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Theorem rngurd 28111
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 2402 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2402 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
3 eqid 2402 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
41, 2, 3dfur2 17368 . 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 2492 . . 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 530 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  (
(  .1.  .x.  x
)  =  x  /\  ( x  .x.  .1.  )  =  x ) )
1110ralrimiva 2817 . . . 4  |-  ( ph  ->  A. x  e.  B  ( (  .1.  .x.  x )  =  x  /\  ( x  .x.  .1.  )  =  x
) )
12 rngurd.p . . . . . . . . 9  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
1312adantr 463 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  .x.  =  ( .r `  R ) )
1413oveqd 6251 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (  .1.  .x.  x )  =  (  .1.  ( .r
`  R ) x ) )
1514eqeq1d 2404 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (
(  .1.  .x.  x
)  =  x  <->  (  .1.  ( .r `  R ) x )  =  x ) )
1613oveqd 6251 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .x.  .1.  )  =  ( x ( .r `  R )  .1.  ) )
1716eqeq1d 2404 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (
( x  .x.  .1.  )  =  x  <->  ( x
( .r `  R
)  .1.  )  =  x ) )
1815, 17anbi12d 709 . . . . 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 3019 . . . 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 ) )
216eleq2d 2472 . . . . . . . 8  |-  ( ph  ->  ( e  e.  B  <->  e  e.  ( Base `  R
) ) )
2213oveqd 6251 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  B )  ->  (
e  .x.  x )  =  ( e ( .r `  R ) x ) )
2322eqeq1d 2404 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (
( e  .x.  x
)  =  x  <->  ( e
( .r `  R
) x )  =  x ) )
2413oveqd 6251 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .x.  e )  =  ( x ( .r `  R ) e ) )
2524eqeq1d 2404 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (
( x  .x.  e
)  =  x  <->  ( x
( .r `  R
) e )  =  x ) )
2623, 25anbi12d 709 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  (
( ( e  .x.  x )  =  x  /\  ( x  .x.  e )  =  x )  <->  ( ( e ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
) e )  =  x ) ) )
276, 26raleqbidva 3019 . . . . . . . 8  |-  ( 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 ) ) )
2821, 27anbi12d 709 . . . . . . 7  |-  ( 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 ) ) ) )
298ralrimiva 2817 . . . . . . . . . . . 12  |-  ( ph  ->  A. x  e.  B  (  .1.  .x.  x )  =  x )
3029adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  e  e.  B )  ->  A. x  e.  B  (  .1.  .x.  x )  =  x )
31 simpr 459 . . . . . . . . . . . 12  |-  ( (
ph  /\  e  e.  B )  ->  e  e.  B )
32 simpr 459 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  B )  /\  x  =  e )  ->  x  =  e )
3332oveq2d 6250 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  e  e.  B )  /\  x  =  e )  -> 
(  .1.  .x.  x
)  =  (  .1. 
.x.  e ) )
3433, 32eqeq12d 2424 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  e  e.  B )  /\  x  =  e )  -> 
( (  .1.  .x.  x )  =  x  <-> 
(  .1.  .x.  e
)  =  e ) )
3531, 34rspcdv 3162 . . . . . . . . . . 11  |-  ( (
ph  /\  e  e.  B )  ->  ( A. x  e.  B  (  .1.  .x.  x )  =  x  ->  (  .1. 
.x.  e )  =  e ) )
3630, 35mpd 15 . . . . . . . . . 10  |-  ( (
ph  /\  e  e.  B )  ->  (  .1.  .x.  e )  =  e )
3736adantrr 715 . . . . . . . . 9  |-  ( (
ph  /\  ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) ) )  ->  (  .1.  .x.  e )  =  e )
385adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) ) )  ->  .1.  e.  B
)
39 simprr 758 . . . . . . . . . 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 ) )
40 oveq2 6242 . . . . . . . . . . . . . 14  |-  ( x  =  .1.  ->  (
e  .x.  x )  =  ( e  .x.  .1.  ) )
41 id 22 . . . . . . . . . . . . . 14  |-  ( x  =  .1.  ->  x  =  .1.  )
4240, 41eqeq12d 2424 . . . . . . . . . . . . 13  |-  ( x  =  .1.  ->  (
( e  .x.  x
)  =  x  <->  ( e  .x.  .1.  )  =  .1.  ) )
43 oveq1 6241 . . . . . . . . . . . . . 14  |-  ( x  =  .1.  ->  (
x  .x.  e )  =  (  .1.  .x.  e
) )
4443, 41eqeq12d 2424 . . . . . . . . . . . . 13  |-  ( x  =  .1.  ->  (
( x  .x.  e
)  =  x  <->  (  .1.  .x.  e )  =  .1.  ) )
4542, 44anbi12d 709 . . . . . . . . . . . 12  |-  ( x  =  .1.  ->  (
( ( e  .x.  x )  =  x  /\  ( x  .x.  e )  =  x )  <->  ( ( e 
.x.  .1.  )  =  .1.  /\  (  .1.  .x.  e )  =  .1.  ) ) )
4645rspcva 3157 . . . . . . . . . . 11  |-  ( (  .1.  e.  B  /\  A. x  e.  B  ( ( e  .x.  x
)  =  x  /\  ( x  .x.  e )  =  x ) )  ->  ( ( e 
.x.  .1.  )  =  .1.  /\  (  .1.  .x.  e )  =  .1.  ) )
4746simprd 461 . . . . . . . . . 10  |-  ( (  .1.  e.  B  /\  A. x  e.  B  ( ( e  .x.  x
)  =  x  /\  ( x  .x.  e )  =  x ) )  ->  (  .1.  .x.  e )  =  .1.  )
4838, 39, 47syl2anc 659 . . . . . . . . 9  |-  ( (
ph  /\  ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) ) )  ->  (  .1.  .x.  e )  =  .1.  )
4937, 48eqtr3d 2445 . . . . . . . 8  |-  ( (
ph  /\  ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) ) )  ->  e  =  .1.  )
5049ex 432 . . . . . . 7  |-  ( ph  ->  ( ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) )  -> 
e  =  .1.  )
)
5128, 50sylbird 235 . . . . . 6  |-  ( ph  ->  ( ( e  e.  ( Base `  R
)  /\  A. x  e.  ( Base `  R
) ( ( e ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
) e )  =  x ) )  -> 
e  =  .1.  )
)
5251alrimiv 1740 . . . . 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.  ) )
53 eleq1 2474 . . . . . . 7  |-  ( e  =  .1.  ->  (
e  e.  ( Base `  R )  <->  .1.  e.  ( Base `  R )
) )
54 oveq1 6241 . . . . . . . . . 10  |-  ( e  =  .1.  ->  (
e ( .r `  R ) x )  =  (  .1.  ( .r `  R ) x ) )
5554eqeq1d 2404 . . . . . . . . 9  |-  ( e  =  .1.  ->  (
( e ( .r
`  R ) x )  =  x  <->  (  .1.  ( .r `  R ) x )  =  x ) )
56 oveq2 6242 . . . . . . . . . 10  |-  ( e  =  .1.  ->  (
x ( .r `  R ) e )  =  ( x ( .r `  R )  .1.  ) )
5756eqeq1d 2404 . . . . . . . . 9  |-  ( e  =  .1.  ->  (
( x ( .r
`  R ) e )  =  x  <->  ( x
( .r `  R
)  .1.  )  =  x ) )
5855, 57anbi12d 709 . . . . . . . 8  |-  ( e  =  .1.  ->  (
( ( e ( .r `  R ) x )  =  x  /\  ( x ( .r `  R ) e )  =  x )  <->  ( (  .1.  ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
)  .1.  )  =  x ) ) )
5958ralbidv 2842 . . . . . . 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 ) ) )
6053, 59anbi12d 709 . . . . . 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 ) ) ) )
6160eqeu 3219 . . . . 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 ) ) )
627, 7, 20, 52, 61syl121anc 1235 . . . 4  |-  ( ph  ->  E! e ( e  e.  ( Base `  R
)  /\  A. x  e.  ( Base `  R
) ( ( e ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
) e )  =  x ) ) )
6360iota2 5515 . . . 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.  ) )
645, 62, 63syl2anc 659 . . 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.  ) )
657, 20, 64mpbi2and 922 . 2  |-  ( ph  ->  ( iota e ( e  e.  ( Base `  R )  /\  A. x  e.  ( Base `  R ) ( ( e ( .r `  R ) x )  =  x  /\  (
x ( .r `  R ) e )  =  x ) ) )  =  .1.  )
664, 65syl5req 2456 1  |-  ( ph  ->  .1.  =  ( 1r
`  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1403    = wceq 1405    e. wcel 1842   E!weu 2238   A.wral 2753   iotacio 5487   ` cfv 5525  (class class class)co 6234   Basecbs 14733   .rcmulr 14802   1rcur 17365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-recs 6999  df-rdg 7033  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-ndx 14736  df-slot 14737  df-base 14738  df-sets 14739  df-plusg 14814  df-0g 14948  df-mgp 17354  df-ur 17366
This theorem is referenced by:  ress1r  28112
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