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Theorem rngurd 26424
Description: Deduce the unit of a ring from its properties. (Contributed by Thierry Arnoux, 6-Sep-2016.)
Hypotheses
Ref Expression
rngurd.b  |-  ( ph  ->  B  =  ( Base `  R ) )
rngurd.p  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
rngurd.z  |-  ( ph  ->  .1.  e.  B )
rngurd.i  |-  ( (
ph  /\  x  e.  B )  ->  (  .1.  .x.  x )  =  x )
rngurd.j  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .x.  .1.  )  =  x )
Assertion
Ref Expression
rngurd  |-  ( ph  ->  .1.  =  ( 1r
`  R ) )
Distinct variable groups:    x, B    x, R    x,  .1.    x,  .x.    ph, x

Proof of Theorem rngurd
Dummy variable  e is distinct from all other variables.
StepHypRef Expression
1 eqid 2454 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2454 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
3 eqid 2454 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
41, 2, 3dfur2 16738 . 2  |-  ( 1r
`  R )  =  ( iota e ( e  e.  ( Base `  R )  /\  A. x  e.  ( Base `  R ) ( ( e ( .r `  R ) x )  =  x  /\  (
x ( .r `  R ) e )  =  x ) ) )
5 rngurd.z . . . 4  |-  ( ph  ->  .1.  e.  B )
6 rngurd.b . . . 4  |-  ( ph  ->  B  =  ( Base `  R ) )
75, 6eleqtrd 2544 . . 3  |-  ( ph  ->  .1.  e.  ( Base `  R ) )
8 rngurd.i . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (  .1.  .x.  x )  =  x )
9 rngurd.j . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .x.  .1.  )  =  x )
108, 9jca 532 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  (
(  .1.  .x.  x
)  =  x  /\  ( x  .x.  .1.  )  =  x ) )
1110ralrimiva 2830 . . . 4  |-  ( ph  ->  A. x  e.  B  ( (  .1.  .x.  x )  =  x  /\  ( x  .x.  .1.  )  =  x
) )
12 rngurd.p . . . . . . . . 9  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
1312adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  .x.  =  ( .r `  R ) )
1413oveqd 6220 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (  .1.  .x.  x )  =  (  .1.  ( .r
`  R ) x ) )
1514eqeq1d 2456 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (
(  .1.  .x.  x
)  =  x  <->  (  .1.  ( .r `  R ) x )  =  x ) )
1613oveqd 6220 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .x.  .1.  )  =  ( x ( .r `  R )  .1.  ) )
1716eqeq1d 2456 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (
( x  .x.  .1.  )  =  x  <->  ( x
( .r `  R
)  .1.  )  =  x ) )
1815, 17anbi12d 710 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  (
( (  .1.  .x.  x )  =  x  /\  ( x  .x.  .1.  )  =  x
)  <->  ( (  .1.  ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
)  .1.  )  =  x ) ) )
196, 18raleqbidva 3039 . . . 4  |-  ( ph  ->  ( A. x  e.  B  ( (  .1. 
.x.  x )  =  x  /\  ( x 
.x.  .1.  )  =  x )  <->  A. x  e.  ( Base `  R
) ( (  .1.  ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
)  .1.  )  =  x ) ) )
2011, 19mpbid 210 . . 3  |-  ( ph  ->  A. x  e.  (
Base `  R )
( (  .1.  ( .r `  R ) x )  =  x  /\  ( x ( .r
`  R )  .1.  )  =  x ) )
217, 20jca 532 . . . . 5  |-  ( ph  ->  (  .1.  e.  (
Base `  R )  /\  A. x  e.  (
Base `  R )
( (  .1.  ( .r `  R ) x )  =  x  /\  ( x ( .r
`  R )  .1.  )  =  x ) ) )
228ralrimiva 2830 . . . . . . . . . . . 12  |-  ( ph  ->  A. x  e.  B  (  .1.  .x.  x )  =  x )
2322adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  e  e.  B )  ->  A. x  e.  B  (  .1.  .x.  x )  =  x )
24 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  e  e.  B )  ->  e  e.  B )
25 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  B )  /\  x  =  e )  ->  x  =  e )
2625oveq2d 6219 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  e  e.  B )  /\  x  =  e )  -> 
(  .1.  .x.  x
)  =  (  .1. 
.x.  e ) )
2726, 25eqeq12d 2476 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  e  e.  B )  /\  x  =  e )  -> 
( (  .1.  .x.  x )  =  x  <-> 
(  .1.  .x.  e
)  =  e ) )
2824, 27rspcdv 3182 . . . . . . . . . . 11  |-  ( (
ph  /\  e  e.  B )  ->  ( A. x  e.  B  (  .1.  .x.  x )  =  x  ->  (  .1. 
.x.  e )  =  e ) )
2923, 28mpd 15 . . . . . . . . . 10  |-  ( (
ph  /\  e  e.  B )  ->  (  .1.  .x.  e )  =  e )
3029adantrr 716 . . . . . . . . 9  |-  ( (
ph  /\  ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) ) )  ->  (  .1.  .x.  e )  =  e )
315adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) ) )  ->  .1.  e.  B
)
32 simprr 756 . . . . . . . . . 10  |-  ( (
ph  /\  ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) ) )  ->  A. x  e.  B  ( ( e  .x.  x )  =  x  /\  ( x  .x.  e )  =  x ) )
33 oveq2 6211 . . . . . . . . . . . . . . 15  |-  ( x  =  .1.  ->  (
e  .x.  x )  =  ( e  .x.  .1.  ) )
34 id 22 . . . . . . . . . . . . . . 15  |-  ( x  =  .1.  ->  x  =  .1.  )
3533, 34eqeq12d 2476 . . . . . . . . . . . . . 14  |-  ( x  =  .1.  ->  (
( e  .x.  x
)  =  x  <->  ( e  .x.  .1.  )  =  .1.  ) )
36 oveq1 6210 . . . . . . . . . . . . . . 15  |-  ( x  =  .1.  ->  (
x  .x.  e )  =  (  .1.  .x.  e
) )
3736, 34eqeq12d 2476 . . . . . . . . . . . . . 14  |-  ( x  =  .1.  ->  (
( x  .x.  e
)  =  x  <->  (  .1.  .x.  e )  =  .1.  ) )
3835, 37anbi12d 710 . . . . . . . . . . . . 13  |-  ( x  =  .1.  ->  (
( ( e  .x.  x )  =  x  /\  ( x  .x.  e )  =  x )  <->  ( ( e 
.x.  .1.  )  =  .1.  /\  (  .1.  .x.  e )  =  .1.  ) ) )
3938rspcv 3175 . . . . . . . . . . . 12  |-  (  .1. 
e.  B  ->  ( A. x  e.  B  ( ( e  .x.  x )  =  x  /\  ( x  .x.  e )  =  x )  ->  ( (
e  .x.  .1.  )  =  .1.  /\  (  .1. 
.x.  e )  =  .1.  ) ) )
4039imp 429 . . . . . . . . . . 11  |-  ( (  .1.  e.  B  /\  A. x  e.  B  ( ( e  .x.  x
)  =  x  /\  ( x  .x.  e )  =  x ) )  ->  ( ( e 
.x.  .1.  )  =  .1.  /\  (  .1.  .x.  e )  =  .1.  ) )
4140simprd 463 . . . . . . . . . 10  |-  ( (  .1.  e.  B  /\  A. x  e.  B  ( ( e  .x.  x
)  =  x  /\  ( x  .x.  e )  =  x ) )  ->  (  .1.  .x.  e )  =  .1.  )
4231, 32, 41syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) ) )  ->  (  .1.  .x.  e )  =  .1.  )
4330, 42eqtr3d 2497 . . . . . . . 8  |-  ( (
ph  /\  ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) ) )  ->  e  =  .1.  )
4443ex 434 . . . . . . 7  |-  ( ph  ->  ( ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) )  -> 
e  =  .1.  )
)
456eleq2d 2524 . . . . . . . . 9  |-  ( ph  ->  ( e  e.  B  <->  e  e.  ( Base `  R
) ) )
4613oveqd 6220 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  B )  ->  (
e  .x.  x )  =  ( e ( .r `  R ) x ) )
4746eqeq1d 2456 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  B )  ->  (
( e  .x.  x
)  =  x  <->  ( e
( .r `  R
) x )  =  x ) )
4813oveqd 6220 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .x.  e )  =  ( x ( .r `  R ) e ) )
4948eqeq1d 2456 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  B )  ->  (
( x  .x.  e
)  =  x  <->  ( x
( .r `  R
) e )  =  x ) )
5047, 49anbi12d 710 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (
( ( e  .x.  x )  =  x  /\  ( x  .x.  e )  =  x )  <->  ( ( e ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
) e )  =  x ) ) )
516, 50raleqbidva 3039 . . . . . . . . 9  |-  ( ph  ->  ( A. x  e.  B  ( ( e 
.x.  x )  =  x  /\  ( x 
.x.  e )  =  x )  <->  A. x  e.  ( Base `  R
) ( ( e ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
) e )  =  x ) ) )
5245, 51anbi12d 710 . . . . . . . 8  |-  ( ph  ->  ( ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) )  <->  ( e  e.  ( Base `  R
)  /\  A. x  e.  ( Base `  R
) ( ( e ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
) e )  =  x ) ) ) )
5352imbi1d 317 . . . . . . 7  |-  ( ph  ->  ( ( ( e  e.  B  /\  A. x  e.  B  (
( e  .x.  x
)  =  x  /\  ( x  .x.  e )  =  x ) )  ->  e  =  .1.  )  <->  ( ( e  e.  ( Base `  R
)  /\  A. x  e.  ( Base `  R
) ( ( e ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
) e )  =  x ) )  -> 
e  =  .1.  )
) )
5444, 53mpbid 210 . . . . . 6  |-  ( ph  ->  ( ( e  e.  ( Base `  R
)  /\  A. x  e.  ( Base `  R
) ( ( e ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
) e )  =  x ) )  -> 
e  =  .1.  )
)
5554alrimiv 1686 . . . . 5  |-  ( ph  ->  A. e ( ( e  e.  ( Base `  R )  /\  A. x  e.  ( Base `  R ) ( ( e ( .r `  R ) x )  =  x  /\  (
x ( .r `  R ) e )  =  x ) )  ->  e  =  .1.  ) )
56 eleq1 2526 . . . . . . 7  |-  ( e  =  .1.  ->  (
e  e.  ( Base `  R )  <->  .1.  e.  ( Base `  R )
) )
57 oveq1 6210 . . . . . . . . . 10  |-  ( e  =  .1.  ->  (
e ( .r `  R ) x )  =  (  .1.  ( .r `  R ) x ) )
5857eqeq1d 2456 . . . . . . . . 9  |-  ( e  =  .1.  ->  (
( e ( .r
`  R ) x )  =  x  <->  (  .1.  ( .r `  R ) x )  =  x ) )
59 oveq2 6211 . . . . . . . . . 10  |-  ( e  =  .1.  ->  (
x ( .r `  R ) e )  =  ( x ( .r `  R )  .1.  ) )
6059eqeq1d 2456 . . . . . . . . 9  |-  ( e  =  .1.  ->  (
( x ( .r
`  R ) e )  =  x  <->  ( x
( .r `  R
)  .1.  )  =  x ) )
6158, 60anbi12d 710 . . . . . . . 8  |-  ( e  =  .1.  ->  (
( ( e ( .r `  R ) x )  =  x  /\  ( x ( .r `  R ) e )  =  x )  <->  ( (  .1.  ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
)  .1.  )  =  x ) ) )
6261ralbidv 2846 . . . . . . 7  |-  ( e  =  .1.  ->  ( A. x  e.  ( Base `  R ) ( ( e ( .r
`  R ) x )  =  x  /\  ( x ( .r
`  R ) e )  =  x )  <->  A. x  e.  ( Base `  R ) ( (  .1.  ( .r
`  R ) x )  =  x  /\  ( x ( .r
`  R )  .1.  )  =  x ) ) )
6356, 62anbi12d 710 . . . . . 6  |-  ( e  =  .1.  ->  (
( e  e.  (
Base `  R )  /\  A. x  e.  (
Base `  R )
( ( e ( .r `  R ) x )  =  x  /\  ( x ( .r `  R ) e )  =  x ) )  <->  (  .1.  e.  ( Base `  R
)  /\  A. x  e.  ( Base `  R
) ( (  .1.  ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
)  .1.  )  =  x ) ) ) )
6463eqeu 3237 . . . . 5  |-  ( (  .1.  e.  ( Base `  R )  /\  (  .1.  e.  ( Base `  R
)  /\  A. x  e.  ( Base `  R
) ( (  .1.  ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
)  .1.  )  =  x ) )  /\  A. e ( ( e  e.  ( Base `  R
)  /\  A. x  e.  ( Base `  R
) ( ( e ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
) e )  =  x ) )  -> 
e  =  .1.  )
)  ->  E! e
( e  e.  (
Base `  R )  /\  A. x  e.  (
Base `  R )
( ( e ( .r `  R ) x )  =  x  /\  ( x ( .r `  R ) e )  =  x ) ) )
657, 21, 55, 64syl3anc 1219 . . . 4  |-  ( ph  ->  E! e ( e  e.  ( Base `  R
)  /\  A. x  e.  ( Base `  R
) ( ( e ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
) e )  =  x ) ) )
6663iota2 5518 . . . 4  |-  ( (  .1.  e.  B  /\  E! e ( e  e.  ( Base `  R
)  /\  A. x  e.  ( Base `  R
) ( ( e ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
) e )  =  x ) ) )  ->  ( (  .1. 
e.  ( Base `  R
)  /\  A. x  e.  ( Base `  R
) ( (  .1.  ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
)  .1.  )  =  x ) )  <->  ( iota e ( e  e.  ( Base `  R
)  /\  A. x  e.  ( Base `  R
) ( ( e ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
) e )  =  x ) ) )  =  .1.  ) )
675, 65, 66syl2anc 661 . . 3  |-  ( ph  ->  ( (  .1.  e.  ( Base `  R )  /\  A. x  e.  (
Base `  R )
( (  .1.  ( .r `  R ) x )  =  x  /\  ( x ( .r
`  R )  .1.  )  =  x ) )  <->  ( iota e
( e  e.  (
Base `  R )  /\  A. x  e.  (
Base `  R )
( ( e ( .r `  R ) x )  =  x  /\  ( x ( .r `  R ) e )  =  x ) ) )  =  .1.  ) )
687, 20, 67mpbi2and 912 . 2  |-  ( ph  ->  ( iota e ( e  e.  ( Base `  R )  /\  A. x  e.  ( Base `  R ) ( ( e ( .r `  R ) x )  =  x  /\  (
x ( .r `  R ) e )  =  x ) ) )  =  .1.  )
694, 68syl5req 2508 1  |-  ( ph  ->  .1.  =  ( 1r
`  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370    e. wcel 1758   E!weu 2262   A.wral 2799   iotacio 5490   ` cfv 5529  (class class class)co 6203   Basecbs 14296   .rcmulr 14362   1rcur 16735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-plusg 14374  df-0g 14503  df-mgp 16724  df-ur 16736
This theorem is referenced by:  ress1r  26425
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