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Theorem grpidssd 12822
Description: If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then both groups have the same identity element. (Contributed by AV, 15-Mar-2019.)
Hypotheses
Ref Expression
grpidssd.m  |-  ( ph  ->  M  e.  Grp )
grpidssd.s  |-  ( ph  ->  S  e.  Grp )
grpidssd.b  |-  B  =  ( Base `  S
)
grpidssd.c  |-  ( ph  ->  B  C_  ( Base `  M ) )
grpidssd.o  |-  ( ph  ->  A. x  e.  B  A. y  e.  B  ( x ( +g  `  M ) y )  =  ( x ( +g  `  S ) y ) )
Assertion
Ref Expression
grpidssd  |-  ( ph  ->  ( 0g `  M
)  =  ( 0g
`  S ) )
Distinct variable groups:    x, B, y   
x, M, y    x, S, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem grpidssd
StepHypRef Expression
1 grpidssd.s . . . . . 6  |-  ( ph  ->  S  e.  Grp )
2 grpidssd.b . . . . . . 7  |-  B  =  ( Base `  S
)
3 eqid 2177 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
42, 3grpidcl 12781 . . . . . 6  |-  ( S  e.  Grp  ->  ( 0g `  S )  e.  B )
51, 4syl 14 . . . . 5  |-  ( ph  ->  ( 0g `  S
)  e.  B )
6 grpidssd.o . . . . 5  |-  ( ph  ->  A. x  e.  B  A. y  e.  B  ( x ( +g  `  M ) y )  =  ( x ( +g  `  S ) y ) )
7 oveq1 5875 . . . . . . 7  |-  ( x  =  ( 0g `  S )  ->  (
x ( +g  `  M
) y )  =  ( ( 0g `  S ) ( +g  `  M ) y ) )
8 oveq1 5875 . . . . . . 7  |-  ( x  =  ( 0g `  S )  ->  (
x ( +g  `  S
) y )  =  ( ( 0g `  S ) ( +g  `  S ) y ) )
97, 8eqeq12d 2192 . . . . . 6  |-  ( x  =  ( 0g `  S )  ->  (
( x ( +g  `  M ) y )  =  ( x ( +g  `  S ) y )  <->  ( ( 0g `  S ) ( +g  `  M ) y )  =  ( ( 0g `  S
) ( +g  `  S
) y ) ) )
10 oveq2 5876 . . . . . . 7  |-  ( y  =  ( 0g `  S )  ->  (
( 0g `  S
) ( +g  `  M
) y )  =  ( ( 0g `  S ) ( +g  `  M ) ( 0g
`  S ) ) )
11 oveq2 5876 . . . . . . 7  |-  ( y  =  ( 0g `  S )  ->  (
( 0g `  S
) ( +g  `  S
) y )  =  ( ( 0g `  S ) ( +g  `  S ) ( 0g
`  S ) ) )
1210, 11eqeq12d 2192 . . . . . 6  |-  ( y  =  ( 0g `  S )  ->  (
( ( 0g `  S ) ( +g  `  M ) y )  =  ( ( 0g
`  S ) ( +g  `  S ) y )  <->  ( ( 0g `  S ) ( +g  `  M ) ( 0g `  S
) )  =  ( ( 0g `  S
) ( +g  `  S
) ( 0g `  S ) ) ) )
139, 12rspc2va 2855 . . . . 5  |-  ( ( ( ( 0g `  S )  e.  B  /\  ( 0g `  S
)  e.  B )  /\  A. x  e.  B  A. y  e.  B  ( x ( +g  `  M ) y )  =  ( x ( +g  `  S
) y ) )  ->  ( ( 0g
`  S ) ( +g  `  M ) ( 0g `  S
) )  =  ( ( 0g `  S
) ( +g  `  S
) ( 0g `  S ) ) )
145, 5, 6, 13syl21anc 1237 . . . 4  |-  ( ph  ->  ( ( 0g `  S ) ( +g  `  M ) ( 0g
`  S ) )  =  ( ( 0g
`  S ) ( +g  `  S ) ( 0g `  S
) ) )
15 eqid 2177 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
162, 15, 3grplid 12783 . . . . 5  |-  ( ( S  e.  Grp  /\  ( 0g `  S )  e.  B )  -> 
( ( 0g `  S ) ( +g  `  S ) ( 0g
`  S ) )  =  ( 0g `  S ) )
171, 4, 16syl2anc2 412 . . . 4  |-  ( ph  ->  ( ( 0g `  S ) ( +g  `  S ) ( 0g
`  S ) )  =  ( 0g `  S ) )
1814, 17eqtrd 2210 . . 3  |-  ( ph  ->  ( ( 0g `  S ) ( +g  `  M ) ( 0g
`  S ) )  =  ( 0g `  S ) )
19 grpidssd.m . . . 4  |-  ( ph  ->  M  e.  Grp )
20 grpidssd.c . . . . 5  |-  ( ph  ->  B  C_  ( Base `  M ) )
2120, 5sseldd 3156 . . . 4  |-  ( ph  ->  ( 0g `  S
)  e.  ( Base `  M ) )
22 eqid 2177 . . . . 5  |-  ( Base `  M )  =  (
Base `  M )
23 eqid 2177 . . . . 5  |-  ( +g  `  M )  =  ( +g  `  M )
24 eqid 2177 . . . . 5  |-  ( 0g
`  M )  =  ( 0g `  M
)
2522, 23, 24grpidlcan 12812 . . . 4  |-  ( ( M  e.  Grp  /\  ( 0g `  S )  e.  ( Base `  M
)  /\  ( 0g `  S )  e.  (
Base `  M )
)  ->  ( (
( 0g `  S
) ( +g  `  M
) ( 0g `  S ) )  =  ( 0g `  S
)  <->  ( 0g `  S )  =  ( 0g `  M ) ) )
2619, 21, 21, 25syl3anc 1238 . . 3  |-  ( ph  ->  ( ( ( 0g
`  S ) ( +g  `  M ) ( 0g `  S
) )  =  ( 0g `  S )  <-> 
( 0g `  S
)  =  ( 0g
`  M ) ) )
2718, 26mpbid 147 . 2  |-  ( ph  ->  ( 0g `  S
)  =  ( 0g
`  M ) )
2827eqcomd 2183 1  |-  ( ph  ->  ( 0g `  M
)  =  ( 0g
`  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455    C_ wss 3129   ` cfv 5211  (class class class)co 5868   Basecbs 12432   +g cplusg 12505   0gc0g 12640   Grpcgrp 12754
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 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 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-riota 5824  df-ov 5871  df-inn 8896  df-2 8954  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-grp 12757
This theorem is referenced by:  grpinvssd  12823
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