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Theorem abv1z 15903
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 15892 . . . . . . 7  |-  ( F  e.  A  ->  R  e.  Ring )
3 eqid 2430 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
4 abv1.p . . . . . . . 8  |-  .1.  =  ( 1r `  R )
53, 4rngidcl 15667 . . . . . . 7  |-  ( R  e.  Ring  ->  .1.  e.  ( Base `  R )
)
62, 5syl 16 . . . . . 6  |-  ( F  e.  A  ->  .1.  e.  ( Base `  R
) )
71, 3abvcl 15895 . . . . . 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 9098 . . 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 15898 . . . 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 9779 . 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 2430 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
203, 19, 4rnglidm 15670 . . . . . . 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 5718 . . . . 5  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  (  .1.  ( .r `  R
)  .1.  ) )  =  ( F `  .1.  ) )
231, 3, 19abvmul 15900 . . . . . 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 2464 . . . 4  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  .1.  )  =  ( ( F `  .1.  )  x.  ( F `  .1.  ) ) )
2625oveq1d 6082 . . 3  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( ( F `  .1.  )  /  ( F `  .1.  ) )  =  ( ( ( F `  .1.  )  x.  ( F `  .1.  ) )  /  ( F `  .1.  ) ) )
2710, 16dividd 9772 . . 3  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( ( F `  .1.  )  /  ( F `  .1.  ) )  =  1 )
2826, 27eqtr3d 2464 . 2  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( ( ( F `
 .1.  )  x.  ( F `  .1.  ) )  /  ( F `  .1.  ) )  =  1 )
2917, 28eqtr3d 2464 1  |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  -> 
( F `  .1.  )  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2593   ` cfv 5440  (class class class)co 6067   RRcr 8973   0cc0 8974   1c1 8975    x. cmul 8979    / cdiv 9661   Basecbs 13452   .rcmulr 13513   0gc0g 13706   Ringcrg 15643   1rcur 15645  AbsValcabv 15887
This theorem is referenced by:  abv1  15904  abvneg  15905  nm1  18686  qabvle  21302  qabvexp  21303  ostthlem2  21305  ostth3  21315  ostth  21316
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 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051
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 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-riota 6535  df-recs 6619  df-rdg 6654  df-er 6891  df-map 7006  df-en 7096  df-dom 7097  df-sdom 7098  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-div 9662  df-nn 9985  df-2 10042  df-ico 10906  df-ndx 13455  df-slot 13456  df-base 13457  df-sets 13458  df-plusg 13525  df-0g 13710  df-mnd 14673  df-mgp 15632  df-rng 15646  df-ur 15648  df-abv 15888
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