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Theorem divrec 9527
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 956 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  B  e.  CC )
2 simp1 955 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  A  e.  CC )
3 reccl 9518 . . . . 5  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  CC )
433adant1 973 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
1  /  B )  e.  CC )
51, 2, 4mul12d 9108 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( A  x.  ( 1  /  B
) ) )  =  ( A  x.  ( B  x.  ( 1  /  B ) ) ) )
6 recid 9525 . . . . 5  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( B  x.  (
1  /  B ) )  =  1 )
763adant1 973 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( 1  /  B ) )  =  1 )
87oveq2d 5958 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  ( B  x.  ( 1  /  B
) ) )  =  ( A  x.  1 ) )
92mulid1d 8939 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  1 )  =  A )
105, 8, 93eqtrd 2394 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( A  x.  ( 1  /  B
) ) )  =  A )
112, 4mulcld 8942 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  ( 1  /  B ) )  e.  CC )
12 3simpc 954 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  e.  CC  /\  B  =/=  0 ) )
13 divmul 9514 . . 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 1182 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  B
)  =  ( A  x.  ( 1  /  B ) )  <->  ( B  x.  ( A  x.  (
1  /  B ) ) )  =  A ) )
1510, 14mpbird 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521  (class class class)co 5942   CCcc 8822   0cc0 8824   1c1 8825    x. cmul 8829    / cdiv 9510
This theorem is referenced by:  divrec2  9528  divass  9529  divdir  9534  divid  9538  divneg  9542  rec11  9545  divdiv32  9555  redivcl  9566  divreczi  9585  divrecd  9626  ltdiv2  9728  lediv2  9733  qdivcl  10426  expdiv  11242  0.999...  12428  efsub  12471  efival  12523  ef01bndlem  12555  cos01bnd  12557  rpnnen2lem11  12594  prmreclem5  13058  divcn  18469  divccn  18474  subfaclim  24123  lediv2aALT  24417  heiborlem7  25864  ofdivrec  26866  stoweidlem36  27108  wallispilem4  27140
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-po 4393  df-so 4394  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-riota 6388  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511
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