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Theorem subcmnd 13134
Description: A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.)
Hypotheses
Ref Expression
subcmnd.h  |-  ( ph  ->  H  =  ( Gs  S ) )
subcmnd.g  |-  ( ph  ->  G  e. CMnd )
subcmnd.m  |-  ( ph  ->  H  e.  Mnd )
subcmnd.s  |-  ( ph  ->  S  e.  V )
Assertion
Ref Expression
subcmnd  |-  ( ph  ->  H  e. CMnd )

Proof of Theorem subcmnd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2178 . 2  |-  ( ph  ->  ( Base `  H
)  =  ( Base `  H ) )
2 subcmnd.h . . 3  |-  ( ph  ->  H  =  ( Gs  S ) )
3 eqidd 2178 . . 3  |-  ( ph  ->  ( +g  `  G
)  =  ( +g  `  G ) )
4 subcmnd.s . . 3  |-  ( ph  ->  S  e.  V )
5 subcmnd.g . . 3  |-  ( ph  ->  G  e. CMnd )
62, 3, 4, 5ressplusgd 12589 . 2  |-  ( ph  ->  ( +g  `  G
)  =  ( +g  `  H ) )
7 subcmnd.m . 2  |-  ( ph  ->  H  e.  Mnd )
853ad2ant1 1018 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  H )  /\  y  e.  ( Base `  H ) )  ->  G  e. CMnd )
9 eqidd 2178 . . . . . 6  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  G ) )
102, 9, 5, 4ressbasssd 12531 . . . . 5  |-  ( ph  ->  ( Base `  H
)  C_  ( Base `  G ) )
1110sselda 3157 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  H )
)  ->  x  e.  ( Base `  G )
)
12113adant3 1017 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  H )  /\  y  e.  ( Base `  H ) )  ->  x  e.  (
Base `  G )
)
1310sselda 3157 . . . 4  |-  ( (
ph  /\  y  e.  ( Base `  H )
)  ->  y  e.  ( Base `  G )
)
14133adant2 1016 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  H )  /\  y  e.  ( Base `  H ) )  ->  y  e.  (
Base `  G )
)
15 eqid 2177 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
16 eqid 2177 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
1715, 16cmncom 13110 . . 3  |-  ( ( G  e. CMnd  /\  x  e.  ( Base `  G
)  /\  y  e.  ( Base `  G )
)  ->  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
188, 12, 14, 17syl3anc 1238 . 2  |-  ( (
ph  /\  x  e.  ( Base `  H )  /\  y  e.  ( Base `  H ) )  ->  ( x ( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
191, 6, 7, 18iscmnd 13106 1  |-  ( ph  ->  H  e. CMnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 978    = wceq 1353    e. wcel 2148   ` cfv 5218  (class class class)co 5877   Basecbs 12464   ↾s cress 12465   +g cplusg 12538   Mndcmnd 12822  CMndccmn 13093
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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltadd 7929
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-iress 12472  df-plusg 12551  df-cmn 13095
This theorem is referenced by:  unitabl  13291  subrgcrng  13351
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