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Theorem abvdiv 17334
Description: The absolute value distributes under division. (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 )
abvdiv.p  |-  ./  =  (/r
`  R )
Assertion
Ref Expression
abvdiv  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  ( X  ./  Y ) )  =  ( ( F `  X )  /  ( F `  Y )
) )

Proof of Theorem abvdiv
StepHypRef Expression
1 simplr 754 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  F  e.  A )
2 simpr1 1002 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  X  e.  B )
3 simpll 753 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  R  e.  DivRing )
4 simpr2 1003 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  Y  e.  B )
5 simpr3 1004 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  Y  =/=  .0.  )
6 abvneg.b . . . . . 6  |-  B  =  ( Base `  R
)
7 abvrec.z . . . . . 6  |-  .0.  =  ( 0g `  R )
8 eqid 2467 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
96, 7, 8drnginvrcl 17261 . . . . 5  |-  ( ( R  e.  DivRing  /\  Y  e.  B  /\  Y  =/= 
.0.  )  ->  (
( invr `  R ) `  Y )  e.  B
)
103, 4, 5, 9syl3anc 1228 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  (
( invr `  R ) `  Y )  e.  B
)
11 abv0.a . . . . 5  |-  A  =  (AbsVal `  R )
12 eqid 2467 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
1311, 6, 12abvmul 17326 . . . 4  |-  ( ( F  e.  A  /\  X  e.  B  /\  ( ( invr `  R
) `  Y )  e.  B )  ->  ( F `  ( X
( .r `  R
) ( ( invr `  R ) `  Y
) ) )  =  ( ( F `  X )  x.  ( F `  ( ( invr `  R ) `  Y ) ) ) )
141, 2, 10, 13syl3anc 1228 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  ( X
( .r `  R
) ( ( invr `  R ) `  Y
) ) )  =  ( ( F `  X )  x.  ( F `  ( ( invr `  R ) `  Y ) ) ) )
1511, 6, 7, 8abvrec 17333 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  ( ( invr `  R ) `  Y ) )  =  ( 1  /  ( F `  Y )
) )
16153adantr1 1155 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  ( ( invr `  R ) `  Y ) )  =  ( 1  /  ( F `  Y )
) )
1716oveq2d 6310 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  (
( F `  X
)  x.  ( F `
 ( ( invr `  R ) `  Y
) ) )  =  ( ( F `  X )  x.  (
1  /  ( F `
 Y ) ) ) )
1814, 17eqtrd 2508 . 2  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  ( X
( .r `  R
) ( ( invr `  R ) `  Y
) ) )  =  ( ( F `  X )  x.  (
1  /  ( F `
 Y ) ) ) )
19 eqid 2467 . . . . . . 7  |-  (Unit `  R )  =  (Unit `  R )
206, 19, 7drngunit 17249 . . . . . 6  |-  ( R  e.  DivRing  ->  ( Y  e.  (Unit `  R )  <->  ( Y  e.  B  /\  Y  =/=  .0.  ) ) )
213, 20syl 16 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( Y  e.  (Unit `  R
)  <->  ( Y  e.  B  /\  Y  =/= 
.0.  ) ) )
224, 5, 21mpbir2and 920 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  Y  e.  (Unit `  R )
)
23 abvdiv.p . . . . 5  |-  ./  =  (/r
`  R )
246, 12, 19, 8, 23dvrval 17183 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  (Unit `  R
) )  ->  ( X  ./  Y )  =  ( X ( .r
`  R ) ( ( invr `  R
) `  Y )
) )
252, 22, 24syl2anc 661 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( X  ./  Y )  =  ( X ( .r
`  R ) ( ( invr `  R
) `  Y )
) )
2625fveq2d 5875 . 2  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  ( X  ./  Y ) )  =  ( F `  ( X ( .r `  R ) ( (
invr `  R ) `  Y ) ) ) )
2711, 6abvcl 17321 . . . . 5  |-  ( ( F  e.  A  /\  X  e.  B )  ->  ( F `  X
)  e.  RR )
281, 2, 27syl2anc 661 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  X )  e.  RR )
2928recnd 9632 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  X )  e.  CC )
3011, 6abvcl 17321 . . . . 5  |-  ( ( F  e.  A  /\  Y  e.  B )  ->  ( F `  Y
)  e.  RR )
311, 4, 30syl2anc 661 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  Y )  e.  RR )
3231recnd 9632 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  Y )  e.  CC )
3311, 6, 7abvne0 17324 . . . 4  |-  ( ( F  e.  A  /\  Y  e.  B  /\  Y  =/=  .0.  )  -> 
( F `  Y
)  =/=  0 )
341, 4, 5, 33syl3anc 1228 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  Y )  =/=  0 )
3529, 32, 34divrecd 10333 . 2  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  (
( F `  X
)  /  ( F `
 Y ) )  =  ( ( F `
 X )  x.  ( 1  /  ( F `  Y )
) ) )
3618, 26, 353eqtr4d 2518 1  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  ( X  ./  Y ) )  =  ( ( F `  X )  /  ( F `  Y )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5593  (class class class)co 6294   RRcr 9501   0cc0 9502   1c1 9503    x. cmul 9507    / cdiv 10216   Basecbs 14502   .rcmulr 14568   0gc0g 14707  Unitcui 17137   invrcinvr 17169  /rcdvr 17180   DivRingcdr 17244  AbsValcabv 17313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-tpos 6965  df-recs 7052  df-rdg 7086  df-er 7321  df-map 7432  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-ico 11545  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-0g 14709  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-grp 15906  df-minusg 15907  df-mgp 16991  df-ur 17003  df-ring 17049  df-oppr 17121  df-dvdsr 17139  df-unit 17140  df-invr 17170  df-dvr 17181  df-drng 17246  df-abv 17314
This theorem is referenced by:  ostthlem1  23655
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