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Theorem rngo2 25255
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 25250 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  E. x  e.  X  ( (
x H A )  =  A  /\  ( A H x )  =  A ) )
5 oveq12 6286 . . . . . . 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 2449 . . . . 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 25253 . . . . . . 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 1229 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  x  e.  X
)  ->  ( (
x G x ) H A )  =  ( ( x H A ) G ( x H A ) ) )
1312eqeq2d 2455 . . . . 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 2916 . 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 1381    e. wcel 1802   E.wrex 2792   ran crn 4986   ` cfv 5574  (class class class)co 6277   1stc1st 6779   2ndc2nd 6780   RingOpscrngo 25242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fv 5582  df-ov 6280  df-1st 6781  df-2nd 6782  df-rngo 25243
This theorem is referenced by: (None)
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