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Theorem abvdiv 18065
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 762 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  F  e.  A )
2 simpr1 1014 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  X  e.  B )
3 simpll 760 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  R  e.  DivRing )
4 simpr2 1015 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  Y  e.  B )
5 simpr3 1016 . . . . 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 2451 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
96, 7, 8drnginvrcl 17992 . . . . 5  |-  ( ( R  e.  DivRing  /\  Y  e.  B  /\  Y  =/= 
.0.  )  ->  (
( invr `  R ) `  Y )  e.  B
)
103, 4, 5, 9syl3anc 1268 . . . 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 2451 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
1311, 6, 12abvmul 18057 . . . 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 1268 . . 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 18064 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  ( ( invr `  R ) `  Y ) )  =  ( 1  /  ( F `  Y )
) )
16153adantr1 1167 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  ( ( invr `  R ) `  Y ) )  =  ( 1  /  ( F `  Y )
) )
1716oveq2d 6306 . . 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 2485 . 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 2451 . . . . . . 7  |-  (Unit `  R )  =  (Unit `  R )
206, 19, 7drngunit 17980 . . . . . 6  |-  ( R  e.  DivRing  ->  ( Y  e.  (Unit `  R )  <->  ( Y  e.  B  /\  Y  =/=  .0.  ) ) )
213, 20syl 17 . . . . 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 933 . . . 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 17913 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  (Unit `  R
) )  ->  ( X  ./  Y )  =  ( X ( .r
`  R ) ( ( invr `  R
) `  Y )
) )
252, 22, 24syl2anc 667 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( X  ./  Y )  =  ( X ( .r
`  R ) ( ( invr `  R
) `  Y )
) )
2625fveq2d 5869 . 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 18052 . . . . 5  |-  ( ( F  e.  A  /\  X  e.  B )  ->  ( F `  X
)  e.  RR )
281, 2, 27syl2anc 667 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  X )  e.  RR )
2928recnd 9669 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  X )  e.  CC )
3011, 6abvcl 18052 . . . . 5  |-  ( ( F  e.  A  /\  Y  e.  B )  ->  ( F `  Y
)  e.  RR )
311, 4, 30syl2anc 667 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  Y )  e.  RR )
3231recnd 9669 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  Y )  e.  CC )
3311, 6, 7abvne0 18055 . . . 4  |-  ( ( F  e.  A  /\  Y  e.  B  /\  Y  =/=  .0.  )  -> 
( F `  Y
)  =/=  0 )
341, 4, 5, 33syl3anc 1268 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  Y )  =/=  0 )
3529, 32, 34divrecd 10386 . 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 2495 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540    x. cmul 9544    / cdiv 10269   Basecbs 15121   .rcmulr 15191   0gc0g 15338  Unitcui 17867   invrcinvr 17899  /rcdvr 17910   DivRingcdr 17975  AbsValcabv 18044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-tpos 6973  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-ico 11641  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-minusg 16674  df-mgp 17724  df-ur 17736  df-ring 17782  df-oppr 17851  df-dvdsr 17869  df-unit 17870  df-invr 17900  df-dvr 17911  df-drng 17977  df-abv 18045
This theorem is referenced by:  ostthlem1  24465
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