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Theorem ringadd2 13036
Description: A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.)
Hypotheses
Ref Expression
ringadd2.b  |-  B  =  ( Base `  R
)
ringadd2.p  |-  .+  =  ( +g  `  R )
ringadd2.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
ringadd2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  E. x  e.  B  ( X  .+  X )  =  ( ( x  .+  x
)  .x.  X )
)
Distinct variable groups:    x, B    x, R    x, X    x,  .x.
Allowed substitution hint:    .+ ( x)

Proof of Theorem ringadd2
StepHypRef Expression
1 ringadd2.b . . 3  |-  B  =  ( Base `  R
)
2 ringadd2.t . . 3  |-  .x.  =  ( .r `  R )
31, 2ringid 13035 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  E. x  e.  B  ( (
x  .x.  X )  =  X  /\  ( X  .x.  x )  =  X ) )
4 oveq12 5878 . . . . . . 7  |-  ( ( ( x  .x.  X
)  =  X  /\  ( x  .x.  X )  =  X )  -> 
( ( x  .x.  X )  .+  (
x  .x.  X )
)  =  ( X 
.+  X ) )
54anidms 397 . . . . . 6  |-  ( ( x  .x.  X )  =  X  ->  (
( x  .x.  X
)  .+  ( x  .x.  X ) )  =  ( X  .+  X
) )
65eqcomd 2183 . . . . 5  |-  ( ( x  .x.  X )  =  X  ->  ( X  .+  X )  =  ( ( x  .x.  X )  .+  (
x  .x.  X )
) )
7 simpll 527 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  x  e.  B
)  ->  R  e.  Ring )
8 simpr 110 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  x  e.  B
)  ->  x  e.  B )
9 simplr 528 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  x  e.  B
)  ->  X  e.  B )
10 ringadd2.p . . . . . . . 8  |-  .+  =  ( +g  `  R )
111, 10, 2ringdir 13028 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
x  e.  B  /\  x  e.  B  /\  X  e.  B )
)  ->  ( (
x  .+  x )  .x.  X )  =  ( ( x  .x.  X
)  .+  ( x  .x.  X ) ) )
127, 8, 8, 9, 11syl13anc 1240 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  x  e.  B
)  ->  ( (
x  .+  x )  .x.  X )  =  ( ( x  .x.  X
)  .+  ( x  .x.  X ) ) )
1312eqeq2d 2189 . . . . 5  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  x  e.  B
)  ->  ( ( X  .+  X )  =  ( ( x  .+  x )  .x.  X
)  <->  ( X  .+  X )  =  ( ( x  .x.  X
)  .+  ( x  .x.  X ) ) ) )
146, 13syl5ibr 156 . . . 4  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  x  e.  B
)  ->  ( (
x  .x.  X )  =  X  ->  ( X 
.+  X )  =  ( ( x  .+  x )  .x.  X
) ) )
1514adantrd 279 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  x  e.  B
)  ->  ( (
( x  .x.  X
)  =  X  /\  ( X  .x.  x )  =  X )  -> 
( X  .+  X
)  =  ( ( x  .+  x ) 
.x.  X ) ) )
1615reximdva 2579 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( E. x  e.  B  ( ( x  .x.  X )  =  X  /\  ( X  .x.  x )  =  X )  ->  E. x  e.  B  ( X  .+  X )  =  ( ( x  .+  x
)  .x.  X )
) )
173, 16mpd 13 1  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  E. x  e.  B  ( X  .+  X )  =  ( ( x  .+  x
)  .x.  X )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   E.wrex 2456   ` cfv 5212  (class class class)co 5869   Basecbs 12445   +g cplusg 12518   .rcmulr 12519   Ringcrg 13005
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-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-pre-ltirr 7914  ax-pre-ltadd 7918
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-ltxr 7987  df-inn 8909  df-2 8967  df-3 8968  df-ndx 12448  df-slot 12449  df-base 12451  df-sets 12452  df-plusg 12531  df-mulr 12532  df-0g 12655  df-mgm 12667  df-sgrp 12700  df-mnd 12710  df-mgp 12958  df-ur 12969  df-ring 13007
This theorem is referenced by: (None)
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