subm0  16558  xrge00  28293  gsumge0cl  37762
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