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Theorem rngo2 21929
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 21924 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  E. x  e.  X  ( (
x H A )  =  A  /\  ( A H x )  =  A ) )
5 oveq12 6049 . . . . . . 7  |-  ( ( ( x H A )  =  A  /\  ( x H A )  =  A )  ->  ( ( x H A ) G ( x H A ) )  =  ( A G A ) )
65anidms 627 . . . . . 6  |-  ( ( x H A )  =  A  ->  (
( x H A ) G ( x H A ) )  =  ( A G A ) )
76eqcomd 2409 . . . . 5  |-  ( ( x H A )  =  A  ->  ( A G A )  =  ( ( x H A ) G ( x H A ) ) )
8 simpll 731 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  R  e.  RingOps )
9 simpr 448 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  x  e.  X )
10 simplr 732 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  A  e.  X )
111, 2, 3rngodir 21927 . . . . . . 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 1186 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  ( (
x G x ) H A )  =  ( ( x H A ) G ( x H A ) ) )
1312eqeq2d 2415 . . . . 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 213 . . . 4  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  ( (
x H A )  =  A  ->  ( A G A )  =  ( ( x G x ) H A ) ) )
1514adantrd 455 . . 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 2778 . 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   ran crn 4838   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   RingOpscrngo 21916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-1st 6308  df-2nd 6309  df-rngo 21917
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