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Theorem abv1z 15883
Description: The absolute value of one is one in a non-trivial ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
Hypotheses
Ref Expression
abv0.a  |-  A  =  (AbsVal `  R )
abv1.p  |-  .1.  =  ( 1r `  R )
abv1z.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
abv1z  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  .1.  )  =  1 )

Proof of Theorem abv1z
StepHypRef Expression
1 abv0.a . . . . . . . 8  |-  A  =  (AbsVal `  R )
21abvrcl 15872 . . . . . . 7  |-  ( F  e.  A  ->  R  e.  Ring )
3 eqid 2412 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
4 abv1.p . . . . . . . 8  |-  .1.  =  ( 1r `  R )
53, 4rngidcl 15647 . . . . . . 7  |-  ( R  e.  Ring  ->  .1.  e.  ( Base `  R )
)
62, 5syl 16 . . . . . 6  |-  ( F  e.  A  ->  .1.  e.  ( Base `  R
) )
71, 3abvcl 15875 . . . . . 6  |-  ( ( F  e.  A  /\  .1.  e.  ( Base `  R
) )  ->  ( F `  .1.  )  e.  RR )
86, 7mpdan 650 . . . . 5  |-  ( F  e.  A  ->  ( F `  .1.  )  e.  RR )
98adantr 452 . . . 4  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  .1.  )  e.  RR )
109recnd 9078 . . 3  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  .1.  )  e.  CC )
11 simpl 444 . . . 4  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  ->  F  e.  A )
126adantr 452 . . . 4  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  ->  .1.  e.  ( Base `  R
) )
13 simpr 448 . . . 4  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  ->  .1.  =/=  .0.  )
14 abv1z.z . . . . 5  |-  .0.  =  ( 0g `  R )
151, 3, 14abvne0 15878 . . . 4  |-  ( ( F  e.  A  /\  .1.  e.  ( Base `  R
)  /\  .1.  =/=  .0.  )  ->  ( F `
 .1.  )  =/=  0 )
1611, 12, 13, 15syl3anc 1184 . . 3  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  .1.  )  =/=  0 )
1710, 10, 16divcan3d 9759 . 2  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( ( ( F `
 .1.  )  x.  ( F `  .1.  ) )  /  ( F `  .1.  ) )  =  ( F `  .1.  ) )
182adantr 452 . . . . . . 7  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  ->  R  e.  Ring )
19 eqid 2412 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
203, 19, 4rnglidm 15650 . . . . . . 7  |-  ( ( R  e.  Ring  /\  .1.  e.  ( Base `  R
) )  ->  (  .1.  ( .r `  R
)  .1.  )  =  .1.  )
2118, 12, 20syl2anc 643 . . . . . 6  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
(  .1.  ( .r
`  R )  .1.  )  =  .1.  )
2221fveq2d 5699 . . . . 5  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  (  .1.  ( .r `  R
)  .1.  ) )  =  ( F `  .1.  ) )
231, 3, 19abvmul 15880 . . . . . 6  |-  ( ( F  e.  A  /\  .1.  e.  ( Base `  R
)  /\  .1.  e.  ( Base `  R )
)  ->  ( F `  (  .1.  ( .r `  R )  .1.  ) )  =  ( ( F `  .1.  )  x.  ( F `  .1.  ) ) )
2411, 12, 12, 23syl3anc 1184 . . . . 5  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  (  .1.  ( .r `  R
)  .1.  ) )  =  ( ( F `
 .1.  )  x.  ( F `  .1.  ) ) )
2522, 24eqtr3d 2446 . . . 4  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  .1.  )  =  ( ( F `  .1.  )  x.  ( F `  .1.  ) ) )
2625oveq1d 6063 . . 3  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( ( F `  .1.  )  /  ( F `  .1.  ) )  =  ( ( ( F `  .1.  )  x.  ( F `  .1.  ) )  /  ( F `  .1.  ) ) )
2710, 16dividd 9752 . . 3  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( ( F `  .1.  )  /  ( F `  .1.  ) )  =  1 )
2826, 27eqtr3d 2446 . 2  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( ( ( F `
 .1.  )  x.  ( F `  .1.  ) )  /  ( F `  .1.  ) )  =  1 )
2917, 28eqtr3d 2446 1  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  .1.  )  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   ` cfv 5421  (class class class)co 6048   RRcr 8953   0cc0 8954   1c1 8955    x. cmul 8959    / cdiv 9641   Basecbs 13432   .rcmulr 13493   0gc0g 13686   Ringcrg 15623   1rcur 15625  AbsValcabv 15867
This theorem is referenced by:  abv1  15884  abvneg  15885  nm1  18664  qabvle  21280  qabvexp  21281  ostthlem2  21283  ostth3  21293  ostth  21294
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-ico 10886  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-plusg 13505  df-0g 13690  df-mnd 14653  df-mgp 15612  df-rng 15626  df-ur 15628  df-abv 15868
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