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Theorem ressplusf 24173
Description: The group operation function  + f of a structure's restriction is the operation function's restriction to the new base. (Contributed by Thierry Arnoux, 26-Mar-2017.)
Hypotheses
Ref Expression
ressplusf.1  |-  B  =  ( Base `  G
)
ressplusf.2  |-  H  =  ( Gs  A )
ressplusf.3  |-  .+^  =  ( +g  `  G )
ressplusf.4  |-  .+^  Fn  ( B  X.  B )
ressplusf.5  |-  A  C_  B
Assertion
Ref Expression
ressplusf  |-  ( + f `  H )  =  (  .+^  |`  ( A  X.  A ) )

Proof of Theorem ressplusf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ressplusf.5 . . 3  |-  A  C_  B
2 resmpt2 6160 . . 3  |-  ( ( A  C_  B  /\  A  C_  B )  -> 
( ( x  e.  B ,  y  e.  B  |->  ( x  .+^  y ) )  |`  ( A  X.  A
) )  =  ( x  e.  A , 
y  e.  A  |->  ( x  .+^  y )
) )
31, 1, 2mp2an 654 . 2  |-  ( ( x  e.  B , 
y  e.  B  |->  ( x  .+^  y )
)  |`  ( A  X.  A ) )  =  ( x  e.  A ,  y  e.  A  |->  ( x  .+^  y ) )
4 ressplusf.4 . . . 4  |-  .+^  Fn  ( B  X.  B )
5 fnov 6170 . . . 4  |-  (  .+^  Fn  ( B  X.  B
)  <->  .+^  =  ( x  e.  B ,  y  e.  B  |->  ( x 
.+^  y ) ) )
64, 5mpbi 200 . . 3  |-  .+^  =  ( x  e.  B , 
y  e.  B  |->  ( x  .+^  y )
)
76reseq1i 5134 . 2  |-  (  .+^  |`  ( A  X.  A
) )  =  ( ( x  e.  B ,  y  e.  B  |->  ( x  .+^  y ) )  |`  ( A  X.  A ) )
8 ressplusf.2 . . . . 5  |-  H  =  ( Gs  A )
9 ressplusf.1 . . . . 5  |-  B  =  ( Base `  G
)
108, 9ressbas2 13510 . . . 4  |-  ( A 
C_  B  ->  A  =  ( Base `  H
) )
111, 10ax-mp 8 . . 3  |-  A  =  ( Base `  H
)
12 ressplusf.3 . . . 4  |-  .+^  =  ( +g  `  G )
13 fvex 5734 . . . . . . 7  |-  ( Base `  G )  e.  _V
149, 13eqeltri 2505 . . . . . 6  |-  B  e. 
_V
1514, 1ssexi 4340 . . . . 5  |-  A  e. 
_V
16 eqid 2435 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
178, 16ressplusg 13561 . . . . 5  |-  ( A  e.  _V  ->  ( +g  `  G )  =  ( +g  `  H
) )
1815, 17ax-mp 8 . . . 4  |-  ( +g  `  G )  =  ( +g  `  H )
1912, 18eqtri 2455 . . 3  |-  .+^  =  ( +g  `  H )
20 eqid 2435 . . 3  |-  ( + f `  H )  =  ( + f `  H )
2111, 19, 20plusffval 14692 . 2  |-  ( + f `  H )  =  ( x  e.  A ,  y  e.  A  |->  ( x  .+^  y ) )
223, 7, 213eqtr4ri 2466 1  |-  ( + f `  H )  =  (  .+^  |`  ( A  X.  A ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312    X. cxp 4868    |` cres 4872    Fn wfn 5441   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   Basecbs 13459   ↾s cress 13460   +g cplusg 13519   + fcplusf 14677
This theorem is referenced by:  xrge0pluscn  24316  xrge0tmdOLD  24321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-plusf 14681
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