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Theorem abvdom 17361
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 997 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  ->  F  e.  A )
2 simp2l 1023 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  ->  X  e.  B )
3 simp3l 1025 . . . 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 17352 . . . 4  |-  ( ( F  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `  ( X  .x.  Y ) )  =  ( ( F `
 X )  x.  ( F `  Y
) ) )
81, 2, 3, 7syl3anc 1229 . . 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 17347 . . . . . 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 9625 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  X
)  e.  CC )
124, 5abvcl 17347 . . . . . 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 9625 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  Y
)  e.  CC )
15 simp2r 1024 . . . . 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 17350 . . . . 5  |-  ( ( F  e.  A  /\  X  e.  B  /\  X  =/=  .0.  )  -> 
( F `  X
)  =/=  0 )
181, 2, 15, 17syl3anc 1229 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  X
)  =/=  0 )
19 simp3r 1026 . . . . 5  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  ->  Y  =/=  .0.  )
204, 5, 16abvne0 17350 . . . . 5  |-  ( ( F  e.  A  /\  Y  e.  B  /\  Y  =/=  .0.  )  -> 
( F `  Y
)  =/=  0 )
211, 3, 19, 20syl3anc 1229 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  Y
)  =/=  0 )
2211, 14, 18, 21mulne0d 10207 . . 3  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( ( F `  X )  x.  ( F `  Y )
)  =/=  0 )
238, 22eqnetrd 2736 . 2  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  ( X  .x.  Y ) )  =/=  0 )
244, 16abv0 17354 . . . . 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 5856 . . . . 5  |-  ( ( X  .x.  Y )  =  .0.  ->  ( F `  ( X  .x.  Y ) )  =  ( F `  .0.  ) )
2726eqeq1d 2445 . . . 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 2667 . 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   ` cfv 5578  (class class class)co 6281   RRcr 9494   0cc0 9495    x. cmul 9500   Basecbs 14509   .rcmulr 14575   0gc0g 14714  AbsValcabv 17339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-ico 11544  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-ring 17074  df-abv 17340
This theorem is referenced by:  abvn0b  17825
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