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Theorem rngo2 21055
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 21050 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  E. x  e.  X  ( (
x H A )  =  A  /\  ( A H x )  =  A ) )
5 oveq12 5867 . . . . . . 7  |-  ( ( ( x H A )  =  A  /\  ( x H A )  =  A )  ->  ( ( x H A ) G ( x H A ) )  =  ( A G A ) )
65anidms 626 . . . . . 6  |-  ( ( x H A )  =  A  ->  (
( x H A ) G ( x H A ) )  =  ( A G A ) )
76eqcomd 2288 . . . . 5  |-  ( ( x H A )  =  A  ->  ( A G A )  =  ( ( x H A ) G ( x H A ) ) )
8 simpll 730 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  R  e.  RingOps )
9 simpr 447 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  x  e.  X )
10 simplr 731 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  A  e.  X )
111, 2, 3rngodir 21053 . . . . . . 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 1184 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  ( (
x G x ) H A )  =  ( ( x H A ) G ( x H A ) ) )
1312eqeq2d 2294 . . . . 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 212 . . . 4  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  ( (
x H A )  =  A  ->  ( A G A )  =  ( ( x G x ) H A ) ) )
1514adantrd 454 . . 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 2655 . 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 14 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 358    = wceq 1623    e. wcel 1684   E.wrex 2544   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   RingOpscrngo 21042
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-rngo 21043
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