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Theorem ress0g 15823
Description:  0g is unaffected by restriction. This is a bit more generic than submnd0 15824 (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 )
2 ress0g.b . . . 4  |-  B  =  ( Base `  R
)
31, 2ressbas2 14565 . . 3  |-  ( A 
C_  B  ->  A  =  ( Base `  S
) )
433ad2ant3 1020 . 2  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  A  =  ( Base `  S
) )
5 simp3 999 . . . 4  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  A  C_  B )
6 fvex 5866 . . . . 5  |-  ( Base `  R )  e.  _V
72, 6eqeltri 2527 . . . 4  |-  B  e. 
_V
8 ssexg 4583 . . . 4  |-  ( ( A  C_  B  /\  B  e.  _V )  ->  A  e.  _V )
95, 7, 8sylancl 662 . . 3  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  A  e.  _V )
10 eqid 2443 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
111, 10ressplusg 14616 . . 3  |-  ( A  e.  _V  ->  ( +g  `  R )  =  ( +g  `  S
) )
129, 11syl 16 . 2  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  ( +g  `  R )  =  ( +g  `  S
) )
13 simp2 998 . 2  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  .0.  e.  A )
14 simpl1 1000 . . 3  |-  ( ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  /\  x  e.  A )  ->  R  e.  Mnd )
155sselda 3489 . . 3  |-  ( ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  /\  x  e.  A )  ->  x  e.  B )
16 ress0g.0 . . . 4  |-  .0.  =  ( 0g `  R )
172, 10, 16mndlid 15815 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  B )  ->  (  .0.  ( +g  `  R ) x )  =  x )
1814, 15, 17syl2anc 661 . 2  |-  ( ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  /\  x  e.  A )  ->  (  .0.  ( +g  `  R ) x )  =  x )
192, 10, 16mndrid 15816 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  B )  ->  ( x ( +g  `  R )  .0.  )  =  x )
2014, 15, 19syl2anc 661 . 2  |-  ( ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  /\  x  e.  A )  ->  ( x ( +g  `  R )  .0.  )  =  x )
214, 12, 13, 18, 20grpidd 15769 1  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  .0.  =  ( 0g `  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   _Vcvv 3095    C_ wss 3461   ` cfv 5578  (class class class)co 6281   Basecbs 14509   ↾s cress 14510   +g cplusg 14574   0gc0g 14714   Mndcmnd 15793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795
This theorem is referenced by:  nn0srg  18360  rge0srg  18361  zring0  18372  zrng0  18378  re0g  18521  ressnm  27512  xrge0slmod  27707  2zrng0  32454
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