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Theorem rngoueqz 21097
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 21065 . . 3  |-  ( R  e.  RingOps  ->  Z  e.  X
)
5 en1eqsn 7088 . . . . . 6  |-  ( ( Z  e.  X  /\  X  ~~  1o )  ->  X  =  { Z } )
61rneqi 4905 . . . . . . . 8  |-  ran  G  =  ran  ( 1st `  R
)
7 uznzr.2 . . . . . . . 8  |-  H  =  ( 2nd `  R
)
8 uznzr.4 . . . . . . . 8  |-  U  =  (GId `  H )
96, 7, 8rngo1cl 21096 . . . . . . 7  |-  ( R  e.  RingOps  ->  U  e.  ran  G )
10 eleq2 2344 . . . . . . . . . 10  |-  ( X  =  { Z }  ->  ( U  e.  X  <->  U  e.  { Z }
) )
1110biimpd 198 . . . . . . . . 9  |-  ( X  =  { Z }  ->  ( U  e.  X  ->  U  e.  { Z } ) )
12 elsni 3664 . . . . . . . . 9  |-  ( U  e.  { Z }  ->  U  =  Z )
1311, 12syl6com 31 . . . . . . . 8  |-  ( U  e.  X  ->  ( X  =  { Z }  ->  U  =  Z ) )
142eqcomi 2287 . . . . . . . 8  |-  ran  G  =  X
1513, 14eleq2s 2375 . . . . . . 7  |-  ( U  e.  ran  G  -> 
( X  =  { Z }  ->  U  =  Z ) )
169, 15syl 15 . . . . . 6  |-  ( R  e.  RingOps  ->  ( X  =  { Z }  ->  U  =  Z ) )
175, 16syl5com 26 . . . . 5  |-  ( ( Z  e.  X  /\  X  ~~  1o )  -> 
( R  e.  RingOps  ->  U  =  Z )
)
1817ex 423 . . . 4  |-  ( Z  e.  X  ->  ( X  ~~  1o  ->  ( R  e.  RingOps  ->  U  =  Z ) ) )
1918com23 72 . . 3  |-  ( Z  e.  X  ->  ( R  e.  RingOps  ->  ( X  ~~  1o  ->  U  =  Z ) ) )
204, 19mpcom 32 . 2  |-  ( R  e.  RingOps  ->  ( X  ~~  1o  ->  U  =  Z ) )
211, 2rngon0 21083 . . 3  |-  ( R  e.  RingOps  ->  X  =/=  (/) )
22 oveq2 5866 . . . . . 6  |-  ( U  =  Z  ->  (
x H U )  =  ( x H Z ) )
2322ralrimivw 2627 . . . . 5  |-  ( U  =  Z  ->  A. x  e.  X  ( x H U )  =  ( x H Z ) )
243, 2, 1, 7rngorz 21069 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
x H Z )  =  Z )
2524ralrimiva 2626 . . . . . 6  |-  ( R  e.  RingOps  ->  A. x  e.  X  ( x H Z )  =  Z )
262, 6eqtri 2303 . . . . . . . . 9  |-  X  =  ran  ( 1st `  R
)
277, 26, 8rngoridm 21092 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
x H U )  =  x )
2827ralrimiva 2626 . . . . . . 7  |-  ( R  e.  RingOps  ->  A. x  e.  X  ( x H U )  =  x )
29 r19.26 2675 . . . . . . . . . 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 2675 . . . . . . . . . . . 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 2300 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  ( x H U )  /\  ( x H U )  =  ( x H Z ) )  ->  x  =  ( x H Z ) )
32 eqtr 2300 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  =  ( x H Z )  /\  ( x H Z )  =  Z )  ->  x  =  Z )
3332ex 423 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( x H Z )  ->  (
( x H Z )  =  Z  ->  x  =  Z )
)
3431, 33syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( x  =  ( x H U )  /\  ( x H U )  =  ( x H Z ) )  ->  ( ( x H Z )  =  Z  ->  x  =  Z ) )
3534ex 423 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( x H U )  ->  (
( x H U )  =  ( x H Z )  -> 
( ( x H Z )  =  Z  ->  x  =  Z ) ) )
3635eqcoms 2286 . . . . . . . . . . . . . . 15  |-  ( ( x H U )  =  x  ->  (
( x H U )  =  ( x H Z )  -> 
( ( x H Z )  =  Z  ->  x  =  Z ) ) )
3736imp31 421 . . . . . . . . . . . . . 14  |-  ( ( ( ( x H U )  =  x  /\  ( x H U )  =  ( x H Z ) )  /\  ( x H Z )  =  Z )  ->  x  =  Z )
3837ralimi 2618 . . . . . . . . . . . . 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 3775 . . . . . . . . . . . . . . 15  |-  ( X  =/=  (/)  ->  ( X  =  { Z }  <->  A. x  e.  X  x  =  Z ) )
40 ensn1g 6926 . . . . . . . . . . . . . . . . 17  |-  ( Z  e.  X  ->  { Z }  ~~  1o )
414, 40syl 15 . . . . . . . . . . . . . . . 16  |-  ( R  e.  RingOps  ->  { Z }  ~~  1o )
42 breq1 4026 . . . . . . . . . . . . . . . 16  |-  ( X  =  { Z }  ->  ( X  ~~  1o  <->  { Z }  ~~  1o ) )
4341, 42syl5ibr 212 . . . . . . . . . . . . . . 15  |-  ( X  =  { Z }  ->  ( R  e.  RingOps  ->  X  ~~  1o ) )
4439, 43syl6bir 220 . . . . . . . . . . . . . 14  |-  ( X  =/=  (/)  ->  ( A. x  e.  X  x  =  Z  ->  ( R  e.  RingOps  ->  X  ~~  1o ) ) )
4544com3l 75 . . . . . . . . . . . . 13  |-  ( A. x  e.  X  x  =  Z  ->  ( R  e.  RingOps  ->  ( X  =/=  (/)  ->  X  ~~  1o ) ) )
4638, 45syl 15 . . . . . . . . . . . 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 204 . . . . . . . . . . 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 423 . . . . . . . . . 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 204 . . . . . . . . 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 423 . . . . . . . 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 81 . . . . . . 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 32 . . . . . 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 14 . . . . 5  |-  ( R  e.  RingOps  ->  ( A. x  e.  X  ( x H U )  =  ( x H Z )  ->  ( X  =/=  (/)  ->  X  ~~  1o ) ) )
5423, 53syl5com 26 . . . 4  |-  ( U  =  Z  ->  ( R  e.  RingOps  ->  ( X  =/=  (/)  ->  X  ~~  1o ) ) )
5554com13 74 . . 3  |-  ( X  =/=  (/)  ->  ( R  e.  RingOps  ->  ( U  =  Z  ->  X  ~~  1o ) ) )
5621, 55mpcom 32 . 2  |-  ( R  e.  RingOps  ->  ( U  =  Z  ->  X  ~~  1o ) )
5720, 56impbid 183 1  |-  ( R  e.  RingOps  ->  ( X  ~~  1o 
<->  U  =  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   (/)c0 3455   {csn 3640   class class class wbr 4023   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   1oc1o 6472    ~~ cen 6860  GIdcgi 20854   RingOpscrngo 21042
This theorem is referenced by:  dvrunz  21100  isdmn3  26699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-grpo 20858  df-gid 20859  df-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043
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