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Theorem divrecap 8097
Description: Relationship between division and reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
Assertion
Ref Expression
divrecap  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  /  B )  =  ( A  x.  (
1  /  B ) ) )

Proof of Theorem divrecap
StepHypRef Expression
1 simp2 942 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  B  e.  CC )
2 simp1 941 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  A  e.  CC )
3 recclap 8088 . . . . 5  |-  ( ( B  e.  CC  /\  B #  0 )  ->  (
1  /  B )  e.  CC )
433adant1 959 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  (
1  /  B )  e.  CC )
51, 2, 4mul12d 7581 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( B  x.  ( A  x.  ( 1  /  B
) ) )  =  ( A  x.  ( B  x.  ( 1  /  B ) ) ) )
6 recidap 8095 . . . . 5  |-  ( ( B  e.  CC  /\  B #  0 )  ->  ( B  x.  ( 1  /  B ) )  =  1 )
763adant1 959 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( B  x.  ( 1  /  B ) )  =  1 )
87oveq2d 5631 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  x.  ( B  x.  ( 1  /  B
) ) )  =  ( A  x.  1 ) )
92mulid1d 7452 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  x.  1 )  =  A )
105, 8, 93eqtrd 2121 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( B  x.  ( A  x.  ( 1  /  B
) ) )  =  A )
112, 4mulcld 7455 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  x.  ( 1  /  B ) )  e.  CC )
12 3simpc 940 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( B  e.  CC  /\  B #  0 ) )
13 divmulap 8084 . . 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 1172 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  (
( A  /  B
)  =  ( A  x.  ( 1  /  B ) )  <->  ( B  x.  ( A  x.  (
1  /  B ) ) )  =  A ) )
1510, 14mpbird 165 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    /\ wa 102    <-> wb 103    /\ w3a 922    = wceq 1287    e. wcel 1436   class class class wbr 3822  (class class class)co 5615   CCcc 7295   0cc0 7297   1c1 7298    x. cmul 7302   # cap 8002    / cdiv 8081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-cnex 7383  ax-resscn 7384  ax-1cn 7385  ax-1re 7386  ax-icn 7387  ax-addcl 7388  ax-addrcl 7389  ax-mulcl 7390  ax-mulrcl 7391  ax-addcom 7392  ax-mulcom 7393  ax-addass 7394  ax-mulass 7395  ax-distr 7396  ax-i2m1 7397  ax-0lt1 7398  ax-1rid 7399  ax-0id 7400  ax-rnegex 7401  ax-precex 7402  ax-cnre 7403  ax-pre-ltirr 7404  ax-pre-ltwlin 7405  ax-pre-lttrn 7406  ax-pre-apti 7407  ax-pre-ltadd 7408  ax-pre-mulgt0 7409  ax-pre-mulext 7410
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-id 4096  df-po 4099  df-iso 4100  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-iota 4948  df-fun 4985  df-fv 4991  df-riota 5571  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-pnf 7471  df-mnf 7472  df-xr 7473  df-ltxr 7474  df-le 7475  df-sub 7602  df-neg 7603  df-reap 7996  df-ap 8003  df-div 8082
This theorem is referenced by:  divrecap2  8098  divassap  8099  divdirap  8106  dividap  8110  divnegap  8115  rec11ap  8119  divdiv32ap  8129  redivclap  8140  divrecapzi  8159  divrecapi  8166  divrecapd  8201  expdivap  9926
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