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Theorem rngoid 21963
Description: The multiplication operation of a unital ring has (one or more) identity elements. (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
rngoid  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  E. u  e.  X  ( (
u H A )  =  A  /\  ( A H u )  =  A ) )
Distinct variable groups:    u, G    u, H    u, X    u, A    u, R

Proof of Theorem rngoid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringi.1 . . . . . 6  |-  G  =  ( 1st `  R
)
2 ringi.2 . . . . . 6  |-  H  =  ( 2nd `  R
)
3 ringi.3 . . . . . 6  |-  X  =  ran  G
41, 2, 3rngoi 21960 . . . . 5  |-  ( R  e.  RingOps  ->  ( ( G  e.  AbelOp  /\  H :
( X  X.  X
) --> X )  /\  ( A. u  e.  X  A. x  e.  X  A. y  e.  X  ( ( ( u H x ) H y )  =  ( u H ( x H y ) )  /\  ( u H ( x G y ) )  =  ( ( u H x ) G ( u H y ) )  /\  ( ( u G x ) H y )  =  ( ( u H y ) G ( x H y ) ) )  /\  E. u  e.  X  A. x  e.  X  ( (
u H x )  =  x  /\  (
x H u )  =  x ) ) ) )
54simprd 450 . . . 4  |-  ( R  e.  RingOps  ->  ( A. u  e.  X  A. x  e.  X  A. y  e.  X  ( (
( u H x ) H y )  =  ( u H ( x H y ) )  /\  (
u H ( x G y ) )  =  ( ( u H x ) G ( u H y ) )  /\  (
( u G x ) H y )  =  ( ( u H y ) G ( x H y ) ) )  /\  E. u  e.  X  A. x  e.  X  (
( u H x )  =  x  /\  ( x H u )  =  x ) ) )
65simprd 450 . . 3  |-  ( R  e.  RingOps  ->  E. u  e.  X  A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
7 r19.12 2811 . . 3  |-  ( E. u  e.  X  A. x  e.  X  (
( u H x )  =  x  /\  ( x H u )  =  x )  ->  A. x  e.  X  E. u  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
86, 7syl 16 . 2  |-  ( R  e.  RingOps  ->  A. x  e.  X  E. u  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
9 oveq2 6081 . . . . . 6  |-  ( x  =  A  ->  (
u H x )  =  ( u H A ) )
10 id 20 . . . . . 6  |-  ( x  =  A  ->  x  =  A )
119, 10eqeq12d 2449 . . . . 5  |-  ( x  =  A  ->  (
( u H x )  =  x  <->  ( u H A )  =  A ) )
12 oveq1 6080 . . . . . 6  |-  ( x  =  A  ->  (
x H u )  =  ( A H u ) )
1312, 10eqeq12d 2449 . . . . 5  |-  ( x  =  A  ->  (
( x H u )  =  x  <->  ( A H u )  =  A ) )
1411, 13anbi12d 692 . . . 4  |-  ( x  =  A  ->  (
( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  ( ( u H A )  =  A  /\  ( A H u )  =  A ) ) )
1514rexbidv 2718 . . 3  |-  ( x  =  A  ->  ( E. u  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  E. u  e.  X  ( ( u H A )  =  A  /\  ( A H u )  =  A ) ) )
1615rspccva 3043 . 2  |-  ( ( A. x  e.  X  E. u  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x )  /\  A  e.  X )  ->  E. u  e.  X  ( (
u H A )  =  A  /\  ( A H u )  =  A ) )
178, 16sylan 458 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  E. u  e.  X  ( (
u H A )  =  A  /\  ( A H u )  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    X. cxp 4868   ran crn 4871   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   AbelOpcablo 21861   RingOpscrngo 21955
This theorem is referenced by:  rngo2  21968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-1st 6341  df-2nd 6342  df-rngo 21956
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