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Theorem ringressid 13169
Description: A ring restricted to its base set is a ring. It will usually be the original ring exactly, of course, but to show that needs additional conditions such as those in strressid 12522. (Contributed by Jim Kingdon, 28-Feb-2025.)
Hypothesis
Ref Expression
ringressid.b  |-  B  =  ( Base `  G
)
Assertion
Ref Expression
ringressid  |-  ( G  e.  Ring  ->  ( Gs  B )  e.  Ring )

Proof of Theorem ringressid
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2178 . . 3  |-  ( G  e.  Ring  ->  ( Gs  B )  =  ( Gs  B ) )
2 ringressid.b . . . 4  |-  B  =  ( Base `  G
)
32a1i 9 . . 3  |-  ( G  e.  Ring  ->  B  =  ( Base `  G
) )
4 id 19 . . 3  |-  ( G  e.  Ring  ->  G  e. 
Ring )
5 ssidd 3176 . . 3  |-  ( G  e.  Ring  ->  B  C_  B )
61, 3, 4, 5ressbas2d 12520 . 2  |-  ( G  e.  Ring  ->  B  =  ( Base `  ( Gs  B ) ) )
7 eqidd 2178 . . 3  |-  ( G  e.  Ring  ->  ( +g  `  G )  =  ( +g  `  G ) )
8 basfn 12512 . . . . 5  |-  Base  Fn  _V
9 elex 2748 . . . . 5  |-  ( G  e.  Ring  ->  G  e. 
_V )
10 funfvex 5531 . . . . . 6  |-  ( ( Fun  Base  /\  G  e. 
dom  Base )  ->  ( Base `  G )  e. 
_V )
1110funfni 5315 . . . . 5  |-  ( (
Base  Fn  _V  /\  G  e.  _V )  ->  ( Base `  G )  e. 
_V )
128, 9, 11sylancr 414 . . . 4  |-  ( G  e.  Ring  ->  ( Base `  G )  e.  _V )
132, 12eqeltrid 2264 . . 3  |-  ( G  e.  Ring  ->  B  e. 
_V )
141, 7, 13, 4ressplusgd 12579 . 2  |-  ( G  e.  Ring  ->  ( +g  `  G )  =  ( +g  `  ( Gs  B ) ) )
15 eqid 2177 . . . 4  |-  ( Gs  B )  =  ( Gs  B )
16 eqid 2177 . . . 4  |-  ( .r
`  G )  =  ( .r `  G
)
1715, 16ressmulrg 12595 . . 3  |-  ( ( B  e.  _V  /\  G  e.  Ring )  -> 
( .r `  G
)  =  ( .r
`  ( Gs  B ) ) )
1813, 17mpancom 422 . 2  |-  ( G  e.  Ring  ->  ( .r
`  G )  =  ( .r `  ( Gs  B ) ) )
19 ringgrp 13115 . . 3  |-  ( G  e.  Ring  ->  G  e. 
Grp )
202grpressid 12863 . . 3  |-  ( G  e.  Grp  ->  ( Gs  B )  e.  Grp )
2119, 20syl 14 . 2  |-  ( G  e.  Ring  ->  ( Gs  B )  e.  Grp )
222, 16ringcl 13127 . 2  |-  ( ( G  e.  Ring  /\  x  e.  B  /\  y  e.  B )  ->  (
x ( .r `  G ) y )  e.  B )
232, 16ringass 13130 . 2  |-  ( ( G  e.  Ring  /\  (
x  e.  B  /\  y  e.  B  /\  z  e.  B )
)  ->  ( (
x ( .r `  G ) y ) ( .r `  G
) z )  =  ( x ( .r
`  G ) ( y ( .r `  G ) z ) ) )
24 eqid 2177 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
252, 24, 16ringdi 13132 . 2  |-  ( ( G  e.  Ring  /\  (
x  e.  B  /\  y  e.  B  /\  z  e.  B )
)  ->  ( x
( .r `  G
) ( y ( +g  `  G ) z ) )  =  ( ( x ( .r `  G ) y ) ( +g  `  G ) ( x ( .r `  G
) z ) ) )
262, 24, 16ringdir 13133 . 2  |-  ( ( G  e.  Ring  /\  (
x  e.  B  /\  y  e.  B  /\  z  e.  B )
)  ->  ( (
x ( +g  `  G
) y ) ( .r `  G ) z )  =  ( ( x ( .r
`  G ) z ) ( +g  `  G
) ( y ( .r `  G ) z ) ) )
27 eqid 2177 . . 3  |-  ( 1r
`  G )  =  ( 1r `  G
)
282, 27ringidcl 13134 . 2  |-  ( G  e.  Ring  ->  ( 1r
`  G )  e.  B )
292, 16, 27ringlidm 13137 . 2  |-  ( ( G  e.  Ring  /\  x  e.  B )  ->  (
( 1r `  G
) ( .r `  G ) x )  =  x )
302, 16, 27ringridm 13138 . 2  |-  ( ( G  e.  Ring  /\  x  e.  B )  ->  (
x ( .r `  G ) ( 1r
`  G ) )  =  x )
316, 14, 18, 21, 22, 23, 25, 26, 28, 29, 30isringd 13151 1  |-  ( G  e.  Ring  ->  ( Gs  B )  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   _Vcvv 2737    Fn wfn 5210   ` cfv 5215  (class class class)co 5872   Basecbs 12454   ↾s cress 12455   +g cplusg 12528   .rcmulr 12529   Grpcgrp 12809   1rcur 13073   Ringcrg 13110
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-coll 4117  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-lttrn 7922  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-iun 3888  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-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  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-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-mgp 13062  df-ur 13074  df-ring 13112
This theorem is referenced by:  subrgid  13282
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