MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rngrghm Unicode version

Theorem rngrghm 15389
Description: Right-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.)
Hypotheses
Ref Expression
rnglghm.b  |-  B  =  ( Base `  R
)
rnglghm.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
rngrghm  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
x  e.  B  |->  ( x  .x.  X ) )  e.  ( R 
GrpHom  R ) )
Distinct variable groups:    x, B    x, R    x,  .x.    x, X

Proof of Theorem rngrghm
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnglghm.b . 2  |-  B  =  ( Base `  R
)
2 eqid 2283 . 2  |-  ( +g  `  R )  =  ( +g  `  R )
3 rnggrp 15346 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
43adantr 451 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  R  e.  Grp )
5 rnglghm.t . . . . . 6  |-  .x.  =  ( .r `  R )
61, 5rngcl 15354 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  B  /\  X  e.  B )  ->  (
x  .x.  X )  e.  B )
763expa 1151 . . . 4  |-  ( ( ( R  e.  Ring  /\  x  e.  B )  /\  X  e.  B
)  ->  ( x  .x.  X )  e.  B
)
87an32s 779 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  x  e.  B
)  ->  ( x  .x.  X )  e.  B
)
9 eqid 2283 . . 3  |-  ( x  e.  B  |->  ( x 
.x.  X ) )  =  ( x  e.  B  |->  ( x  .x.  X ) )
108, 9fmptd 5684 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
x  e.  B  |->  ( x  .x.  X ) ) : B --> B )
11 df-3an 936 . . . . 5  |-  ( ( y  e.  B  /\  z  e.  B  /\  X  e.  B )  <->  ( ( y  e.  B  /\  z  e.  B
)  /\  X  e.  B ) )
121, 2, 5rngdir 15360 . . . . 5  |-  ( ( R  e.  Ring  /\  (
y  e.  B  /\  z  e.  B  /\  X  e.  B )
)  ->  ( (
y ( +g  `  R
) z )  .x.  X )  =  ( ( y  .x.  X
) ( +g  `  R
) ( z  .x.  X ) ) )
1311, 12sylan2br 462 . . . 4  |-  ( ( R  e.  Ring  /\  (
( y  e.  B  /\  z  e.  B
)  /\  X  e.  B ) )  -> 
( ( y ( +g  `  R ) z )  .x.  X
)  =  ( ( y  .x.  X ) ( +g  `  R
) ( z  .x.  X ) ) )
1413anass1rs 782 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( ( y ( +g  `  R ) z )  .x.  X
)  =  ( ( y  .x.  X ) ( +g  `  R
) ( z  .x.  X ) ) )
151, 2rngacl 15368 . . . . . 6  |-  ( ( R  e.  Ring  /\  y  e.  B  /\  z  e.  B )  ->  (
y ( +g  `  R
) z )  e.  B )
16153expb 1152 . . . . 5  |-  ( ( R  e.  Ring  /\  (
y  e.  B  /\  z  e.  B )
)  ->  ( y
( +g  `  R ) z )  e.  B
)
1716adantlr 695 . . . 4  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( y ( +g  `  R ) z )  e.  B )
18 oveq1 5865 . . . . 5  |-  ( x  =  ( y ( +g  `  R ) z )  ->  (
x  .x.  X )  =  ( ( y ( +g  `  R
) z )  .x.  X ) )
19 ovex 5883 . . . . 5  |-  ( ( y ( +g  `  R
) z )  .x.  X )  e.  _V
2018, 9, 19fvmpt 5602 . . . 4  |-  ( ( y ( +g  `  R
) z )  e.  B  ->  ( (
x  e.  B  |->  ( x  .x.  X ) ) `  ( y ( +g  `  R
) z ) )  =  ( ( y ( +g  `  R
) z )  .x.  X ) )
2117, 20syl 15 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( ( x  e.  B  |->  ( x  .x.  X ) ) `  ( y ( +g  `  R ) z ) )  =  ( ( y ( +g  `  R
) z )  .x.  X ) )
22 oveq1 5865 . . . . . 6  |-  ( x  =  y  ->  (
x  .x.  X )  =  ( y  .x.  X ) )
23 ovex 5883 . . . . . 6  |-  ( y 
.x.  X )  e. 
_V
2422, 9, 23fvmpt 5602 . . . . 5  |-  ( y  e.  B  ->  (
( x  e.  B  |->  ( x  .x.  X
) ) `  y
)  =  ( y 
.x.  X ) )
25 oveq1 5865 . . . . . 6  |-  ( x  =  z  ->  (
x  .x.  X )  =  ( z  .x.  X ) )
26 ovex 5883 . . . . . 6  |-  ( z 
.x.  X )  e. 
_V
2725, 9, 26fvmpt 5602 . . . . 5  |-  ( z  e.  B  ->  (
( x  e.  B  |->  ( x  .x.  X
) ) `  z
)  =  ( z 
.x.  X ) )
2824, 27oveqan12d 5877 . . . 4  |-  ( ( y  e.  B  /\  z  e.  B )  ->  ( ( ( x  e.  B  |->  ( x 
.x.  X ) ) `
 y ) ( +g  `  R ) ( ( x  e.  B  |->  ( x  .x.  X ) ) `  z ) )  =  ( ( y  .x.  X ) ( +g  `  R ) ( z 
.x.  X ) ) )
2928adantl 452 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( ( ( x  e.  B  |->  ( x 
.x.  X ) ) `
 y ) ( +g  `  R ) ( ( x  e.  B  |->  ( x  .x.  X ) ) `  z ) )  =  ( ( y  .x.  X ) ( +g  `  R ) ( z 
.x.  X ) ) )
3014, 21, 293eqtr4d 2325 . 2  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( ( x  e.  B  |->  ( x  .x.  X ) ) `  ( y ( +g  `  R ) z ) )  =  ( ( ( x  e.  B  |->  ( x  .x.  X
) ) `  y
) ( +g  `  R
) ( ( x  e.  B  |->  ( x 
.x.  X ) ) `
 z ) ) )
311, 1, 2, 2, 4, 4, 10, 30isghmd 14692 1  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
x  e.  B  |->  ( x  .x.  X ) )  e.  ( R 
GrpHom  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209   Grpcgrp 14362    GrpHom cghm 14680   Ringcrg 15337
This theorem is referenced by:  gsummulc1  15390  fidomndrnglem  16047
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-mnd 14367  df-grp 14489  df-ghm 14681  df-mgp 15326  df-rng 15340
  Copyright terms: Public domain W3C validator