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Theorem divrec 10212
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 992 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  B  e.  CC )
2 simp1 991 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  A  e.  CC )
3 reccl 10203 . . . . 5  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  CC )
433adant1 1009 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
1  /  B )  e.  CC )
51, 2, 4mul12d 9777 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( A  x.  ( 1  /  B
) ) )  =  ( A  x.  ( B  x.  ( 1  /  B ) ) ) )
6 recid 10210 . . . . 5  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( B  x.  (
1  /  B ) )  =  1 )
763adant1 1009 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( 1  /  B ) )  =  1 )
87oveq2d 6291 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  ( B  x.  ( 1  /  B
) ) )  =  ( A  x.  1 ) )
92mulid1d 9602 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  1 )  =  A )
105, 8, 93eqtrd 2505 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( A  x.  ( 1  /  B
) ) )  =  A )
112, 4mulcld 9605 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  ( 1  /  B ) )  e.  CC )
12 3simpc 990 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  e.  CC  /\  B  =/=  0 ) )
13 divmul 10199 . . 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 1223 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  B
)  =  ( A  x.  ( 1  /  B ) )  <->  ( B  x.  ( A  x.  (
1  /  B ) ) )  =  A ) )
1510, 14mpbird 232 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( A  x.  (
1  /  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655  (class class class)co 6275   CCcc 9479   0cc0 9481   1c1 9482    x. cmul 9486    / cdiv 10195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196
This theorem is referenced by:  divrec2  10213  divass  10214  divdir  10219  divid  10223  divneg  10228  rec11  10231  divdiv32  10241  redivcl  10252  divreczi  10271  divrecd  10312  ltdiv2  10419  lediv2  10424  qdivcl  11192  expdiv  12171  0.999...  13642  efsub  13685  efival  13737  ef01bndlem  13769  cos01bnd  13771  rpnnen2lem11  13808  prmreclem5  14286  divcn  21100  divccn  21105  subfaclim  28258  lediv2aALT  28504  heiborlem7  29903  ofdivrec  30786
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