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Theorem submnd0 14725
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 2436 . 2  |-  ( Base `  H )  =  (
Base `  H )
2 eqid 2436 . 2  |-  ( 0g
`  H )  =  ( 0g `  H
)
3 eqid 2436 . 2  |-  ( +g  `  H )  =  ( +g  `  H )
4 simprr 734 . . 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 13520 . . . 4  |-  ( S 
C_  B  ->  S  =  ( Base `  H
) )
87ad2antrl 709 . . 3  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  ->  S  =  ( Base `  H )
)
94, 8eleqtrd 2512 . 2  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  ->  .0.  e.  ( Base `  H )
)
10 fvex 5742 . . . . . . 7  |-  ( Base `  H )  e.  _V
118, 10syl6eqel 2524 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  ->  S  e.  _V )
1211adantr 452 . . . . 5  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  /\  x  e.  ( Base `  H ) )  ->  S  e.  _V )
13 eqid 2436 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
145, 13ressplusg 13571 . . . . 5  |-  ( S  e.  _V  ->  ( +g  `  G )  =  ( +g  `  H
) )
1512, 14syl 16 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  /\  x  e.  ( Base `  H ) )  -> 
( +g  `  G )  =  ( +g  `  H
) )
1615oveqd 6098 . . 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 731 . . . 4  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  ->  G  e.  Mnd )
185, 6ressbasss 13521 . . . . 5  |-  ( Base `  H )  C_  B
1918sseli 3344 . . . 4  |-  ( x  e.  ( Base `  H
)  ->  x  e.  B )
20 submnd0.z . . . . 5  |-  .0.  =  ( 0g `  G )
216, 13, 20mndlid 14716 . . . 4  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  (  .0.  ( +g  `  G ) x )  =  x )
2217, 19, 21syl2an 464 . . 3  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  /\  x  e.  ( Base `  H ) )  -> 
(  .0.  ( +g  `  G ) x )  =  x )
2316, 22eqtr3d 2470 . 2  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  /\  x  e.  ( Base `  H ) )  -> 
(  .0.  ( +g  `  H ) x )  =  x )
2415oveqd 6098 . . 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 14717 . . . 4  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  ( x ( +g  `  G )  .0.  )  =  x )
2617, 19, 25syl2an 464 . . 3  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  /\  x  e.  ( Base `  H ) )  -> 
( x ( +g  `  G )  .0.  )  =  x )
2724, 26eqtr3d 2470 . 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 14713 1  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S ) )  ->  .0.  =  ( 0g `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    C_ wss 3320   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470   +g cplusg 13529   0gc0g 13723   Mndcmnd 14684
This theorem is referenced by:  subm0  14756  xrge00  24208
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-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-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-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  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-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690
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