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Theorem rngoueqz 21971
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 21939 . . 3  |-  ( R  e.  RingOps  ->  Z  e.  X
)
5 en1eqsn 7297 . . . . . 6  |-  ( ( Z  e.  X  /\  X  ~~  1o )  ->  X  =  { Z } )
61rneqi 5055 . . . . . . . 8  |-  ran  G  =  ran  ( 1st `  R
)
7 uznzr.2 . . . . . . . 8  |-  H  =  ( 2nd `  R
)
8 uznzr.4 . . . . . . . 8  |-  U  =  (GId `  H )
96, 7, 8rngo1cl 21970 . . . . . . 7  |-  ( R  e.  RingOps  ->  U  e.  ran  G )
10 eleq2 2465 . . . . . . . . . 10  |-  ( X  =  { Z }  ->  ( U  e.  X  <->  U  e.  { Z }
) )
1110biimpd 199 . . . . . . . . 9  |-  ( X  =  { Z }  ->  ( U  e.  X  ->  U  e.  { Z } ) )
12 elsni 3798 . . . . . . . . 9  |-  ( U  e.  { Z }  ->  U  =  Z )
1311, 12syl6com 33 . . . . . . . 8  |-  ( U  e.  X  ->  ( X  =  { Z }  ->  U  =  Z ) )
142eqcomi 2408 . . . . . . . 8  |-  ran  G  =  X
1513, 14eleq2s 2496 . . . . . . 7  |-  ( U  e.  ran  G  -> 
( X  =  { Z }  ->  U  =  Z ) )
169, 15syl 16 . . . . . 6  |-  ( R  e.  RingOps  ->  ( X  =  { Z }  ->  U  =  Z ) )
175, 16syl5com 28 . . . . 5  |-  ( ( Z  e.  X  /\  X  ~~  1o )  -> 
( R  e.  RingOps  ->  U  =  Z )
)
1817ex 424 . . . 4  |-  ( Z  e.  X  ->  ( X  ~~  1o  ->  ( R  e.  RingOps  ->  U  =  Z ) ) )
1918com23 74 . . 3  |-  ( Z  e.  X  ->  ( R  e.  RingOps  ->  ( X  ~~  1o  ->  U  =  Z ) ) )
204, 19mpcom 34 . 2  |-  ( R  e.  RingOps  ->  ( X  ~~  1o  ->  U  =  Z ) )
211, 2rngon0 21957 . . 3  |-  ( R  e.  RingOps  ->  X  =/=  (/) )
22 oveq2 6048 . . . . . 6  |-  ( U  =  Z  ->  (
x H U )  =  ( x H Z ) )
2322ralrimivw 2750 . . . . 5  |-  ( U  =  Z  ->  A. x  e.  X  ( x H U )  =  ( x H Z ) )
243, 2, 1, 7rngorz 21943 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
x H Z )  =  Z )
2524ralrimiva 2749 . . . . . 6  |-  ( R  e.  RingOps  ->  A. x  e.  X  ( x H Z )  =  Z )
262, 6eqtri 2424 . . . . . . . . 9  |-  X  =  ran  ( 1st `  R
)
277, 26, 8rngoridm 21966 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
x H U )  =  x )
2827ralrimiva 2749 . . . . . . 7  |-  ( R  e.  RingOps  ->  A. x  e.  X  ( x H U )  =  x )
29 r19.26 2798 . . . . . . . . . 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 2798 . . . . . . . . . . . 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 2421 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  ( x H U )  /\  ( x H U )  =  ( x H Z ) )  ->  x  =  ( x H Z ) )
32 eqtr 2421 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  =  ( x H Z )  /\  ( x H Z )  =  Z )  ->  x  =  Z )
3332ex 424 . . . . . . . . . . . . . . . . . 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 424 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( x H U )  ->  (
( x H U )  =  ( x H Z )  -> 
( ( x H Z )  =  Z  ->  x  =  Z ) ) )
3635eqcoms 2407 . . . . . . . . . . . . . . 15  |-  ( ( x H U )  =  x  ->  (
( x H U )  =  ( x H Z )  -> 
( ( x H Z )  =  Z  ->  x  =  Z ) ) )
3736imp31 422 . . . . . . . . . . . . . 14  |-  ( ( ( ( x H U )  =  x  /\  ( x H U )  =  ( x H Z ) )  /\  ( x H Z )  =  Z )  ->  x  =  Z )
3837ralimi 2741 . . . . . . . . . . . . 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 3920 . . . . . . . . . . . . . . 15  |-  ( X  =/=  (/)  ->  ( X  =  { Z }  <->  A. x  e.  X  x  =  Z ) )
40 ensn1g 7131 . . . . . . . . . . . . . . . . 17  |-  ( Z  e.  X  ->  { Z }  ~~  1o )
414, 40syl 16 . . . . . . . . . . . . . . . 16  |-  ( R  e.  RingOps  ->  { Z }  ~~  1o )
42 breq1 4175 . . . . . . . . . . . . . . . 16  |-  ( X  =  { Z }  ->  ( X  ~~  1o  <->  { Z }  ~~  1o ) )
4341, 42syl5ibr 213 . . . . . . . . . . . . . . 15  |-  ( X  =  { Z }  ->  ( R  e.  RingOps  ->  X  ~~  1o ) )
4439, 43syl6bir 221 . . . . . . . . . . . . . 14  |-  ( X  =/=  (/)  ->  ( A. x  e.  X  x  =  Z  ->  ( R  e.  RingOps  ->  X  ~~  1o ) ) )
4544com3l 77 . . . . . . . . . . . . 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 205 . . . . . . . . . . 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 424 . . . . . . . . . 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 205 . . . . . . . . 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 424 . . . . . . . 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 83 . . . . . . 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 34 . . . . . 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 28 . . . 4  |-  ( U  =  Z  ->  ( R  e.  RingOps  ->  ( X  =/=  (/)  ->  X  ~~  1o ) ) )
5554com13 76 . . 3  |-  ( X  =/=  (/)  ->  ( R  e.  RingOps  ->  ( U  =  Z  ->  X  ~~  1o ) ) )
5621, 55mpcom 34 . 2  |-  ( R  e.  RingOps  ->  ( U  =  Z  ->  X  ~~  1o ) )
5720, 56impbid 184 1  |-  ( R  e.  RingOps  ->  ( X  ~~  1o 
<->  U  =  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   (/)c0 3588   {csn 3774   class class class wbr 4172   ran crn 4838   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   1oc1o 6676    ~~ cen 7065  GIdcgi 21728   RingOpscrngo 21916
This theorem is referenced by:  dvrunz  21974  isdmn3  26574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-1st 6308  df-2nd 6309  df-riota 6508  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-grpo 21732  df-gid 21733  df-ablo 21823  df-ass 21854  df-exid 21856  df-mgm 21860  df-sgr 21872  df-mndo 21879  df-rngo 21917
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