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Theorem divrec 9654
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 958 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  B  e.  CC )
2 simp1 957 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  A  e.  CC )
3 reccl 9645 . . . . 5  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  CC )
433adant1 975 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
1  /  B )  e.  CC )
51, 2, 4mul12d 9235 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( A  x.  ( 1  /  B
) ) )  =  ( A  x.  ( B  x.  ( 1  /  B ) ) ) )
6 recid 9652 . . . . 5  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( B  x.  (
1  /  B ) )  =  1 )
763adant1 975 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( 1  /  B ) )  =  1 )
87oveq2d 6060 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  ( B  x.  ( 1  /  B
) ) )  =  ( A  x.  1 ) )
92mulid1d 9065 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  1 )  =  A )
105, 8, 93eqtrd 2444 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( A  x.  ( 1  /  B
) ) )  =  A )
112, 4mulcld 9068 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  ( 1  /  B ) )  e.  CC )
12 3simpc 956 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  e.  CC  /\  B  =/=  0 ) )
13 divmul 9641 . . 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 1184 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  B
)  =  ( A  x.  ( 1  /  B ) )  <->  ( B  x.  ( A  x.  (
1  /  B ) ) )  =  A ) )
1510, 14mpbird 224 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571  (class class class)co 6044   CCcc 8948   0cc0 8950   1c1 8951    x. cmul 8955    / cdiv 9637
This theorem is referenced by:  divrec2  9655  divass  9656  divdir  9661  divid  9665  divneg  9669  rec11  9672  divdiv32  9682  redivcl  9693  divreczi  9712  divrecd  9753  ltdiv2  9855  lediv2  9860  qdivcl  10555  expdiv  11389  0.999...  12617  efsub  12660  efival  12712  ef01bndlem  12744  cos01bnd  12746  rpnnen2lem11  12783  prmreclem5  13247  divcn  18855  divccn  18860  subfaclim  24831  lediv2aALT  25074  heiborlem7  26420  ofdivrec  27415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-po 4467  df-so 4468  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-riota 6512  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638
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