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Theorem zlidlring 40201
Description: The zero (left) ideal of a non-unital ring is a unital ring (the zero ring). (Contributed by AV, 16-Feb-2020.)
Hypotheses
Ref Expression
lidlabl.l  |-  L  =  (LIdeal `  R )
lidlabl.i  |-  I  =  ( Rs  U )
zlidlring.b  |-  B  =  ( Base `  R
)
zlidlring.0  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
zlidlring  |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  I  e.  Ring )

Proof of Theorem zlidlring
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 463 . . . . 5  |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  R  e.  Ring )
2 lidlabl.l . . . . . . . 8  |-  L  =  (LIdeal `  R )
3 zlidlring.0 . . . . . . . 8  |-  .0.  =  ( 0g `  R )
42, 3lidl0 18492 . . . . . . 7  |-  ( R  e.  Ring  ->  {  .0.  }  e.  L )
54adantr 471 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  {  .0.  }  e.  L )
6 eleq1 2528 . . . . . . 7  |-  ( U  =  {  .0.  }  ->  ( U  e.  L  <->  {  .0.  }  e.  L
) )
76adantl 472 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  ( U  e.  L  <->  {  .0.  }  e.  L ) )
85, 7mpbird 240 . . . . 5  |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  U  e.  L
)
91, 8jca 539 . . . 4  |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  ( R  e. 
Ring  /\  U  e.  L
) )
10 lidlabl.i . . . . 5  |-  I  =  ( Rs  U )
112, 10lidlrng 40200 . . . 4  |-  ( ( R  e.  Ring  /\  U  e.  L )  ->  I  e. Rng )
129, 11syl 17 . . 3  |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  I  e. Rng )
13 eleq1 2528 . . . . . . 7  |-  ( {  .0.  }  =  U  ->  ( {  .0.  }  e.  L  <->  U  e.  L ) )
1413eqcoms 2470 . . . . . 6  |-  ( U  =  {  .0.  }  ->  ( {  .0.  }  e.  L  <->  U  e.  L
) )
1514adantl 472 . . . . 5  |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  ( {  .0.  }  e.  L  <->  U  e.  L ) )
16 id 22 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  R  e. 
Ring )
17 eqid 2462 . . . . . . . . . . . . . 14  |-  ( Base `  R )  =  (
Base `  R )
1817, 3ring0cl 17851 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  .0.  e.  ( Base `  R )
)
1916, 18jca 539 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  ( R  e.  Ring  /\  .0.  e.  ( Base `  R )
) )
20 eqid 2462 . . . . . . . . . . . . . 14  |-  ( .r
`  R )  =  ( .r `  R
)
2117, 20, 3ringlz 17866 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  .0.  e.  ( Base `  R
) )  ->  (  .0.  ( .r `  R
)  .0.  )  =  .0.  )
2221, 21jca 539 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  .0.  e.  ( Base `  R
) )  ->  (
(  .0.  ( .r
`  R )  .0.  )  =  .0.  /\  (  .0.  ( .r `  R )  .0.  )  =  .0.  ) )
2319, 22syl 17 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  ( (  .0.  ( .r `  R )  .0.  )  =  .0.  /\  (  .0.  ( .r `  R
)  .0.  )  =  .0.  ) )
24 fvex 5898 . . . . . . . . . . . . . 14  |-  ( 0g
`  R )  e. 
_V
253, 24eqeltri 2536 . . . . . . . . . . . . 13  |-  .0.  e.  _V
2625a1i 11 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  .0.  e.  _V )
27 oveq2 6323 . . . . . . . . . . . . . . 15  |-  ( y  =  .0.  ->  (  .0.  ( .r `  R
) y )  =  (  .0.  ( .r
`  R )  .0.  ) )
28 id 22 . . . . . . . . . . . . . . 15  |-  ( y  =  .0.  ->  y  =  .0.  )
2927, 28eqeq12d 2477 . . . . . . . . . . . . . 14  |-  ( y  =  .0.  ->  (
(  .0.  ( .r
`  R ) y )  =  y  <->  (  .0.  ( .r `  R )  .0.  )  =  .0.  ) )
30 oveq1 6322 . . . . . . . . . . . . . . 15  |-  ( y  =  .0.  ->  (
y ( .r `  R )  .0.  )  =  (  .0.  ( .r `  R )  .0.  ) )
3130, 28eqeq12d 2477 . . . . . . . . . . . . . 14  |-  ( y  =  .0.  ->  (
( y ( .r
`  R )  .0.  )  =  y  <->  (  .0.  ( .r `  R )  .0.  )  =  .0.  ) )
3229, 31anbi12d 722 . . . . . . . . . . . . 13  |-  ( y  =  .0.  ->  (
( (  .0.  ( .r `  R ) y )  =  y  /\  ( y ( .r
`  R )  .0.  )  =  y )  <-> 
( (  .0.  ( .r `  R )  .0.  )  =  .0.  /\  (  .0.  ( .r `  R )  .0.  )  =  .0.  ) ) )
3332ralsng 4018 . . . . . . . . . . . 12  |-  (  .0. 
e.  _V  ->  ( A. y  e.  {  .0.  }  ( (  .0.  ( .r `  R ) y )  =  y  /\  ( y ( .r
`  R )  .0.  )  =  y )  <-> 
( (  .0.  ( .r `  R )  .0.  )  =  .0.  /\  (  .0.  ( .r `  R )  .0.  )  =  .0.  ) ) )
3426, 33syl 17 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  ( A. y  e.  {  .0.  }  ( (  .0.  ( .r `  R ) y )  =  y  /\  ( y ( .r
`  R )  .0.  )  =  y )  <-> 
( (  .0.  ( .r `  R )  .0.  )  =  .0.  /\  (  .0.  ( .r `  R )  .0.  )  =  .0.  ) ) )
3523, 34mpbird 240 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  A. y  e.  {  .0.  }  (
(  .0.  ( .r
`  R ) y )  =  y  /\  ( y ( .r
`  R )  .0.  )  =  y ) )
36 oveq1 6322 . . . . . . . . . . . . . . 15  |-  ( x  =  .0.  ->  (
x ( .r `  R ) y )  =  (  .0.  ( .r `  R ) y ) )
3736eqeq1d 2464 . . . . . . . . . . . . . 14  |-  ( x  =  .0.  ->  (
( x ( .r
`  R ) y )  =  y  <->  (  .0.  ( .r `  R ) y )  =  y ) )
38 oveq2 6323 . . . . . . . . . . . . . . 15  |-  ( x  =  .0.  ->  (
y ( .r `  R ) x )  =  ( y ( .r `  R )  .0.  ) )
3938eqeq1d 2464 . . . . . . . . . . . . . 14  |-  ( x  =  .0.  ->  (
( y ( .r
`  R ) x )  =  y  <->  ( y
( .r `  R
)  .0.  )  =  y ) )
4037, 39anbi12d 722 . . . . . . . . . . . . 13  |-  ( x  =  .0.  ->  (
( ( x ( .r `  R ) y )  =  y  /\  ( y ( .r `  R ) x )  =  y )  <->  ( (  .0.  ( .r `  R
) y )  =  y  /\  ( y ( .r `  R
)  .0.  )  =  y ) ) )
4140ralbidv 2839 . . . . . . . . . . . 12  |-  ( x  =  .0.  ->  ( A. y  e.  {  .0.  }  ( ( x ( .r `  R ) y )  =  y  /\  ( y ( .r `  R ) x )  =  y )  <->  A. y  e.  {  .0.  }  ( (  .0.  ( .r `  R
) y )  =  y  /\  ( y ( .r `  R
)  .0.  )  =  y ) ) )
4241rexsng 4019 . . . . . . . . . . 11  |-  (  .0. 
e.  _V  ->  ( E. x  e.  {  .0.  } A. y  e.  {  .0.  }  ( ( x ( .r `  R
) y )  =  y  /\  ( y ( .r `  R
) x )  =  y )  <->  A. y  e.  {  .0.  }  (
(  .0.  ( .r
`  R ) y )  =  y  /\  ( y ( .r
`  R )  .0.  )  =  y ) ) )
4326, 42syl 17 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  ( E. x  e.  {  .0.  } A. y  e.  {  .0.  }  ( ( x ( .r `  R
) y )  =  y  /\  ( y ( .r `  R
) x )  =  y )  <->  A. y  e.  {  .0.  }  (
(  .0.  ( .r
`  R ) y )  =  y  /\  ( y ( .r
`  R )  .0.  )  =  y ) ) )
4435, 43mpbird 240 . . . . . . . . 9  |-  ( R  e.  Ring  ->  E. x  e.  {  .0.  } A. y  e.  {  .0.  }  ( ( x ( .r `  R ) y )  =  y  /\  ( y ( .r `  R ) x )  =  y ) )
4544adantr 471 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  E. x  e.  {  .0.  } A. y  e. 
{  .0.  }  (
( x ( .r
`  R ) y )  =  y  /\  ( y ( .r
`  R ) x )  =  y ) )
4645adantr 471 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  U  =  {  .0.  } )  /\  U  e.  L )  ->  E. x  e.  {  .0.  } A. y  e.  {  .0.  }  ( ( x ( .r `  R ) y )  =  y  /\  ( y ( .r `  R ) x )  =  y ) )
47 simpr 467 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  U  =  {  .0.  } )  /\  U  e.  L )  ->  U  e.  L )
482, 10lidlbas 40196 . . . . . . . . . 10  |-  ( U  e.  L  ->  ( Base `  I )  =  U )
4947, 48syl 17 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  U  =  {  .0.  } )  /\  U  e.  L )  ->  ( Base `  I )  =  U )
50 simpr 467 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  U  =  {  .0.  } )
5150adantr 471 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  U  =  {  .0.  } )  /\  U  e.  L )  ->  U  =  {  .0.  } )
5249, 51eqtrd 2496 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  U  =  {  .0.  } )  /\  U  e.  L )  ->  ( Base `  I )  =  {  .0.  } )
5310, 20ressmulr 15299 . . . . . . . . . . . . . 14  |-  ( U  e.  L  ->  ( .r `  R )  =  ( .r `  I
) )
5453eqcomd 2468 . . . . . . . . . . . . 13  |-  ( U  e.  L  ->  ( .r `  I )  =  ( .r `  R
) )
5554adantl 472 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  U  =  {  .0.  } )  /\  U  e.  L )  ->  ( .r `  I )  =  ( .r `  R
) )
5655oveqd 6332 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  U  =  {  .0.  } )  /\  U  e.  L )  ->  (
x ( .r `  I ) y )  =  ( x ( .r `  R ) y ) )
5756eqeq1d 2464 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  U  =  {  .0.  } )  /\  U  e.  L )  ->  (
( x ( .r
`  I ) y )  =  y  <->  ( x
( .r `  R
) y )  =  y ) )
5855oveqd 6332 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  U  =  {  .0.  } )  /\  U  e.  L )  ->  (
y ( .r `  I ) x )  =  ( y ( .r `  R ) x ) )
5958eqeq1d 2464 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  U  =  {  .0.  } )  /\  U  e.  L )  ->  (
( y ( .r
`  I ) x )  =  y  <->  ( y
( .r `  R
) x )  =  y ) )
6057, 59anbi12d 722 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  U  =  {  .0.  } )  /\  U  e.  L )  ->  (
( ( x ( .r `  I ) y )  =  y  /\  ( y ( .r `  I ) x )  =  y )  <->  ( ( x ( .r `  R
) y )  =  y  /\  ( y ( .r `  R
) x )  =  y ) ) )
6152, 60raleqbidv 3013 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  U  =  {  .0.  } )  /\  U  e.  L )  ->  ( A. y  e.  ( Base `  I ) ( ( x ( .r
`  I ) y )  =  y  /\  ( y ( .r
`  I ) x )  =  y )  <->  A. y  e.  {  .0.  }  ( ( x ( .r `  R ) y )  =  y  /\  ( y ( .r `  R ) x )  =  y ) ) )
6252, 61rexeqbidv 3014 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  U  =  {  .0.  } )  /\  U  e.  L )  ->  ( E. x  e.  ( Base `  I ) A. y  e.  ( Base `  I ) ( ( x ( .r `  I ) y )  =  y  /\  (
y ( .r `  I ) x )  =  y )  <->  E. x  e.  {  .0.  } A. y  e.  {  .0.  }  ( ( x ( .r `  R ) y )  =  y  /\  ( y ( .r `  R ) x )  =  y ) ) )
6346, 62mpbird 240 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  U  =  {  .0.  } )  /\  U  e.  L )  ->  E. x  e.  ( Base `  I
) A. y  e.  ( Base `  I
) ( ( x ( .r `  I
) y )  =  y  /\  ( y ( .r `  I
) x )  =  y ) )
6463ex 440 . . . . 5  |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  ( U  e.  L  ->  E. x  e.  ( Base `  I
) A. y  e.  ( Base `  I
) ( ( x ( .r `  I
) y )  =  y  /\  ( y ( .r `  I
) x )  =  y ) ) )
6515, 64sylbid 223 . . . 4  |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  ( {  .0.  }  e.  L  ->  E. x  e.  ( Base `  I
) A. y  e.  ( Base `  I
) ( ( x ( .r `  I
) y )  =  y  /\  ( y ( .r `  I
) x )  =  y ) ) )
665, 65mpd 15 . . 3  |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  E. x  e.  (
Base `  I ) A. y  e.  ( Base `  I ) ( ( x ( .r
`  I ) y )  =  y  /\  ( y ( .r
`  I ) x )  =  y ) )
6712, 66jca 539 . 2  |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  ( I  e. Rng  /\  E. x  e.  (
Base `  I ) A. y  e.  ( Base `  I ) ( ( x ( .r
`  I ) y )  =  y  /\  ( y ( .r
`  I ) x )  =  y ) ) )
68 eqid 2462 . . 3  |-  ( Base `  I )  =  (
Base `  I )
69 eqid 2462 . . 3  |-  ( .r
`  I )  =  ( .r `  I
)
7068, 69isringrng 40154 . 2  |-  ( I  e.  Ring  <->  ( I  e. Rng  /\  E. x  e.  (
Base `  I ) A. y  e.  ( Base `  I ) ( ( x ( .r
`  I ) y )  =  y  /\  ( y ( .r
`  I ) x )  =  y ) ) )
7167, 70sylibr 217 1  |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  I  e.  Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   A.wral 2749   E.wrex 2750   _Vcvv 3057   {csn 3980   ` cfv 5601  (class class class)co 6315   Basecbs 15170   ↾s cress 15171   .rcmulr 15240   0gc0g 15387   Ringcrg 17829  LIdealclidl 18442  Rngcrng 40147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-2 10696  df-3 10697  df-4 10698  df-5 10699  df-6 10700  df-7 10701  df-8 10702  df-ndx 15173  df-slot 15174  df-base 15175  df-sets 15176  df-ress 15177  df-plusg 15252  df-mulr 15253  df-sca 15255  df-vsca 15256  df-ip 15257  df-0g 15389  df-mgm 16537  df-sgrp 16576  df-mnd 16586  df-grp 16722  df-minusg 16723  df-sbg 16724  df-subg 16863  df-cmn 17481  df-abl 17482  df-mgp 17773  df-ur 17785  df-ring 17831  df-subrg 18055  df-lmod 18142  df-lss 18205  df-sra 18444  df-rgmod 18445  df-lidl 18446  df-rng0 40148
This theorem is referenced by:  uzlidlring  40202
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