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Theorem abvdom 15855
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 15846 . . . 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 15841 . . . . . 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 9049 . . . 4  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  X
)  e.  CC )
124, 5abvcl 15841 . . . . . 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 9049 . . . 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 15844 . . . . 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 15844 . . . . 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 9608 . . 3  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( ( F `  X )  x.  ( F `  Y )
)  =/=  0 )
238, 22eqnetrd 2570 . 2  |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/= 
.0.  ) )  -> 
( F `  ( X  .x.  Y ) )  =/=  0 )
244, 16abv0 15848 . . . . 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 5670 . . . . 5  |-  ( ( X  .x.  Y )  =  .0.  ->  ( F `  ( X  .x.  Y ) )  =  ( F `  .0.  ) )
2726eqeq1d 2397 . . . 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 2590 . 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 1649    e. wcel 1717    =/= wne 2552   ` cfv 5396  (class class class)co 6022   RRcr 8924   0cc0 8925    x. cmul 8930   Basecbs 13398   .rcmulr 13459   0gc0g 13652  AbsValcabv 15833
This theorem is referenced by:  abvn0b  16291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-po 4446  df-so 4447  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-ico 10856  df-0g 13656  df-mnd 14619  df-grp 14741  df-rng 15592  df-abv 15834
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