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Theorem abvdiv 15925
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 732 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  F  e.  A )
2 simpr1 963 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  X  e.  B )
3 simpll 731 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  R  e.  DivRing )
4 simpr2 964 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  Y  e.  B )
5 simpr3 965 . . . . 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 2436 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
96, 7, 8drnginvrcl 15852 . . . . 5  |-  ( ( R  e.  DivRing  /\  Y  e.  B  /\  Y  =/= 
.0.  )  ->  (
( invr `  R ) `  Y )  e.  B
)
103, 4, 5, 9syl3anc 1184 . . . 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 2436 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
1311, 6, 12abvmul 15917 . . . 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 1184 . . 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 15924 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  ( ( invr `  R ) `  Y ) )  =  ( 1  /  ( F `  Y )
) )
16153adantr1 1116 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  ( ( invr `  R ) `  Y ) )  =  ( 1  /  ( F `  Y )
) )
1716oveq2d 6097 . . 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 2468 . 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 2436 . . . . . . 7  |-  (Unit `  R )  =  (Unit `  R )
206, 19, 7drngunit 15840 . . . . . 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 889 . . . 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 15790 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  (Unit `  R
) )  ->  ( X  ./  Y )  =  ( X ( .r
`  R ) ( ( invr `  R
) `  Y )
) )
252, 22, 24syl2anc 643 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( X  ./  Y )  =  ( X ( .r
`  R ) ( ( invr `  R
) `  Y )
) )
2625fveq2d 5732 . 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 15912 . . . . 5  |-  ( ( F  e.  A  /\  X  e.  B )  ->  ( F `  X
)  e.  RR )
281, 2, 27syl2anc 643 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  X )  e.  RR )
2928recnd 9114 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  X )  e.  CC )
3011, 6abvcl 15912 . . . . 5  |-  ( ( F  e.  A  /\  Y  e.  B )  ->  ( F `  Y
)  e.  RR )
311, 4, 30syl2anc 643 . . . 4  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  Y )  e.  RR )
3231recnd 9114 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  Y )  e.  CC )
3311, 6, 7abvne0 15915 . . . 4  |-  ( ( F  e.  A  /\  Y  e.  B  /\  Y  =/=  .0.  )  -> 
( F `  Y
)  =/=  0 )
341, 4, 5, 33syl3anc 1184 . . 3  |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( F `  Y )  =/=  0 )
3529, 32, 34divrecd 9793 . 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 2478 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   ` cfv 5454  (class class class)co 6081   RRcr 8989   0cc0 8990   1c1 8991    x. cmul 8995    / cdiv 9677   Basecbs 13469   .rcmulr 13530   0gc0g 13723  Unitcui 15744   invrcinvr 15776  /rcdvr 15787   DivRingcdr 15835  AbsValcabv 15904
This theorem is referenced by:  ostthlem1  21321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-ico 10922  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-dvr 15788  df-drng 15837  df-abv 15905
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