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Theorem ress0g 15756
Description:  0g is unaffected by restriction. This is a bit more generic than submnd0 15757 (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 14535 . . 3  |-  ( A 
C_  B  ->  A  =  ( Base `  S
) )
433ad2ant3 1014 . 2  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  A  =  ( Base `  S
) )
5 simp3 993 . . . 4  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  A  C_  B )
6 fvex 5867 . . . . 5  |-  ( Base `  R )  e.  _V
72, 6eqeltri 2544 . . . 4  |-  B  e. 
_V
8 ssexg 4586 . . . 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 2460 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
111, 10ressplusg 14586 . . 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 992 . 2  |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  .0.  e.  A )
14 simpl1 994 . . 3  |-  ( ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  /\  x  e.  A )  ->  R  e.  Mnd )
155sselda 3497 . . 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 15747 . . 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 15748 . . 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 15749 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 968    = wceq 1374    e. wcel 1762   _Vcvv 3106    C_ wss 3469   ` cfv 5579  (class class class)co 6275   Basecbs 14479   ↾s cress 14480   +g cplusg 14544   0gc0g 14684   Mndcmnd 15715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-0g 14686  df-mnd 15721
This theorem is referenced by:  nn0srg  18247  rge0srg  18248  zring0  18259  zrng0  18265  re0g  18408  ressnm  27151  xrge0slmod  27347
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