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Theorem abv1z 15951
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 15940 . . . . . . 7  |-  ( F  e.  A  ->  R  e.  Ring )
3 eqid 2442 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
4 abv1.p . . . . . . . 8  |-  .1.  =  ( 1r `  R )
53, 4rngidcl 15715 . . . . . . 7  |-  ( R  e.  Ring  ->  .1.  e.  ( Base `  R )
)
62, 5syl 16 . . . . . 6  |-  ( F  e.  A  ->  .1.  e.  ( Base `  R
) )
71, 3abvcl 15943 . . . . . 6  |-  ( ( F  e.  A  /\  .1.  e.  ( Base `  R
) )  ->  ( F `  .1.  )  e.  RR )
86, 7mpdan 651 . . . . 5  |-  ( F  e.  A  ->  ( F `  .1.  )  e.  RR )
98adantr 453 . . . 4  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  .1.  )  e.  RR )
109recnd 9145 . . 3  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  .1.  )  e.  CC )
11 simpl 445 . . . 4  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  ->  F  e.  A )
126adantr 453 . . . 4  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  ->  .1.  e.  ( Base `  R
) )
13 simpr 449 . . . 4  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  ->  .1.  =/=  .0.  )
14 abv1z.z . . . . 5  |-  .0.  =  ( 0g `  R )
151, 3, 14abvne0 15946 . . . 4  |-  ( ( F  e.  A  /\  .1.  e.  ( Base `  R
)  /\  .1.  =/=  .0.  )  ->  ( F `
 .1.  )  =/=  0 )
1611, 12, 13, 15syl3anc 1185 . . 3  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  .1.  )  =/=  0 )
1710, 10, 16divcan3d 9826 . 2  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( ( ( F `
 .1.  )  x.  ( F `  .1.  ) )  /  ( F `  .1.  ) )  =  ( F `  .1.  ) )
182adantr 453 . . . . . . 7  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  ->  R  e.  Ring )
19 eqid 2442 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
203, 19, 4rnglidm 15718 . . . . . . 7  |-  ( ( R  e.  Ring  /\  .1.  e.  ( Base `  R
) )  ->  (  .1.  ( .r `  R
)  .1.  )  =  .1.  )
2118, 12, 20syl2anc 644 . . . . . 6  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
(  .1.  ( .r
`  R )  .1.  )  =  .1.  )
2221fveq2d 5761 . . . . 5  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  (  .1.  ( .r `  R
)  .1.  ) )  =  ( F `  .1.  ) )
231, 3, 19abvmul 15948 . . . . . 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 1185 . . . . 5  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  (  .1.  ( .r `  R
)  .1.  ) )  =  ( ( F `
 .1.  )  x.  ( F `  .1.  ) ) )
2522, 24eqtr3d 2476 . . . 4  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  .1.  )  =  ( ( F `  .1.  )  x.  ( F `  .1.  ) ) )
2625oveq1d 6125 . . 3  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( ( F `  .1.  )  /  ( F `  .1.  ) )  =  ( ( ( F `  .1.  )  x.  ( F `  .1.  ) )  /  ( F `  .1.  ) ) )
2710, 16dividd 9819 . . 3  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( ( F `  .1.  )  /  ( F `  .1.  ) )  =  1 )
2826, 27eqtr3d 2476 . 2  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( ( ( F `
 .1.  )  x.  ( F `  .1.  ) )  /  ( F `  .1.  ) )  =  1 )
2917, 28eqtr3d 2476 1  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  .1.  )  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605   ` cfv 5483  (class class class)co 6110   RRcr 9020   0cc0 9021   1c1 9022    x. cmul 9026    / cdiv 9708   Basecbs 13500   .rcmulr 13561   0gc0g 13754   Ringcrg 15691   1rcur 15693  AbsValcabv 15935
This theorem is referenced by:  abv1  15952  abvneg  15953  nm1  18734  qabvle  21350  qabvexp  21351  ostthlem2  21353  ostth3  21363  ostth  21364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-ico 10953  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-plusg 13573  df-0g 13758  df-mnd 14721  df-mgp 15680  df-rng 15694  df-ur 15696  df-abv 15936
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