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Theorem abvrec 16844
Description: The absolute value distributes under reciprocal. (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 )
abvrec.p  |-  I  =  ( invr `  R
)
Assertion
Ref Expression
abvrec  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( I `  X ) )  =  ( 1  /  ( F `  X )
) )

Proof of Theorem abvrec
StepHypRef Expression
1 simplr 747 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  F  e.  A )
2 simpll 746 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  R  e.  DivRing )
3 simprl 748 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  X  e.  B )
4 simprr 749 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  X  =/=  .0.  )
5 abvneg.b . . . . . . 7  |-  B  =  ( Base `  R
)
6 abvrec.z . . . . . . 7  |-  .0.  =  ( 0g `  R )
7 abvrec.p . . . . . . 7  |-  I  =  ( invr `  R
)
85, 6, 7drnginvrcl 16772 . . . . . 6  |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  (
I `  X )  e.  B )
92, 3, 4, 8syl3anc 1211 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  (
I `  X )  e.  B )
10 abv0.a . . . . . 6  |-  A  =  (AbsVal `  R )
1110, 5abvcl 16832 . . . . 5  |-  ( ( F  e.  A  /\  ( I `  X
)  e.  B )  ->  ( F `  ( I `  X
) )  e.  RR )
121, 9, 11syl2anc 654 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( I `  X ) )  e.  RR )
1312recnd 9399 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( I `  X ) )  e.  CC )
1410, 5abvcl 16832 . . . . 5  |-  ( ( F  e.  A  /\  X  e.  B )  ->  ( F `  X
)  e.  RR )
151, 3, 14syl2anc 654 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  X )  e.  RR )
1615recnd 9399 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  X )  e.  CC )
1710, 5, 6abvne0 16835 . . . 4  |-  ( ( F  e.  A  /\  X  e.  B  /\  X  =/=  .0.  )  -> 
( F `  X
)  =/=  0 )
181, 3, 4, 17syl3anc 1211 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  X )  =/=  0 )
1913, 16, 18divcan3d 10099 . 2  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  (
( ( F `  X )  x.  ( F `  ( I `  X ) ) )  /  ( F `  X ) )  =  ( F `  (
I `  X )
) )
20 eqid 2433 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
21 eqid 2433 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
225, 6, 20, 21, 7drnginvrr 16775 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  ( X ( .r `  R ) ( I `
 X ) )  =  ( 1r `  R ) )
232, 3, 4, 22syl3anc 1211 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( X ( .r `  R ) ( I `
 X ) )  =  ( 1r `  R ) )
2423fveq2d 5683 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( X
( .r `  R
) ( I `  X ) ) )  =  ( F `  ( 1r `  R ) ) )
2510, 5, 20abvmul 16837 . . . . . 6  |-  ( ( F  e.  A  /\  X  e.  B  /\  ( I `  X
)  e.  B )  ->  ( F `  ( X ( .r `  R ) ( I `
 X ) ) )  =  ( ( F `  X )  x.  ( F `  ( I `  X
) ) ) )
261, 3, 9, 25syl3anc 1211 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( X
( .r `  R
) ( I `  X ) ) )  =  ( ( F `
 X )  x.  ( F `  (
I `  X )
) ) )
2724, 26eqtr3d 2467 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( 1r `  R ) )  =  ( ( F `  X )  x.  ( F `  ( I `  X ) ) ) )
2810, 21abv1 16841 . . . . 5  |-  ( ( R  e.  DivRing  /\  F  e.  A )  ->  ( F `  ( 1r `  R ) )  =  1 )
2928adantr 462 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( 1r `  R ) )  =  1 )
3027, 29eqtr3d 2467 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  (
( F `  X
)  x.  ( F `
 ( I `  X ) ) )  =  1 )
3130oveq1d 6095 . 2  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  (
( ( F `  X )  x.  ( F `  ( I `  X ) ) )  /  ( F `  X ) )  =  ( 1  /  ( F `  X )
) )
3219, 31eqtr3d 2467 1  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( I `  X ) )  =  ( 1  /  ( F `  X )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755    =/= wne 2596   ` cfv 5406  (class class class)co 6080   RRcr 9268   0cc0 9269   1c1 9270    x. cmul 9274    / cdiv 9980   Basecbs 14156   .rcmulr 14221   0gc0g 14360   1rcur 16578   invrcinvr 16696   DivRingcdr 16755  AbsValcabv 16824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-tpos 6734  df-recs 6818  df-rdg 6852  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-ico 11293  df-ndx 14159  df-slot 14160  df-base 14161  df-sets 14162  df-ress 14163  df-plusg 14233  df-mulr 14234  df-0g 14362  df-mnd 15397  df-grp 15524  df-minusg 15525  df-mgp 16565  df-rng 16579  df-ur 16581  df-oppr 16648  df-dvdsr 16666  df-unit 16667  df-invr 16697  df-drng 16757  df-abv 16825
This theorem is referenced by:  abvdiv  16845
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