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Theorem divdivdivd 9578
Description: Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
div1d.1  |-  ( ph  ->  A  e.  CC )
divcld.2  |-  ( ph  ->  B  e.  CC )
divmuld.3  |-  ( ph  ->  C  e.  CC )
divmuldivd.4  |-  ( ph  ->  D  e.  CC )
divmuldivd.5  |-  ( ph  ->  B  =/=  0 )
divmuldivd.6  |-  ( ph  ->  D  =/=  0 )
divdivdivd.7  |-  ( ph  ->  C  =/=  0 )
Assertion
Ref Expression
divdivdivd  |-  ( ph  ->  ( ( A  /  B )  /  ( C  /  D ) )  =  ( ( A  x.  D )  / 
( B  x.  C
) ) )

Proof of Theorem divdivdivd
StepHypRef Expression
1 div1d.1 . 2  |-  ( ph  ->  A  e.  CC )
2 divcld.2 . . 3  |-  ( ph  ->  B  e.  CC )
3 divmuldivd.5 . . 3  |-  ( ph  ->  B  =/=  0 )
42, 3jca 520 . 2  |-  ( ph  ->  ( B  e.  CC  /\  B  =/=  0 ) )
5 divmuld.3 . . 3  |-  ( ph  ->  C  e.  CC )
6 divdivdivd.7 . . 3  |-  ( ph  ->  C  =/=  0 )
75, 6jca 520 . 2  |-  ( ph  ->  ( C  e.  CC  /\  C  =/=  0 ) )
8 divmuldivd.4 . . 3  |-  ( ph  ->  D  e.  CC )
9 divmuldivd.6 . . 3  |-  ( ph  ->  D  =/=  0 )
108, 9jca 520 . 2  |-  ( ph  ->  ( D  e.  CC  /\  D  =/=  0 ) )
11 divdivdiv 9456 . 2  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  (
( A  /  B
)  /  ( C  /  D ) )  =  ( ( A  x.  D )  / 
( B  x.  C
) ) )
121, 4, 7, 10, 11syl22anc 1185 1  |-  ( ph  ->  ( ( A  /  B )  /  ( C  /  D ) )  =  ( ( A  x.  D )  / 
( B  x.  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685    =/= wne 2447  (class class class)co 5819   CCcc 8730   0cc0 8732    x. cmul 8737    / cdiv 9418
This theorem is referenced by:  pcadd  12931  pnt  20757  wallispilem4  27216  stirlinglem4  27225  stirlinglem10  27231
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-iota 6252  df-riota 6299  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419
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