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Theorem ringinvnzdiv 13040
Description: In a unital ring, a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.) (Revised by Jeff Madsen, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)
Hypotheses
Ref Expression
ringinvnzdiv.b  |-  B  =  ( Base `  R
)
ringinvnzdiv.t  |-  .x.  =  ( .r `  R )
ringinvnzdiv.u  |-  .1.  =  ( 1r `  R )
ringinvnzdiv.z  |-  .0.  =  ( 0g `  R )
ringinvnzdiv.r  |-  ( ph  ->  R  e.  Ring )
ringinvnzdiv.x  |-  ( ph  ->  X  e.  B )
ringinvnzdiv.a  |-  ( ph  ->  E. a  e.  B  ( a  .x.  X
)  =  .1.  )
ringinvnzdiv.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
ringinvnzdiv  |-  ( ph  ->  ( ( X  .x.  Y )  =  .0.  <->  Y  =  .0.  ) )
Distinct variable groups:    X, a    .0. , a    .1. , a    .x. , a    ph, a    Y, a
Allowed substitution hints:    B( a)    R( a)

Proof of Theorem ringinvnzdiv
StepHypRef Expression
1 ringinvnzdiv.a . . 3  |-  ( ph  ->  E. a  e.  B  ( a  .x.  X
)  =  .1.  )
2 ringinvnzdiv.r . . . . . . . . 9  |-  ( ph  ->  R  e.  Ring )
3 ringinvnzdiv.y . . . . . . . . 9  |-  ( ph  ->  Y  e.  B )
4 ringinvnzdiv.b . . . . . . . . . 10  |-  B  =  ( Base `  R
)
5 ringinvnzdiv.t . . . . . . . . . 10  |-  .x.  =  ( .r `  R )
6 ringinvnzdiv.u . . . . . . . . . 10  |-  .1.  =  ( 1r `  R )
74, 5, 6ringlidm 13019 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  Y  e.  B )  ->  (  .1.  .x.  Y )  =  Y )
82, 3, 7syl2anc 411 . . . . . . . 8  |-  ( ph  ->  (  .1.  .x.  Y
)  =  Y )
98eqcomd 2183 . . . . . . 7  |-  ( ph  ->  Y  =  (  .1. 
.x.  Y ) )
109ad3antrrr 492 . . . . . 6  |-  ( ( ( ( ph  /\  a  e.  B )  /\  ( a  .x.  X
)  =  .1.  )  /\  ( X  .x.  Y
)  =  .0.  )  ->  Y  =  (  .1. 
.x.  Y ) )
11 oveq1 5875 . . . . . . . . . 10  |-  (  .1.  =  ( a  .x.  X )  ->  (  .1.  .x.  Y )  =  ( ( a  .x.  X )  .x.  Y
) )
1211eqcoms 2180 . . . . . . . . 9  |-  ( ( a  .x.  X )  =  .1.  ->  (  .1.  .x.  Y )  =  ( ( a  .x.  X )  .x.  Y
) )
1312adantl 277 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  B )  /\  (
a  .x.  X )  =  .1.  )  ->  (  .1.  .x.  Y )  =  ( ( a  .x.  X )  .x.  Y
) )
142adantr 276 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  B )  ->  R  e.  Ring )
15 simpr 110 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  B )  ->  a  e.  B )
16 ringinvnzdiv.x . . . . . . . . . . . . 13  |-  ( ph  ->  X  e.  B )
1716adantr 276 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  B )  ->  X  e.  B )
183adantr 276 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  B )  ->  Y  e.  B )
1915, 17, 183jca 1177 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  B )  ->  (
a  e.  B  /\  X  e.  B  /\  Y  e.  B )
)
2014, 19jca 306 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  B )  ->  ( R  e.  Ring  /\  (
a  e.  B  /\  X  e.  B  /\  Y  e.  B )
) )
2120adantr 276 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  B )  /\  (
a  .x.  X )  =  .1.  )  ->  ( R  e.  Ring  /\  (
a  e.  B  /\  X  e.  B  /\  Y  e.  B )
) )
224, 5ringass 13012 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
a  e.  B  /\  X  e.  B  /\  Y  e.  B )
)  ->  ( (
a  .x.  X )  .x.  Y )  =  ( a  .x.  ( X 
.x.  Y ) ) )
2321, 22syl 14 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  B )  /\  (
a  .x.  X )  =  .1.  )  ->  (
( a  .x.  X
)  .x.  Y )  =  ( a  .x.  ( X  .x.  Y ) ) )
2413, 23eqtrd 2210 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  B )  /\  (
a  .x.  X )  =  .1.  )  ->  (  .1.  .x.  Y )  =  ( a  .x.  ( X  .x.  Y ) ) )
2524adantr 276 . . . . . 6  |-  ( ( ( ( ph  /\  a  e.  B )  /\  ( a  .x.  X
)  =  .1.  )  /\  ( X  .x.  Y
)  =  .0.  )  ->  (  .1.  .x.  Y
)  =  ( a 
.x.  ( X  .x.  Y ) ) )
26 oveq2 5876 . . . . . . 7  |-  ( ( X  .x.  Y )  =  .0.  ->  (
a  .x.  ( X  .x.  Y ) )  =  ( a  .x.  .0.  ) )
27 ringinvnzdiv.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  R )
284, 5, 27ringrz 13036 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  a  e.  B )  ->  (
a  .x.  .0.  )  =  .0.  )
292, 28sylan 283 . . . . . . . 8  |-  ( (
ph  /\  a  e.  B )  ->  (
a  .x.  .0.  )  =  .0.  )
3029adantr 276 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  B )  /\  (
a  .x.  X )  =  .1.  )  ->  (
a  .x.  .0.  )  =  .0.  )
3126, 30sylan9eqr 2232 . . . . . 6  |-  ( ( ( ( ph  /\  a  e.  B )  /\  ( a  .x.  X
)  =  .1.  )  /\  ( X  .x.  Y
)  =  .0.  )  ->  ( a  .x.  ( X  .x.  Y ) )  =  .0.  )
3210, 25, 313eqtrd 2214 . . . . 5  |-  ( ( ( ( ph  /\  a  e.  B )  /\  ( a  .x.  X
)  =  .1.  )  /\  ( X  .x.  Y
)  =  .0.  )  ->  Y  =  .0.  )
3332exp31 364 . . . 4  |-  ( (
ph  /\  a  e.  B )  ->  (
( a  .x.  X
)  =  .1.  ->  ( ( X  .x.  Y
)  =  .0.  ->  Y  =  .0.  ) ) )
3433rexlimdva 2594 . . 3  |-  ( ph  ->  ( E. a  e.  B  ( a  .x.  X )  =  .1. 
->  ( ( X  .x.  Y )  =  .0. 
->  Y  =  .0.  ) ) )
351, 34mpd 13 . 2  |-  ( ph  ->  ( ( X  .x.  Y )  =  .0. 
->  Y  =  .0.  ) )
36 oveq2 5876 . . . 4  |-  ( Y  =  .0.  ->  ( X  .x.  Y )  =  ( X  .x.  .0.  ) )
374, 5, 27ringrz 13036 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .0.  )  =  .0.  )
382, 16, 37syl2anc 411 . . . 4  |-  ( ph  ->  ( X  .x.  .0.  )  =  .0.  )
3936, 38sylan9eqr 2232 . . 3  |-  ( (
ph  /\  Y  =  .0.  )  ->  ( X 
.x.  Y )  =  .0.  )
4039ex 115 . 2  |-  ( ph  ->  ( Y  =  .0. 
->  ( X  .x.  Y
)  =  .0.  )
)
4135, 40impbid 129 1  |-  ( ph  ->  ( ( X  .x.  Y )  =  .0.  <->  Y  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   E.wrex 2456   ` cfv 5211  (class class class)co 5868   Basecbs 12432   .rcmulr 12506   0gc0g 12640   1rcur 12955   Ringcrg 12992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-pre-ltirr 7901  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-pnf 7971  df-mnf 7972  df-ltxr 7974  df-inn 8896  df-2 8954  df-3 8955  df-ndx 12435  df-slot 12436  df-base 12438  df-sets 12439  df-plusg 12518  df-mulr 12519  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-grp 12757  df-mgp 12945  df-ur 12956  df-ring 12994
This theorem is referenced by: (None)
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