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Theorem rngo2 23880
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.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringi.1  |-  G  =  ( 1st `  R
)
ringi.2  |-  H  =  ( 2nd `  R
)
ringi.3  |-  X  =  ran  G
Assertion
Ref Expression
rngo2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  E. x  e.  X  ( A G A )  =  ( ( x G x ) H A ) )
Distinct variable groups:    x, G    x, H    x, X    x, A    x, R

Proof of Theorem rngo2
StepHypRef Expression
1 ringi.1 . . 3  |-  G  =  ( 1st `  R
)
2 ringi.2 . . 3  |-  H  =  ( 2nd `  R
)
3 ringi.3 . . 3  |-  X  =  ran  G
41, 2, 3rngoid 23875 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  E. x  e.  X  ( (
x H A )  =  A  /\  ( A H x )  =  A ) )
5 oveq12 6105 . . . . . . 7  |-  ( ( ( x H A )  =  A  /\  ( x H A )  =  A )  ->  ( ( x H A ) G ( x H A ) )  =  ( A G A ) )
65anidms 645 . . . . . 6  |-  ( ( x H A )  =  A  ->  (
( x H A ) G ( x H A ) )  =  ( A G A ) )
76eqcomd 2448 . . . . 5  |-  ( ( x H A )  =  A  ->  ( A G A )  =  ( ( x H A ) G ( x H A ) ) )
8 simpll 753 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  R  e.  RingOps )
9 simpr 461 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  x  e.  X )
10 simplr 754 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  A  e.  X )
111, 2, 3rngodir 23878 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  (
x  e.  X  /\  x  e.  X  /\  A  e.  X )
)  ->  ( (
x G x ) H A )  =  ( ( x H A ) G ( x H A ) ) )
128, 9, 9, 10, 11syl13anc 1220 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  ( (
x G x ) H A )  =  ( ( x H A ) G ( x H A ) ) )
1312eqeq2d 2454 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  ( ( A G A )  =  ( ( x G x ) H A )  <->  ( A G A )  =  ( ( x H A ) G ( x H A ) ) ) )
147, 13syl5ibr 221 . . . 4  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  ( (
x H A )  =  A  ->  ( A G A )  =  ( ( x G x ) H A ) ) )
1514adantrd 468 . . 3  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  ( (
( x H A )  =  A  /\  ( A H x )  =  A )  -> 
( A G A )  =  ( ( x G x ) H A ) ) )
1615reximdva 2833 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( E. x  e.  X  ( ( x H A )  =  A  /\  ( A H x )  =  A )  ->  E. x  e.  X  ( A G A )  =  ( ( x G x ) H A ) ) )
174, 16mpd 15 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  E. x  e.  X  ( A G A )  =  ( ( x G x ) H A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2721   ran crn 4846   ` cfv 5423  (class class class)co 6096   1stc1st 6580   2ndc2nd 6581   RingOpscrngo 23867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-1st 6582  df-2nd 6583  df-rngo 23868
This theorem is referenced by: (None)
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