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Theorem abvdom 15597
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 957 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  ->  F  e.  A )
2 simp2l 983 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  ->  X  e.  B )
3 simp3l 985 . . . 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 15588 . . . 4  |-  ( ( F  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `  ( X  .x.  Y ) )  =  ( ( F `
 X )  x.  ( F `  Y
) ) )
81, 2, 3, 7syl3anc 1184 . . 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 15583 . . . . . 6  |-  ( ( F  e.  A  /\  X  e.  B )  ->  ( F `  X
)  e.  RR )
101, 2, 9syl2anc 644 . . . . 5  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  X
)  e.  RR )
1110recnd 8856 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  X
)  e.  CC )
124, 5abvcl 15583 . . . . . 6  |-  ( ( F  e.  A  /\  Y  e.  B )  ->  ( F `  Y
)  e.  RR )
131, 3, 12syl2anc 644 . . . . 5  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  Y
)  e.  RR )
1413recnd 8856 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  Y
)  e.  CC )
15 simp2r 984 . . . . 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 15586 . . . . 5  |-  ( ( F  e.  A  /\  X  e.  B  /\  X  =/=  .0.  )  -> 
( F `  X
)  =/=  0 )
181, 2, 15, 17syl3anc 1184 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  X
)  =/=  0 )
19 simp3r 986 . . . . 5  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  ->  Y  =/=  .0.  )
204, 5, 16abvne0 15586 . . . . 5  |-  ( ( F  e.  A  /\  Y  e.  B  /\  Y  =/=  .0.  )  -> 
( F `  Y
)  =/=  0 )
211, 3, 19, 20syl3anc 1184 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  Y
)  =/=  0 )
2211, 14, 18, 21mulne0d 9415 . . 3  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( ( F `  X )  x.  ( F `  Y )
)  =/=  0 )
238, 22eqnetrd 2465 . 2  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  ( X  .x.  Y ) )  =/=  0 )
244, 16abv0 15590 . . . . 5  |-  ( F  e.  A  ->  ( F `  .0.  )  =  0 )
251, 24syl 17 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  .0.  )  =  0 )
26 fveq2 5485 . . . . 5  |-  ( ( X  .x.  Y )  =  .0.  ->  ( F `  ( X  .x.  Y ) )  =  ( F `  .0.  ) )
2726eqeq1d 2292 . . . 4  |-  ( ( X  .x.  Y )  =  .0.  ->  (
( F `  ( X  .x.  Y ) )  =  0  <->  ( F `  .0.  )  =  0 ) )
2825, 27syl5ibrcom 215 . . 3  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( ( X  .x.  Y )  =  .0. 
->  ( F `  ( X  .x.  Y ) )  =  0 ) )
2928necon3d 2485 . 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 16 1  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( X  .x.  Y
)  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2447   ` cfv 5221  (class class class)co 5819   RRcr 8731   0cc0 8732    x. cmul 8737   Basecbs 13142   .rcmulr 13203   0gc0g 13394  AbsValcabv 15575
This theorem is referenced by:  abvn0b  16037
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-iota 6252  df-riota 6299  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-ico 10656  df-0g 13398  df-mnd 14361  df-grp 14483  df-rng 15334  df-abv 15576
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