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Theorem abvdiv 15602
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 731 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  F  e.  A )
2 simpr1 961 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  X  e.  B )
3 simpll 730 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  R  e.  DivRing )
4 simpr2 962 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  Y  e.  B )
5 simpr3 963 . . . . 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 2283 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
96, 7, 8drnginvrcl 15529 . . . . 5  |-  ( ( R  e.  DivRing  /\  Y  e.  B  /\  Y  =/= 
.0.  )  ->  (
( invr `  R ) `  Y )  e.  B
)
103, 4, 5, 9syl3anc 1182 . . . 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 2283 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
1311, 6, 12abvmul 15594 . . . 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 1182 . . 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 15601 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  ( ( invr `  R ) `  Y ) )  =  ( 1  /  ( F `  Y )
) )
16153adantr1 1114 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  ( ( invr `  R ) `  Y ) )  =  ( 1  /  ( F `  Y )
) )
1716oveq2d 5874 . . 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 2315 . 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 2283 . . . . . . 7  |-  (Unit `  R )  =  (Unit `  R )
206, 19, 7drngunit 15517 . . . . . 6  |-  ( R  e.  DivRing  ->  ( Y  e.  (Unit `  R )  <->  ( Y  e.  B  /\  Y  =/=  .0.  ) ) )
213, 20syl 15 . . . . 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 888 . . . 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 15467 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  (Unit `  R
) )  ->  ( X  ./  Y )  =  ( X ( .r
`  R ) ( ( invr `  R
) `  Y )
) )
252, 22, 24syl2anc 642 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( X  ./  Y )  =  ( X ( .r
`  R ) ( ( invr `  R
) `  Y )
) )
2625fveq2d 5529 . 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 15589 . . . . 5  |-  ( ( F  e.  A  /\  X  e.  B )  ->  ( F `  X
)  e.  RR )
281, 2, 27syl2anc 642 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  X )  e.  RR )
2928recnd 8861 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  X )  e.  CC )
3011, 6abvcl 15589 . . . . 5  |-  ( ( F  e.  A  /\  Y  e.  B )  ->  ( F `  Y
)  e.  RR )
311, 4, 30syl2anc 642 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  Y )  e.  RR )
3231recnd 8861 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  Y )  e.  CC )
3311, 6, 7abvne0 15592 . . . 4  |-  ( ( F  e.  A  /\  Y  e.  B  /\  Y  =/=  .0.  )  -> 
( F `  Y
)  =/=  0 )
341, 4, 5, 33syl3anc 1182 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  Y )  =/=  0 )
3529, 32, 34divrecd 9539 . 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 2325 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    / cdiv 9423   Basecbs 13148   .rcmulr 13209   0gc0g 13400  Unitcui 15421   invrcinvr 15453  /rcdvr 15464   DivRingcdr 15512  AbsValcabv 15581
This theorem is referenced by:  ostthlem1  20776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-ico 10662  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-abv 15582
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