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Theorem abvdiv 15598
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 2284 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
96, 7, 8drnginvrcl 15525 . . . . 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 2284 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
1311, 6, 12abvmul 15590 . . . 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 15597 . . . . 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 5836 . . 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 2316 . 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 2284 . . . . . . 7  |-  (Unit `  R )  =  (Unit `  R )
206, 19, 7drngunit 15513 . . . . . 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 15463 . . . 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 5490 . 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 15585 . . . . 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 8857 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  X )  e.  CC )
3011, 6abvcl 15585 . . . . 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 8857 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  Y )  e.  CC )
3311, 6, 7abvne0 15588 . . . 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 9535 . 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 2326 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 1685    =/= wne 2447   ` cfv 5221  (class class class)co 5820   RRcr 8732   0cc0 8733   1c1 8734    x. cmul 8738    / cdiv 9419   Basecbs 13144   .rcmulr 13205   0gc0g 13396  Unitcui 15417   invrcinvr 15449  /rcdvr 15460   DivRingcdr 15508  AbsValcabv 15577
This theorem is referenced by:  ostthlem1  20772
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-tpos 6196  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-ico 10658  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-0g 13400  df-mnd 14363  df-grp 14485  df-minusg 14486  df-mgp 15322  df-rng 15336  df-ur 15338  df-oppr 15401  df-dvdsr 15419  df-unit 15420  df-invr 15450  df-dvr 15461  df-drng 15510  df-abv 15578
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