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Theorem rngoueqz 22049
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 22017 . . 3  |-  ( R  e.  RingOps  ->  Z  e.  X
)
5 en1eqsn 7367 . . . . . 6  |-  ( ( Z  e.  X  /\  X  ~~  1o )  ->  X  =  { Z } )
61rneqi 5125 . . . . . . . 8  |-  ran  G  =  ran  ( 1st `  R
)
7 uznzr.2 . . . . . . . 8  |-  H  =  ( 2nd `  R
)
8 uznzr.4 . . . . . . . 8  |-  U  =  (GId `  H )
96, 7, 8rngo1cl 22048 . . . . . . 7  |-  ( R  e.  RingOps  ->  U  e.  ran  G )
10 eleq2 2503 . . . . . . . . . 10  |-  ( X  =  { Z }  ->  ( U  e.  X  <->  U  e.  { Z }
) )
1110biimpd 200 . . . . . . . . 9  |-  ( X  =  { Z }  ->  ( U  e.  X  ->  U  e.  { Z } ) )
12 elsni 3862 . . . . . . . . 9  |-  ( U  e.  { Z }  ->  U  =  Z )
1311, 12syl6com 34 . . . . . . . 8  |-  ( U  e.  X  ->  ( X  =  { Z }  ->  U  =  Z ) )
142eqcomi 2446 . . . . . . . 8  |-  ran  G  =  X
1513, 14eleq2s 2534 . . . . . . 7  |-  ( U  e.  ran  G  -> 
( X  =  { Z }  ->  U  =  Z ) )
169, 15syl 16 . . . . . 6  |-  ( R  e.  RingOps  ->  ( X  =  { Z }  ->  U  =  Z ) )
175, 16syl5com 29 . . . . 5  |-  ( ( Z  e.  X  /\  X  ~~  1o )  -> 
( R  e.  RingOps  ->  U  =  Z )
)
1817ex 425 . . . 4  |-  ( Z  e.  X  ->  ( X  ~~  1o  ->  ( R  e.  RingOps  ->  U  =  Z ) ) )
1918com23 75 . . 3  |-  ( Z  e.  X  ->  ( R  e.  RingOps  ->  ( X  ~~  1o  ->  U  =  Z ) ) )
204, 19mpcom 35 . 2  |-  ( R  e.  RingOps  ->  ( X  ~~  1o  ->  U  =  Z ) )
211, 2rngon0 22035 . . 3  |-  ( R  e.  RingOps  ->  X  =/=  (/) )
22 oveq2 6118 . . . . . 6  |-  ( U  =  Z  ->  (
x H U )  =  ( x H Z ) )
2322ralrimivw 2796 . . . . 5  |-  ( U  =  Z  ->  A. x  e.  X  ( x H U )  =  ( x H Z ) )
243, 2, 1, 7rngorz 22021 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
x H Z )  =  Z )
2524ralrimiva 2795 . . . . . 6  |-  ( R  e.  RingOps  ->  A. x  e.  X  ( x H Z )  =  Z )
262, 6eqtri 2462 . . . . . . . . 9  |-  X  =  ran  ( 1st `  R
)
277, 26, 8rngoridm 22044 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
x H U )  =  x )
2827ralrimiva 2795 . . . . . . 7  |-  ( R  e.  RingOps  ->  A. x  e.  X  ( x H U )  =  x )
29 r19.26 2844 . . . . . . . . . 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 2844 . . . . . . . . . . . 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 2459 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  ( x H U )  /\  ( x H U )  =  ( x H Z ) )  ->  x  =  ( x H Z ) )
32 eqtr 2459 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  =  ( x H Z )  /\  ( x H Z )  =  Z )  ->  x  =  Z )
3332ex 425 . . . . . . . . . . . . . . . . . 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 425 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( x H U )  ->  (
( x H U )  =  ( x H Z )  -> 
( ( x H Z )  =  Z  ->  x  =  Z ) ) )
3635eqcoms 2445 . . . . . . . . . . . . . . 15  |-  ( ( x H U )  =  x  ->  (
( x H U )  =  ( x H Z )  -> 
( ( x H Z )  =  Z  ->  x  =  Z ) ) )
3736imp31 423 . . . . . . . . . . . . . 14  |-  ( ( ( ( x H U )  =  x  /\  ( x H U )  =  ( x H Z ) )  /\  ( x H Z )  =  Z )  ->  x  =  Z )
3837ralimi 2787 . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . . . 15  |-  ( X  =/=  (/)  ->  ( X  =  { Z }  <->  A. x  e.  X  x  =  Z ) )
40 ensn1g 7201 . . . . . . . . . . . . . . . . 17  |-  ( Z  e.  X  ->  { Z }  ~~  1o )
414, 40syl 16 . . . . . . . . . . . . . . . 16  |-  ( R  e.  RingOps  ->  { Z }  ~~  1o )
42 breq1 4240 . . . . . . . . . . . . . . . 16  |-  ( X  =  { Z }  ->  ( X  ~~  1o  <->  { Z }  ~~  1o ) )
4341, 42syl5ibr 214 . . . . . . . . . . . . . . 15  |-  ( X  =  { Z }  ->  ( R  e.  RingOps  ->  X  ~~  1o ) )
4439, 43syl6bir 222 . . . . . . . . . . . . . 14  |-  ( X  =/=  (/)  ->  ( A. x  e.  X  x  =  Z  ->  ( R  e.  RingOps  ->  X  ~~  1o ) ) )
4544com3l 78 . . . . . . . . . . . . 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 206 . . . . . . . . . . 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 425 . . . . . . . . . 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 206 . . . . . . . . 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 425 . . . . . . . 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 84 . . . . . . 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 35 . . . . . 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 29 . . . 4  |-  ( U  =  Z  ->  ( R  e.  RingOps  ->  ( X  =/=  (/)  ->  X  ~~  1o ) ) )
5554com13 77 . . 3  |-  ( X  =/=  (/)  ->  ( R  e.  RingOps  ->  ( U  =  Z  ->  X  ~~  1o ) ) )
5621, 55mpcom 35 . 2  |-  ( R  e.  RingOps  ->  ( U  =  Z  ->  X  ~~  1o ) )
5720, 56impbid 185 1  |-  ( R  e.  RingOps  ->  ( X  ~~  1o 
<->  U  =  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605   A.wral 2711   (/)c0 3613   {csn 3838   class class class wbr 4237   ran crn 4908   ` cfv 5483  (class class class)co 6110   1stc1st 6376   2ndc2nd 6377   1oc1o 6746    ~~ cen 7135  GIdcgi 21806   RingOpscrngo 21994
This theorem is referenced by:  dvrunz  22052  isdmn3  26722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-1st 6378  df-2nd 6379  df-riota 6578  df-1o 6753  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-grpo 21810  df-gid 21811  df-ablo 21901  df-ass 21932  df-exid 21934  df-mgm 21938  df-sgr 21950  df-mndo 21957  df-rngo 21995
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