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Theorem abvdom 17031
Description: Any ring with an absolute value is a domain, which is to say that it contains no zero divisors. (Contributed by Mario Carneiro, 10-Sep-2014.)
Hypotheses
Ref Expression
abv0.a  |-  A  =  (AbsVal `  R )
abvneg.b  |-  B  =  ( Base `  R
)
abvrec.z  |-  .0.  =  ( 0g `  R )
abvdom.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
abvdom  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( X  .x.  Y
)  =/=  .0.  )

Proof of Theorem abvdom
StepHypRef Expression
1 simp1 988 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  ->  F  e.  A )
2 simp2l 1014 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  ->  X  e.  B )
3 simp3l 1016 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  ->  Y  e.  B )
4 abv0.a . . . . 5  |-  A  =  (AbsVal `  R )
5 abvneg.b . . . . 5  |-  B  =  ( Base `  R
)
6 abvdom.t . . . . 5  |-  .x.  =  ( .r `  R )
74, 5, 6abvmul 17022 . . . 4  |-  ( ( F  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `  ( X  .x.  Y ) )  =  ( ( F `
 X )  x.  ( F `  Y
) ) )
81, 2, 3, 7syl3anc 1219 . . 3  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  ( X  .x.  Y ) )  =  ( ( F `
 X )  x.  ( F `  Y
) ) )
94, 5abvcl 17017 . . . . . 6  |-  ( ( F  e.  A  /\  X  e.  B )  ->  ( F `  X
)  e.  RR )
101, 2, 9syl2anc 661 . . . . 5  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  X
)  e.  RR )
1110recnd 9515 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  X
)  e.  CC )
124, 5abvcl 17017 . . . . . 6  |-  ( ( F  e.  A  /\  Y  e.  B )  ->  ( F `  Y
)  e.  RR )
131, 3, 12syl2anc 661 . . . . 5  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  Y
)  e.  RR )
1413recnd 9515 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  Y
)  e.  CC )
15 simp2r 1015 . . . . 5  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  ->  X  =/=  .0.  )
16 abvrec.z . . . . . 6  |-  .0.  =  ( 0g `  R )
174, 5, 16abvne0 17020 . . . . 5  |-  ( ( F  e.  A  /\  X  e.  B  /\  X  =/=  .0.  )  -> 
( F `  X
)  =/=  0 )
181, 2, 15, 17syl3anc 1219 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  X
)  =/=  0 )
19 simp3r 1017 . . . . 5  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  ->  Y  =/=  .0.  )
204, 5, 16abvne0 17020 . . . . 5  |-  ( ( F  e.  A  /\  Y  e.  B  /\  Y  =/=  .0.  )  -> 
( F `  Y
)  =/=  0 )
211, 3, 19, 20syl3anc 1219 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  Y
)  =/=  0 )
2211, 14, 18, 21mulne0d 10091 . . 3  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( ( F `  X )  x.  ( F `  Y )
)  =/=  0 )
238, 22eqnetrd 2741 . 2  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  ( X  .x.  Y ) )  =/=  0 )
244, 16abv0 17024 . . . . 5  |-  ( F  e.  A  ->  ( F `  .0.  )  =  0 )
251, 24syl 16 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  .0.  )  =  0 )
26 fveq2 5791 . . . . 5  |-  ( ( X  .x.  Y )  =  .0.  ->  ( F `  ( X  .x.  Y ) )  =  ( F `  .0.  ) )
2726eqeq1d 2453 . . . 4  |-  ( ( X  .x.  Y )  =  .0.  ->  (
( F `  ( X  .x.  Y ) )  =  0  <->  ( F `  .0.  )  =  0 ) )
2825, 27syl5ibrcom 222 . . 3  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( ( X  .x.  Y )  =  .0. 
->  ( F `  ( X  .x.  Y ) )  =  0 ) )
2928necon3d 2672 . 2  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( ( F `  ( X  .x.  Y ) )  =/=  0  -> 
( X  .x.  Y
)  =/=  .0.  )
)
3023, 29mpd 15 1  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( X  .x.  Y
)  =/=  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   ` cfv 5518  (class class class)co 6192   RRcr 9384   0cc0 9385    x. cmul 9390   Basecbs 14278   .rcmulr 14343   0gc0g 14482  AbsValcabv 17009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-ico 11409  df-0g 14484  df-mnd 15519  df-grp 15649  df-rng 16755  df-abv 17010
This theorem is referenced by:  abvn0b  17482
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