Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngurd Structured version   Visualization version   Unicode version

Theorem rngurd 28551
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 2451 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2451 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
3 eqid 2451 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
41, 2, 3dfur2 17738 . 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 2531 . . 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 535 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  (
(  .1.  .x.  x
)  =  x  /\  ( x  .x.  .1.  )  =  x ) )
1110ralrimiva 2802 . . . 4  |-  ( ph  ->  A. x  e.  B  ( (  .1.  .x.  x )  =  x  /\  ( x  .x.  .1.  )  =  x
) )
12 rngurd.p . . . . . . . . 9  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
1312adantr 467 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  .x.  =  ( .r `  R ) )
1413oveqd 6307 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (  .1.  .x.  x )  =  (  .1.  ( .r
`  R ) x ) )
1514eqeq1d 2453 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (
(  .1.  .x.  x
)  =  x  <->  (  .1.  ( .r `  R ) x )  =  x ) )
1613oveqd 6307 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .x.  .1.  )  =  ( x ( .r `  R )  .1.  ) )
1716eqeq1d 2453 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (
( x  .x.  .1.  )  =  x  <->  ( x
( .r `  R
)  .1.  )  =  x ) )
1815, 17anbi12d 717 . . . . 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 3003 . . . 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 214 . . 3  |-  ( ph  ->  A. x  e.  (
Base `  R )
( (  .1.  ( .r `  R ) x )  =  x  /\  ( x ( .r
`  R )  .1.  )  =  x ) )
216eleq2d 2514 . . . . . . . 8  |-  ( ph  ->  ( e  e.  B  <->  e  e.  ( Base `  R
) ) )
2213oveqd 6307 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  B )  ->  (
e  .x.  x )  =  ( e ( .r `  R ) x ) )
2322eqeq1d 2453 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (
( e  .x.  x
)  =  x  <->  ( e
( .r `  R
) x )  =  x ) )
2413oveqd 6307 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .x.  e )  =  ( x ( .r `  R ) e ) )
2524eqeq1d 2453 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (
( x  .x.  e
)  =  x  <->  ( x
( .r `  R
) e )  =  x ) )
2623, 25anbi12d 717 . . . . . . . . 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 3003 . . . . . . . 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 717 . . . . . . 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 2802 . . . . . . . . . . . 12  |-  ( ph  ->  A. x  e.  B  (  .1.  .x.  x )  =  x )
3029adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  e  e.  B )  ->  A. x  e.  B  (  .1.  .x.  x )  =  x )
31 simpr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  e  e.  B )  ->  e  e.  B )
32 simpr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  B )  /\  x  =  e )  ->  x  =  e )
3332oveq2d 6306 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  e  e.  B )  /\  x  =  e )  -> 
(  .1.  .x.  x
)  =  (  .1. 
.x.  e ) )
3433, 32eqeq12d 2466 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  e  e.  B )  /\  x  =  e )  -> 
( (  .1.  .x.  x )  =  x  <-> 
(  .1.  .x.  e
)  =  e ) )
3531, 34rspcdv 3153 . . . . . . . . . . 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 723 . . . . . . . . 9  |-  ( (
ph  /\  ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) ) )  ->  (  .1.  .x.  e )  =  e )
385adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) ) )  ->  .1.  e.  B
)
39 simprr 766 . . . . . . . . . 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 6298 . . . . . . . . . . . . . 14  |-  ( x  =  .1.  ->  (
e  .x.  x )  =  ( e  .x.  .1.  ) )
41 id 22 . . . . . . . . . . . . . 14  |-  ( x  =  .1.  ->  x  =  .1.  )
4240, 41eqeq12d 2466 . . . . . . . . . . . . 13  |-  ( x  =  .1.  ->  (
( e  .x.  x
)  =  x  <->  ( e  .x.  .1.  )  =  .1.  ) )
43 oveq1 6297 . . . . . . . . . . . . . 14  |-  ( x  =  .1.  ->  (
x  .x.  e )  =  (  .1.  .x.  e
) )
4443, 41eqeq12d 2466 . . . . . . . . . . . . 13  |-  ( x  =  .1.  ->  (
( x  .x.  e
)  =  x  <->  (  .1.  .x.  e )  =  .1.  ) )
4542, 44anbi12d 717 . . . . . . . . . . . 12  |-  ( x  =  .1.  ->  (
( ( e  .x.  x )  =  x  /\  ( x  .x.  e )  =  x )  <->  ( ( e 
.x.  .1.  )  =  .1.  /\  (  .1.  .x.  e )  =  .1.  ) ) )
4645rspcva 3148 . . . . . . . . . . 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 465 . . . . . . . . . 10  |-  ( (  .1.  e.  B  /\  A. x  e.  B  ( ( e  .x.  x
)  =  x  /\  ( x  .x.  e )  =  x ) )  ->  (  .1.  .x.  e )  =  .1.  )
4838, 39, 47syl2anc 667 . . . . . . . . 9  |-  ( (
ph  /\  ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) ) )  ->  (  .1.  .x.  e )  =  .1.  )
4937, 48eqtr3d 2487 . . . . . . . 8  |-  ( (
ph  /\  ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) ) )  ->  e  =  .1.  )
5049ex 436 . . . . . . 7  |-  ( ph  ->  ( ( e  e.  B  /\  A. x  e.  B  ( (
e  .x.  x )  =  x  /\  (
x  .x.  e )  =  x ) )  -> 
e  =  .1.  )
)
5128, 50sylbird 239 . . . . . 6  |-  ( ph  ->  ( ( e  e.  ( Base `  R
)  /\  A. x  e.  ( Base `  R
) ( ( e ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
) e )  =  x ) )  -> 
e  =  .1.  )
)
5251alrimiv 1773 . . . . 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 2517 . . . . . . 7  |-  ( e  =  .1.  ->  (
e  e.  ( Base `  R )  <->  .1.  e.  ( Base `  R )
) )
54 oveq1 6297 . . . . . . . . . 10  |-  ( e  =  .1.  ->  (
e ( .r `  R ) x )  =  (  .1.  ( .r `  R ) x ) )
5554eqeq1d 2453 . . . . . . . . 9  |-  ( e  =  .1.  ->  (
( e ( .r
`  R ) x )  =  x  <->  (  .1.  ( .r `  R ) x )  =  x ) )
56 oveq2 6298 . . . . . . . . . 10  |-  ( e  =  .1.  ->  (
x ( .r `  R ) e )  =  ( x ( .r `  R )  .1.  ) )
5756eqeq1d 2453 . . . . . . . . 9  |-  ( e  =  .1.  ->  (
( x ( .r
`  R ) e )  =  x  <->  ( x
( .r `  R
)  .1.  )  =  x ) )
5855, 57anbi12d 717 . . . . . . . 8  |-  ( e  =  .1.  ->  (
( ( e ( .r `  R ) x )  =  x  /\  ( x ( .r `  R ) e )  =  x )  <->  ( (  .1.  ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
)  .1.  )  =  x ) ) )
5958ralbidv 2827 . . . . . . 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 717 . . . . . 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 3209 . . . . 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 1273 . . . 4  |-  ( ph  ->  E! e ( e  e.  ( Base `  R
)  /\  A. x  e.  ( Base `  R
) ( ( e ( .r `  R
) x )  =  x  /\  ( x ( .r `  R
) e )  =  x ) ) )
6360iota2 5572 . . . 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 667 . . 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 932 . 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 2498 1  |-  ( ph  ->  .1.  =  ( 1r
`  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1442    = wceq 1444    e. wcel 1887   E!weu 2299   A.wral 2737   iotacio 5544   ` cfv 5582  (class class class)co 6290   Basecbs 15121   .rcmulr 15191   1rcur 17735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-plusg 15203  df-0g 15340  df-mgp 17724  df-ur 17736
This theorem is referenced by:  ress1r  28552
  Copyright terms: Public domain W3C validator