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Theorem rngo2 21978
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 21973 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  E. x  e.  X  ( (
x H A )  =  A  /\  ( A H x )  =  A ) )
5 oveq12 6092 . . . . . . 7  |-  ( ( ( x H A )  =  A  /\  ( x H A )  =  A )  ->  ( ( x H A ) G ( x H A ) )  =  ( A G A ) )
65anidms 628 . . . . . 6  |-  ( ( x H A )  =  A  ->  (
( x H A ) G ( x H A ) )  =  ( A G A ) )
76eqcomd 2443 . . . . 5  |-  ( ( x H A )  =  A  ->  ( A G A )  =  ( ( x H A ) G ( x H A ) ) )
8 simpll 732 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  R  e.  RingOps )
9 simpr 449 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  x  e.  X )
10 simplr 733 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  A  e.  X )
111, 2, 3rngodir 21976 . . . . . . 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 1187 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  ( (
x G x ) H A )  =  ( ( x H A ) G ( x H A ) ) )
1312eqeq2d 2449 . . . . 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 214 . . . 4  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  ( (
x H A )  =  A  ->  ( A G A )  =  ( ( x G x ) H A ) ) )
1514adantrd 456 . . 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 2820 . 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 360    = wceq 1653    e. wcel 1726   E.wrex 2708   ran crn 4881   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350   RingOpscrngo 21965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-1st 6351  df-2nd 6352  df-rngo 21966
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