MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rngoueqz Structured version   Unicode version

Theorem rngoueqz 25255
Description: Obsolete as of 23-Jan-2020. Use 0ring01eqbi 17791 instead. In a unital ring the zero equals the unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.) (New usage is discouraged.)
Hypotheses
Ref Expression
uznzr.1  |-  G  =  ( 1st `  R
)
uznzr.2  |-  H  =  ( 2nd `  R
)
uznzr.3  |-  Z  =  (GId `  G )
uznzr.4  |-  U  =  (GId `  H )
uznzr.5  |-  X  =  ran  G
Assertion
Ref Expression
rngoueqz  |-  ( R  e.  RingOps  ->  ( X  ~~  1o 
<->  U  =  Z ) )

Proof of Theorem rngoueqz
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 uznzr.1 . . . 4  |-  G  =  ( 1st `  R
)
2 uznzr.5 . . . 4  |-  X  =  ran  G
3 uznzr.3 . . . 4  |-  Z  =  (GId `  G )
41, 2, 3rngo0cl 25223 . . 3  |-  ( R  e.  RingOps  ->  Z  e.  X
)
5 en1eqsn 7761 . . . . . 6  |-  ( ( Z  e.  X  /\  X  ~~  1o )  ->  X  =  { Z } )
61rneqi 5235 . . . . . . . 8  |-  ran  G  =  ran  ( 1st `  R
)
7 uznzr.2 . . . . . . . 8  |-  H  =  ( 2nd `  R
)
8 uznzr.4 . . . . . . . 8  |-  U  =  (GId `  H )
96, 7, 8rngo1cl 25254 . . . . . . 7  |-  ( R  e.  RingOps  ->  U  e.  ran  G )
10 eleq2 2540 . . . . . . . . . 10  |-  ( X  =  { Z }  ->  ( U  e.  X  <->  U  e.  { Z }
) )
1110biimpd 207 . . . . . . . . 9  |-  ( X  =  { Z }  ->  ( U  e.  X  ->  U  e.  { Z } ) )
12 elsni 4058 . . . . . . . . 9  |-  ( U  e.  { Z }  ->  U  =  Z )
1311, 12syl6com 35 . . . . . . . 8  |-  ( U  e.  X  ->  ( X  =  { Z }  ->  U  =  Z ) )
142eqcomi 2480 . . . . . . . 8  |-  ran  G  =  X
1513, 14eleq2s 2575 . . . . . . 7  |-  ( U  e.  ran  G  -> 
( X  =  { Z }  ->  U  =  Z ) )
169, 15syl 16 . . . . . 6  |-  ( R  e.  RingOps  ->  ( X  =  { Z }  ->  U  =  Z ) )
175, 16syl5com 30 . . . . 5  |-  ( ( Z  e.  X  /\  X  ~~  1o )  -> 
( R  e.  RingOps  ->  U  =  Z )
)
1817ex 434 . . . 4  |-  ( Z  e.  X  ->  ( X  ~~  1o  ->  ( R  e.  RingOps  ->  U  =  Z ) ) )
1918com23 78 . . 3  |-  ( Z  e.  X  ->  ( R  e.  RingOps  ->  ( X  ~~  1o  ->  U  =  Z ) ) )
204, 19mpcom 36 . 2  |-  ( R  e.  RingOps  ->  ( X  ~~  1o  ->  U  =  Z ) )
211, 2rngone0 25241 . . 3  |-  ( R  e.  RingOps  ->  X  =/=  (/) )
22 oveq2 6303 . . . . . 6  |-  ( U  =  Z  ->  (
x H U )  =  ( x H Z ) )
2322ralrimivw 2882 . . . . 5  |-  ( U  =  Z  ->  A. x  e.  X  ( x H U )  =  ( x H Z ) )
243, 2, 1, 7rngorz 25227 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
x H Z )  =  Z )
2524ralrimiva 2881 . . . . . 6  |-  ( R  e.  RingOps  ->  A. x  e.  X  ( x H Z )  =  Z )
262, 6eqtri 2496 . . . . . . . . 9  |-  X  =  ran  ( 1st `  R
)
277, 26, 8rngoridm 25250 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
x H U )  =  x )
2827ralrimiva 2881 . . . . . . 7  |-  ( R  e.  RingOps  ->  A. x  e.  X  ( x H U )  =  x )
29 r19.26 2994 . . . . . . . . . 10  |-  ( A. x  e.  X  (
( x H U )  =  x  /\  ( x H U )  =  ( x H Z ) )  <-> 
( A. x  e.  X  ( x H U )  =  x  /\  A. x  e.  X  ( x H U )  =  ( x H Z ) ) )
30 r19.26 2994 . . . . . . . . . . . 12  |-  ( A. x  e.  X  (
( ( x H U )  =  x  /\  ( x H U )  =  ( x H Z ) )  /\  ( x H Z )  =  Z )  <->  ( A. x  e.  X  (
( x H U )  =  x  /\  ( x H U )  =  ( x H Z ) )  /\  A. x  e.  X  ( x H Z )  =  Z ) )
31 eqtr 2493 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  ( x H U )  /\  ( x H U )  =  ( x H Z ) )  ->  x  =  ( x H Z ) )
32 eqtr 2493 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  =  ( x H Z )  /\  ( x H Z )  =  Z )  ->  x  =  Z )
3332ex 434 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( x H Z )  ->  (
( x H Z )  =  Z  ->  x  =  Z )
)
3431, 33syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( x  =  ( x H U )  /\  ( x H U )  =  ( x H Z ) )  ->  ( ( x H Z )  =  Z  ->  x  =  Z ) )
3534ex 434 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( x H U )  ->  (
( x H U )  =  ( x H Z )  -> 
( ( x H Z )  =  Z  ->  x  =  Z ) ) )
3635eqcoms 2479 . . . . . . . . . . . . . . 15  |-  ( ( x H U )  =  x  ->  (
( x H U )  =  ( x H Z )  -> 
( ( x H Z )  =  Z  ->  x  =  Z ) ) )
3736imp31 432 . . . . . . . . . . . . . 14  |-  ( ( ( ( x H U )  =  x  /\  ( x H U )  =  ( x H Z ) )  /\  ( x H Z )  =  Z )  ->  x  =  Z )
3837ralimi 2860 . . . . . . . . . . . . 13  |-  ( A. x  e.  X  (
( ( x H U )  =  x  /\  ( x H U )  =  ( x H Z ) )  /\  ( x H Z )  =  Z )  ->  A. x  e.  X  x  =  Z )
39 eqsn 4194 . . . . . . . . . . . . . . 15  |-  ( X  =/=  (/)  ->  ( X  =  { Z }  <->  A. x  e.  X  x  =  Z ) )
40 ensn1g 7592 . . . . . . . . . . . . . . . . 17  |-  ( Z  e.  X  ->  { Z }  ~~  1o )
414, 40syl 16 . . . . . . . . . . . . . . . 16  |-  ( R  e.  RingOps  ->  { Z }  ~~  1o )
42 breq1 4456 . . . . . . . . . . . . . . . 16  |-  ( X  =  { Z }  ->  ( X  ~~  1o  <->  { Z }  ~~  1o ) )
4341, 42syl5ibr 221 . . . . . . . . . . . . . . 15  |-  ( X  =  { Z }  ->  ( R  e.  RingOps  ->  X  ~~  1o ) )
4439, 43syl6bir 229 . . . . . . . . . . . . . 14  |-  ( X  =/=  (/)  ->  ( A. x  e.  X  x  =  Z  ->  ( R  e.  RingOps  ->  X  ~~  1o ) ) )
4544com3l 81 . . . . . . . . . . . . 13  |-  ( A. x  e.  X  x  =  Z  ->  ( R  e.  RingOps  ->  ( X  =/=  (/)  ->  X  ~~  1o ) ) )
4638, 45syl 16 . . . . . . . . . . . 12  |-  ( A. x  e.  X  (
( ( x H U )  =  x  /\  ( x H U )  =  ( x H Z ) )  /\  ( x H Z )  =  Z )  ->  ( R  e.  RingOps  ->  ( X  =/=  (/)  ->  X  ~~  1o ) ) )
4730, 46sylbir 213 . . . . . . . . . . 11  |-  ( ( A. x  e.  X  ( ( x H U )  =  x  /\  ( x H U )  =  ( x H Z ) )  /\  A. x  e.  X  ( x H Z )  =  Z )  ->  ( R  e.  RingOps  ->  ( X  =/=  (/)  ->  X  ~~  1o ) ) )
4847ex 434 . . . . . . . . . 10  |-  ( A. x  e.  X  (
( x H U )  =  x  /\  ( x H U )  =  ( x H Z ) )  ->  ( A. x  e.  X  ( x H Z )  =  Z  ->  ( R  e.  RingOps 
->  ( X  =/=  (/)  ->  X  ~~  1o ) ) ) )
4929, 48sylbir 213 . . . . . . . . 9  |-  ( ( A. x  e.  X  ( x H U )  =  x  /\  A. x  e.  X  ( x H U )  =  ( x H Z ) )  -> 
( A. x  e.  X  ( x H Z )  =  Z  ->  ( R  e.  RingOps 
->  ( X  =/=  (/)  ->  X  ~~  1o ) ) ) )
5049ex 434 . . . . . . . 8  |-  ( A. x  e.  X  (
x H U )  =  x  ->  ( A. x  e.  X  ( x H U )  =  ( x H Z )  -> 
( A. x  e.  X  ( x H Z )  =  Z  ->  ( R  e.  RingOps 
->  ( X  =/=  (/)  ->  X  ~~  1o ) ) ) ) )
5150com24 87 . . . . . . 7  |-  ( A. x  e.  X  (
x H U )  =  x  ->  ( R  e.  RingOps  ->  ( A. x  e.  X  ( x H Z )  =  Z  -> 
( A. x  e.  X  ( x H U )  =  ( x H Z )  ->  ( X  =/=  (/)  ->  X  ~~  1o ) ) ) ) )
5228, 51mpcom 36 . . . . . 6  |-  ( R  e.  RingOps  ->  ( A. x  e.  X  ( x H Z )  =  Z  ->  ( A. x  e.  X  ( x H U )  =  ( x H Z )  ->  ( X  =/=  (/)  ->  X  ~~  1o ) ) ) )
5325, 52mpd 15 . . . . 5  |-  ( R  e.  RingOps  ->  ( A. x  e.  X  ( x H U )  =  ( x H Z )  ->  ( X  =/=  (/)  ->  X  ~~  1o ) ) )
5423, 53syl5com 30 . . . 4  |-  ( U  =  Z  ->  ( R  e.  RingOps  ->  ( X  =/=  (/)  ->  X  ~~  1o ) ) )
5554com13 80 . . 3  |-  ( X  =/=  (/)  ->  ( R  e.  RingOps  ->  ( U  =  Z  ->  X  ~~  1o ) ) )
5621, 55mpcom 36 . 2  |-  ( R  e.  RingOps  ->  ( U  =  Z  ->  X  ~~  1o ) )
5720, 56impbid 191 1  |-  ( R  e.  RingOps  ->  ( X  ~~  1o 
<->  U  =  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   (/)c0 3790   {csn 4033   class class class wbr 4453   ran crn 5006   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   1oc1o 7135    ~~ cen 7525  GIdcgi 25012   RingOpscrngo 25200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-om 6696  df-1st 6795  df-2nd 6796  df-1o 7142  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-grpo 25016  df-gid 25017  df-ablo 25107  df-ass 25138  df-exid 25140  df-mgmOLD 25144  df-sgrOLD 25156  df-mndo 25163  df-rngo 25201
This theorem is referenced by:  dvrunz  25258  isdmn3  30398
  Copyright terms: Public domain W3C validator