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Theorem abvdom 15916
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 15907 . . . 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 15902 . . . . . 6  |-  ( ( F  e.  A  /\  X  e.  B )  ->  ( F `  X
)  e.  RR )
101, 2, 9syl2anc 643 . . . . 5  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  X
)  e.  RR )
1110recnd 9104 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  X
)  e.  CC )
124, 5abvcl 15902 . . . . . 6  |-  ( ( F  e.  A  /\  Y  e.  B )  ->  ( F `  Y
)  e.  RR )
131, 3, 12syl2anc 643 . . . . 5  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  Y
)  e.  RR )
1413recnd 9104 . . . 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 15905 . . . . 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 15905 . . . . 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 9664 . . 3  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( ( F `  X )  x.  ( F `  Y )
)  =/=  0 )
238, 22eqnetrd 2616 . 2  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  ( X  .x.  Y ) )  =/=  0 )
244, 16abv0 15909 . . . . 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 5720 . . . . 5  |-  ( ( X  .x.  Y )  =  .0.  ->  ( F `  ( X  .x.  Y ) )  =  ( F `  .0.  ) )
2726eqeq1d 2443 . . . 4  |-  ( ( X  .x.  Y )  =  .0.  ->  (
( F `  ( X  .x.  Y ) )  =  0  <->  ( F `  .0.  )  =  0 ) )
2825, 27syl5ibrcom 214 . . 3  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( ( X  .x.  Y )  =  .0. 
->  ( F `  ( X  .x.  Y ) )  =  0 ) )
2928necon3d 2636 . 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   ` cfv 5446  (class class class)co 6073   RRcr 8979   0cc0 8980    x. cmul 8985   Basecbs 13459   .rcmulr 13520   0gc0g 13713  AbsValcabv 15894
This theorem is referenced by:  abvn0b  16352
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-ico 10912  df-0g 13717  df-mnd 14680  df-grp 14802  df-rng 15653  df-abv 15895
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