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Theorem abvrec 17805
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 754 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  F  e.  A )
2 simprl 756 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  X  e.  B )
3 abv0.a . . . . 5  |-  A  =  (AbsVal `  R )
4 abvneg.b . . . . 5  |-  B  =  ( Base `  R
)
53, 4abvcl 17793 . . . 4  |-  ( ( F  e.  A  /\  X  e.  B )  ->  ( F `  X
)  e.  RR )
61, 2, 5syl2anc 659 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  X )  e.  RR )
76recnd 9652 . 2  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  X )  e.  CC )
8 simpll 752 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  R  e.  DivRing )
9 simprr 758 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  X  =/=  .0.  )
10 abvrec.z . . . . . 6  |-  .0.  =  ( 0g `  R )
11 abvrec.p . . . . . 6  |-  I  =  ( invr `  R
)
124, 10, 11drnginvrcl 17733 . . . . 5  |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  (
I `  X )  e.  B )
138, 2, 9, 12syl3anc 1230 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  (
I `  X )  e.  B )
143, 4abvcl 17793 . . . 4  |-  ( ( F  e.  A  /\  ( I `  X
)  e.  B )  ->  ( F `  ( I `  X
) )  e.  RR )
151, 13, 14syl2anc 659 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( I `  X ) )  e.  RR )
1615recnd 9652 . 2  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( I `  X ) )  e.  CC )
173, 4, 10abvne0 17796 . . 3  |-  ( ( F  e.  A  /\  X  e.  B  /\  X  =/=  .0.  )  -> 
( F `  X
)  =/=  0 )
181, 2, 9, 17syl3anc 1230 . 2  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  X )  =/=  0 )
19 eqid 2402 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
20 eqid 2402 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
214, 10, 19, 20, 11drnginvrr 17736 . . . . 5  |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  ( X ( .r `  R ) ( I `
 X ) )  =  ( 1r `  R ) )
228, 2, 9, 21syl3anc 1230 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( X ( .r `  R ) ( I `
 X ) )  =  ( 1r `  R ) )
2322fveq2d 5853 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( X
( .r `  R
) ( I `  X ) ) )  =  ( F `  ( 1r `  R ) ) )
243, 4, 19abvmul 17798 . . . 4  |-  ( ( F  e.  A  /\  X  e.  B  /\  ( I `  X
)  e.  B )  ->  ( F `  ( X ( .r `  R ) ( I `
 X ) ) )  =  ( ( F `  X )  x.  ( F `  ( I `  X
) ) ) )
251, 2, 13, 24syl3anc 1230 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( X
( .r `  R
) ( I `  X ) ) )  =  ( ( F `
 X )  x.  ( F `  (
I `  X )
) ) )
263, 20abv1 17802 . . . 4  |-  ( ( R  e.  DivRing  /\  F  e.  A )  ->  ( F `  ( 1r `  R ) )  =  1 )
2726adantr 463 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( 1r `  R ) )  =  1 )
2823, 25, 273eqtr3d 2451 . 2  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  (
( F `  X
)  x.  ( F `
 ( I `  X ) ) )  =  1 )
297, 16, 18, 28mvllmuld 10417 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 367    = wceq 1405    e. wcel 1842    =/= wne 2598   ` cfv 5569  (class class class)co 6278   RRcr 9521   0cc0 9522   1c1 9523    x. cmul 9527    / cdiv 10247   Basecbs 14841   .rcmulr 14910   0gc0g 15054   1rcur 17473   invrcinvr 17640   DivRingcdr 17716  AbsValcabv 17785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-tpos 6958  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-ico 11588  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-grp 16381  df-minusg 16382  df-mgp 17462  df-ur 17474  df-ring 17520  df-oppr 17592  df-dvdsr 17610  df-unit 17611  df-invr 17641  df-drng 17718  df-abv 17786
This theorem is referenced by:  abvdiv  17806
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