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Theorem divrec 9725
Description: Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
divrec  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( A  x.  (
1  /  B ) ) )

Proof of Theorem divrec
StepHypRef Expression
1 simp2 959 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  B  e.  CC )
2 simp1 958 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  A  e.  CC )
3 reccl 9716 . . . . 5  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  CC )
433adant1 976 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
1  /  B )  e.  CC )
51, 2, 4mul12d 9306 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( A  x.  ( 1  /  B
) ) )  =  ( A  x.  ( B  x.  ( 1  /  B ) ) ) )
6 recid 9723 . . . . 5  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( B  x.  (
1  /  B ) )  =  1 )
763adant1 976 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( 1  /  B ) )  =  1 )
87oveq2d 6126 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  ( B  x.  ( 1  /  B
) ) )  =  ( A  x.  1 ) )
92mulid1d 9136 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  1 )  =  A )
105, 8, 93eqtrd 2478 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( A  x.  ( 1  /  B
) ) )  =  A )
112, 4mulcld 9139 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  ( 1  /  B ) )  e.  CC )
12 3simpc 957 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  e.  CC  /\  B  =/=  0 ) )
13 divmul 9712 . . 3  |-  ( ( A  e.  CC  /\  ( A  x.  (
1  /  B ) )  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( ( A  /  B )  =  ( A  x.  (
1  /  B ) )  <->  ( B  x.  ( A  x.  (
1  /  B ) ) )  =  A ) )
142, 11, 12, 13syl3anc 1185 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  B
)  =  ( A  x.  ( 1  /  B ) )  <->  ( B  x.  ( A  x.  (
1  /  B ) ) )  =  A ) )
1510, 14mpbird 225 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( A  x.  (
1  /  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605  (class class class)co 6110   CCcc 9019   0cc0 9021   1c1 9022    x. cmul 9026    / cdiv 9708
This theorem is referenced by:  divrec2  9726  divass  9727  divdir  9732  divid  9736  divneg  9740  rec11  9743  divdiv32  9753  redivcl  9764  divreczi  9783  divrecd  9824  ltdiv2  9926  lediv2  9931  qdivcl  10626  expdiv  11461  0.999...  12689  efsub  12732  efival  12784  ef01bndlem  12816  cos01bnd  12818  rpnnen2lem11  12855  prmreclem5  13319  divcn  18929  divccn  18934  subfaclim  24905  lediv2aALT  25148  heiborlem7  26564  ofdivrec  27558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-po 4532  df-so 4533  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-riota 6578  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709
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