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Theorem mndissubm 12727
Description: If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the the other monoid is contained in the base set of the monoid, then the (base set of the) monoid is a submonoid of the other monoid. (Contributed by AV, 17-Feb-2024.)
Hypotheses
Ref Expression
mndissubm.b  |-  B  =  ( Base `  G
)
mndissubm.s  |-  S  =  ( Base `  H
)
mndissubm.z  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
mndissubm  |-  ( ( G  e.  Mnd  /\  H  e.  Mnd )  ->  ( ( S  C_  B  /\  .0.  e.  S  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) )  ->  S  e.  (SubMnd `  G
) ) )

Proof of Theorem mndissubm
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr1 1003 . . 3  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S  /\  ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) ) ) )  ->  S  C_  B )
2 simpr2 1004 . . 3  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S  /\  ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) ) ) )  ->  .0.  e.  S )
3 mndmgm 12688 . . . . . . 7  |-  ( G  e.  Mnd  ->  G  e. Mgm )
4 mndmgm 12688 . . . . . . 7  |-  ( H  e.  Mnd  ->  H  e. Mgm )
53, 4anim12i 338 . . . . . 6  |-  ( ( G  e.  Mnd  /\  H  e.  Mnd )  ->  ( G  e. Mgm  /\  H  e. Mgm ) )
65ad2antrr 488 . . . . 5  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S  /\  ( +g  `  H )  =  ( ( +g  `  G
)  |`  ( S  X.  S ) ) ) )  /\  ( a  e.  S  /\  b  e.  S ) )  -> 
( G  e. Mgm  /\  H  e. Mgm ) )
7 3simpb 995 . . . . . 6  |-  ( ( S  C_  B  /\  .0.  e.  S  /\  ( +g  `  H )  =  ( ( +g  `  G
)  |`  ( S  X.  S ) ) )  ->  ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )
87ad2antlr 489 . . . . 5  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S  /\  ( +g  `  H )  =  ( ( +g  `  G
)  |`  ( S  X.  S ) ) ) )  /\  ( a  e.  S  /\  b  e.  S ) )  -> 
( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )
9 simpr 110 . . . . 5  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S  /\  ( +g  `  H )  =  ( ( +g  `  G
)  |`  ( S  X.  S ) ) ) )  /\  ( a  e.  S  /\  b  e.  S ) )  -> 
( a  e.  S  /\  b  e.  S
) )
10 mndissubm.b . . . . . 6  |-  B  =  ( Base `  G
)
11 mndissubm.s . . . . . 6  |-  S  =  ( Base `  H
)
1210, 11mgmsscl 12645 . . . . 5  |-  ( ( ( G  e. Mgm  /\  H  e. Mgm )  /\  ( S  C_  B  /\  ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) ) )  /\  ( a  e.  S  /\  b  e.  S ) )  -> 
( a ( +g  `  G ) b )  e.  S )
136, 8, 9, 12syl3anc 1238 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S  /\  ( +g  `  H )  =  ( ( +g  `  G
)  |`  ( S  X.  S ) ) ) )  /\  ( a  e.  S  /\  b  e.  S ) )  -> 
( a ( +g  `  G ) b )  e.  S )
1413ralrimivva 2557 . . 3  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S  /\  ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) ) ) )  ->  A. a  e.  S  A. b  e.  S  ( a
( +g  `  G ) b )  e.  S
)
15 mndissubm.z . . . . 5  |-  .0.  =  ( 0g `  G )
16 eqid 2175 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
1710, 15, 16issubm 12725 . . . 4  |-  ( G  e.  Mnd  ->  ( S  e.  (SubMnd `  G
)  <->  ( S  C_  B  /\  .0.  e.  S  /\  A. a  e.  S  A. b  e.  S  ( a ( +g  `  G ) b )  e.  S ) ) )
1817ad2antrr 488 . . 3  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S  /\  ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) ) ) )  ->  ( S  e.  (SubMnd `  G
)  <->  ( S  C_  B  /\  .0.  e.  S  /\  A. a  e.  S  A. b  e.  S  ( a ( +g  `  G ) b )  e.  S ) ) )
191, 2, 14, 18mpbir3and 1180 . 2  |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S  C_  B  /\  .0.  e.  S  /\  ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) ) ) )  ->  S  e.  (SubMnd `  G )
)
2019ex 115 1  |-  ( ( G  e.  Mnd  /\  H  e.  Mnd )  ->  ( ( S  C_  B  /\  .0.  e.  S  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) )  ->  S  e.  (SubMnd `  G
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2146   A.wral 2453    C_ wss 3127    X. cxp 4618    |` cres 4622   ` cfv 5208  (class class class)co 5865   Basecbs 12428   +g cplusg 12492   0gc0g 12626  Mgmcmgm 12638   Mndcmnd 12682  SubMndcsubmnd 12712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-ov 5868  df-inn 8891  df-2 8949  df-ndx 12431  df-slot 12432  df-base 12434  df-plusg 12505  df-mgm 12640  df-sgrp 12673  df-mnd 12683  df-submnd 12714
This theorem is referenced by: (None)
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