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Theorem rngoueqz 23917
Description: 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 23885 . . 3  |-  ( R  e.  RingOps  ->  Z  e.  X
)
5 en1eqsn 7542 . . . . . 6  |-  ( ( Z  e.  X  /\  X  ~~  1o )  ->  X  =  { Z } )
61rneqi 5066 . . . . . . . 8  |-  ran  G  =  ran  ( 1st `  R
)
7 uznzr.2 . . . . . . . 8  |-  H  =  ( 2nd `  R
)
8 uznzr.4 . . . . . . . 8  |-  U  =  (GId `  H )
96, 7, 8rngo1cl 23916 . . . . . . 7  |-  ( R  e.  RingOps  ->  U  e.  ran  G )
10 eleq2 2504 . . . . . . . . . 10  |-  ( X  =  { Z }  ->  ( U  e.  X  <->  U  e.  { Z }
) )
1110biimpd 207 . . . . . . . . 9  |-  ( X  =  { Z }  ->  ( U  e.  X  ->  U  e.  { Z } ) )
12 elsni 3902 . . . . . . . . 9  |-  ( U  e.  { Z }  ->  U  =  Z )
1311, 12syl6com 35 . . . . . . . 8  |-  ( U  e.  X  ->  ( X  =  { Z }  ->  U  =  Z ) )
142eqcomi 2447 . . . . . . . 8  |-  ran  G  =  X
1513, 14eleq2s 2535 . . . . . . 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, 2rngon0 23903 . . 3  |-  ( R  e.  RingOps  ->  X  =/=  (/) )
22 oveq2 6099 . . . . . 6  |-  ( U  =  Z  ->  (
x H U )  =  ( x H Z ) )
2322ralrimivw 2800 . . . . 5  |-  ( U  =  Z  ->  A. x  e.  X  ( x H U )  =  ( x H Z ) )
243, 2, 1, 7rngorz 23889 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
x H Z )  =  Z )
2524ralrimiva 2799 . . . . . 6  |-  ( R  e.  RingOps  ->  A. x  e.  X  ( x H Z )  =  Z )
262, 6eqtri 2463 . . . . . . . . 9  |-  X  =  ran  ( 1st `  R
)
277, 26, 8rngoridm 23912 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
x H U )  =  x )
2827ralrimiva 2799 . . . . . . 7  |-  ( R  e.  RingOps  ->  A. x  e.  X  ( x H U )  =  x )
29 r19.26 2849 . . . . . . . . . 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 2849 . . . . . . . . . . . 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 2460 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  ( x H U )  /\  ( x H U )  =  ( x H Z ) )  ->  x  =  ( x H Z ) )
32 eqtr 2460 . . . . . . . . . . . . . . . . . . 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 2446 . . . . . . . . . . . . . . 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 2791 . . . . . . . . . . . . 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 4034 . . . . . . . . . . . . . . 15  |-  ( X  =/=  (/)  ->  ( X  =  { Z }  <->  A. x  e.  X  x  =  Z ) )
40 ensn1g 7374 . . . . . . . . . . . . . . . . 17  |-  ( Z  e.  X  ->  { Z }  ~~  1o )
414, 40syl 16 . . . . . . . . . . . . . . . 16  |-  ( R  e.  RingOps  ->  { Z }  ~~  1o )
42 breq1 4295 . . . . . . . . . . . . . . . 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 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   (/)c0 3637   {csn 3877   class class class wbr 4292   ran crn 4841   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576   1oc1o 6913    ~~ cen 7307  GIdcgi 23674   RingOpscrngo 23862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-om 6477  df-1st 6577  df-2nd 6578  df-1o 6920  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-grpo 23678  df-gid 23679  df-ablo 23769  df-ass 23800  df-exid 23802  df-mgm 23806  df-sgr 23818  df-mndo 23825  df-rngo 23863
This theorem is referenced by:  dvrunz  23920  isdmn3  28874
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