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Theorem ress0g 12776
Description:  0g is unaffected by restriction. This is a bit more generic than submnd0 12777. (Contributed by Thierry Arnoux, 23-Oct-2017.)
Hypotheses
Ref Expression
ress0g.s  |-  S  =  ( Rs  A )
ress0g.b  |-  B  =  ( Base `  R
)
ress0g.0  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
ress0g  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  .0.  =  ( 0g `  S ) )

Proof of Theorem ress0g
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ress0g.s . . . 4  |-  S  =  ( Rs  A )
21a1i 9 . . 3  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  S  =  ( Rs  A ) )
3 ress0g.b . . . 4  |-  B  =  ( Base `  R
)
43a1i 9 . . 3  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  B  =  ( Base `  R
) )
5 simp1 997 . . 3  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  R  e.  Mnd )
6 simp3 999 . . 3  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  A  C_  B )
72, 4, 5, 6ressbas2d 12520 . 2  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  A  =  ( Base `  S
) )
8 eqidd 2178 . . 3  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  ( +g  `  R )  =  ( +g  `  R
) )
9 basfn 12512 . . . . . 6  |-  Base  Fn  _V
105elexd 2750 . . . . . 6  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  R  e.  _V )
11 funfvex 5531 . . . . . . 7  |-  ( ( Fun  Base  /\  R  e. 
dom  Base )  ->  ( Base `  R )  e. 
_V )
1211funfni 5315 . . . . . 6  |-  ( (
Base  Fn  _V  /\  R  e.  _V )  ->  ( Base `  R )  e. 
_V )
139, 10, 12sylancr 414 . . . . 5  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  ( Base `  R )  e. 
_V )
143, 13eqeltrid 2264 . . . 4  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  B  e.  _V )
1514, 6ssexd 4142 . . 3  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  A  e.  _V )
162, 8, 15, 5ressplusgd 12579 . 2  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  ( +g  `  R )  =  ( +g  `  S
) )
17 simp2 998 . 2  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  .0.  e.  A )
18 simpl1 1000 . . 3  |-  ( ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  /\  x  e.  A )  ->  R  e.  Mnd )
196sselda 3155 . . 3  |-  ( ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  /\  x  e.  A )  ->  x  e.  B )
20 eqid 2177 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
21 ress0g.0 . . . 4  |-  .0.  =  ( 0g `  R )
223, 20, 21mndlid 12768 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  B )  ->  (  .0.  ( +g  `  R ) x )  =  x )
2318, 19, 22syl2anc 411 . 2  |-  ( ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  /\  x  e.  A )  ->  (  .0.  ( +g  `  R ) x )  =  x )
243, 20, 21mndrid 12769 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  B )  ->  ( x ( +g  `  R )  .0.  )  =  x )
2518, 19, 24syl2anc 411 . 2  |-  ( ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  /\  x  e.  A )  ->  ( x ( +g  `  R )  .0.  )  =  x )
267, 16, 17, 23, 25grpidd 12734 1  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  .0.  =  ( 0g `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148   _Vcvv 2737    C_ wss 3129    Fn wfn 5210   ` cfv 5215  (class class class)co 5872   Basecbs 12454   ↾s cress 12455   +g cplusg 12528   0gc0g 12693   Mndcmnd 12749
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 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-ltadd 7924
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-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750
This theorem is referenced by:  submnd0  12777  zring0  13359
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