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Theorem abvrec 17029
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 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  F  e.  A )
2 simpll 753 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  R  e.  DivRing )
3 simprl 755 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  X  e.  B )
4 simprr 756 . . . . . 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 16957 . . . . . 6  |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  (
I `  X )  e.  B )
92, 3, 4, 8syl3anc 1219 . . . . 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 17017 . . . . 5  |-  ( ( F  e.  A  /\  ( I `  X
)  e.  B )  ->  ( F `  ( I `  X
) )  e.  RR )
121, 9, 11syl2anc 661 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( I `  X ) )  e.  RR )
1312recnd 9515 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( I `  X ) )  e.  CC )
1410, 5abvcl 17017 . . . . 5  |-  ( ( F  e.  A  /\  X  e.  B )  ->  ( F `  X
)  e.  RR )
151, 3, 14syl2anc 661 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  X )  e.  RR )
1615recnd 9515 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  X )  e.  CC )
1710, 5, 6abvne0 17020 . . . 4  |-  ( ( F  e.  A  /\  X  e.  B  /\  X  =/=  .0.  )  -> 
( F `  X
)  =/=  0 )
181, 3, 4, 17syl3anc 1219 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  X )  =/=  0 )
1913, 16, 18divcan3d 10215 . 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 2451 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
21 eqid 2451 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
225, 6, 20, 21, 7drnginvrr 16960 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  ( X ( .r `  R ) ( I `
 X ) )  =  ( 1r `  R ) )
232, 3, 4, 22syl3anc 1219 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( X ( .r `  R ) ( I `
 X ) )  =  ( 1r `  R ) )
2423fveq2d 5795 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( X
( .r `  R
) ( I `  X ) ) )  =  ( F `  ( 1r `  R ) ) )
2510, 5, 20abvmul 17022 . . . . . 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 1219 . . . . 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 2494 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( 1r `  R ) )  =  ( ( F `  X )  x.  ( F `  ( I `  X ) ) ) )
2810, 21abv1 17026 . . . . 5  |-  ( ( R  e.  DivRing  /\  F  e.  A )  ->  ( F `  ( 1r `  R ) )  =  1 )
2928adantr 465 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( 1r `  R ) )  =  1 )
3027, 29eqtr3d 2494 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  (
( F `  X
)  x.  ( F `
 ( I `  X ) ) )  =  1 )
3130oveq1d 6207 . 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 2494 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 1370    e. wcel 1758    =/= wne 2644   ` cfv 5518  (class class class)co 6192   RRcr 9384   0cc0 9385   1c1 9386    x. cmul 9390    / cdiv 10096   Basecbs 14278   .rcmulr 14343   0gc0g 14482   1rcur 16710   invrcinvr 16871   DivRingcdr 16940  AbsValcabv 17009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-tpos 6847  df-recs 6934  df-rdg 6968  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-ico 11409  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-mulr 14356  df-0g 14484  df-mnd 15519  df-grp 15649  df-minusg 15650  df-mgp 16699  df-ur 16711  df-rng 16755  df-oppr 16823  df-dvdsr 16841  df-unit 16842  df-invr 16872  df-drng 16942  df-abv 17010
This theorem is referenced by:  abvdiv  17030
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