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Theorem rng1eq0 15702
Description: If one and zero are equal, then any two elements of a ring are equal. Alternatively, every ring has one distinct from zero except the zero ring containing the single element  { 0 }. (Contributed by Mario Carneiro, 10-Sep-2014.)
Hypotheses
Ref Expression
rng1eq0.b  |-  B  =  ( Base `  R
)
rng1eq0.u  |-  .1.  =  ( 1r `  R )
rng1eq0.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
rng1eq0  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (  .1.  =  .0.  ->  X  =  Y ) )

Proof of Theorem rng1eq0
StepHypRef Expression
1 simpr 448 . . . . 5  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  /\  .1.  =  .0.  )  ->  .1.  =  .0.  )
21oveq1d 6096 . . . 4  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  /\  .1.  =  .0.  )  ->  (  .1.  ( .r
`  R ) X )  =  (  .0.  ( .r `  R
) X ) )
31oveq1d 6096 . . . . 5  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  /\  .1.  =  .0.  )  ->  (  .1.  ( .r
`  R ) Y )  =  (  .0.  ( .r `  R
) Y ) )
4 simpl1 960 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  /\  .1.  =  .0.  )  ->  R  e.  Ring )
5 simpl2 961 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  /\  .1.  =  .0.  )  ->  X  e.  B )
6 rng1eq0.b . . . . . . . 8  |-  B  =  ( Base `  R
)
7 eqid 2436 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
8 rng1eq0.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
96, 7, 8rnglz 15700 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  ( .r `  R
) X )  =  .0.  )
104, 5, 9syl2anc 643 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  /\  .1.  =  .0.  )  ->  (  .0.  ( .r
`  R ) X )  =  .0.  )
11 simpl3 962 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  /\  .1.  =  .0.  )  ->  Y  e.  B )
126, 7, 8rnglz 15700 . . . . . . 7  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (  .0.  ( .r `  R
) Y )  =  .0.  )
134, 11, 12syl2anc 643 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  /\  .1.  =  .0.  )  ->  (  .0.  ( .r
`  R ) Y )  =  .0.  )
1410, 13eqtr4d 2471 . . . . 5  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  /\  .1.  =  .0.  )  ->  (  .0.  ( .r
`  R ) X )  =  (  .0.  ( .r `  R
) Y ) )
153, 14eqtr4d 2471 . . . 4  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  /\  .1.  =  .0.  )  ->  (  .1.  ( .r
`  R ) Y )  =  (  .0.  ( .r `  R
) X ) )
162, 15eqtr4d 2471 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  /\  .1.  =  .0.  )  ->  (  .1.  ( .r
`  R ) X )  =  (  .1.  ( .r `  R
) Y ) )
17 rng1eq0.u . . . . 5  |-  .1.  =  ( 1r `  R )
186, 7, 17rnglidm 15687 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .1.  ( .r `  R
) X )  =  X )
194, 5, 18syl2anc 643 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  /\  .1.  =  .0.  )  ->  (  .1.  ( .r
`  R ) X )  =  X )
206, 7, 17rnglidm 15687 . . . 4  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (  .1.  ( .r `  R
) Y )  =  Y )
214, 11, 20syl2anc 643 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  /\  .1.  =  .0.  )  ->  (  .1.  ( .r
`  R ) Y )  =  Y )
2216, 19, 213eqtr3d 2476 . 2  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  /\  .1.  =  .0.  )  ->  X  =  Y )
2322ex 424 1  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (  .1.  =  .0.  ->  X  =  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   .rcmulr 13530   0gc0g 13723   Ringcrg 15660   1rcur 15662
This theorem is referenced by:  abvneg  15922  isnzr2  16334  rngelnzr  16336  nrginvrcn  18727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-plusg 13542  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-mgp 15649  df-rng 15663  df-ur 15665
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