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Theorem divdiv1 10325
Description: Division into a fraction. (Contributed by NM, 31-Dec-2007.)
Assertion
Ref Expression
divdiv1  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  B )  /  C
)  =  ( A  /  ( B  x.  C ) ) )

Proof of Theorem divdiv1
StepHypRef Expression
1 ax-1cn 9604 . . . . 5  |-  1  e.  CC
2 ax-1ne0 9615 . . . . 5  |-  1  =/=  0
31, 2pm3.2i 456 . . . 4  |-  ( 1  e.  CC  /\  1  =/=  0 )
4 divdivdiv 10315 . . . 4  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( 1  e.  CC  /\  1  =/=  0 ) ) )  ->  (
( A  /  B
)  /  ( C  /  1 ) )  =  ( ( A  x.  1 )  / 
( B  x.  C
) ) )
53, 4mpanr2 688 . . 3  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  B )  /  ( C  /  1 ) )  =  ( ( A  x.  1 )  / 
( B  x.  C
) ) )
653impa 1200 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  B )  /  ( C  /  1 ) )  =  ( ( A  x.  1 )  / 
( B  x.  C
) ) )
7 div1 10306 . . . . 5  |-  ( C  e.  CC  ->  ( C  /  1 )  =  C )
87oveq2d 6321 . . . 4  |-  ( C  e.  CC  ->  (
( A  /  B
)  /  ( C  /  1 ) )  =  ( ( A  /  B )  /  C ) )
98ad2antrl 732 . . 3  |-  ( ( ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  B )  /  ( C  /  1 ) )  =  ( ( A  /  B )  /  C ) )
1093adant1 1023 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  B )  /  ( C  /  1 ) )  =  ( ( A  /  B )  /  C ) )
11 mulid1 9647 . . . 4  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
1211oveq1d 6320 . . 3  |-  ( A  e.  CC  ->  (
( A  x.  1 )  /  ( B  x.  C ) )  =  ( A  / 
( B  x.  C
) ) )
13123ad2ant1 1026 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  x.  1 )  /  ( B  x.  C )
)  =  ( A  /  ( B  x.  C ) ) )
146, 10, 133eqtr3d 2471 1  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  B )  /  C
)  =  ( A  /  ( B  x.  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614  (class class class)co 6305   CCcc 9544   0cc0 9546   1c1 9547    x. cmul 9551    / cdiv 10276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277
This theorem is referenced by:  recdiv2  10327  divdiv1d  10421  fldiv2  12094  sin01bnd  14238  pythagtriplem12  14775  pythagtriplem14  14777  pythagtriplem16  14779  coseq1  23475  efeq1  23476  ang180lem1  23736  atan1  23852  fsumdvdscom  24112  bposlem8  24217  rplogsumlem2  24321  dchrvmasum2lem  24332  dchrisum0lem2  24354  dchrisum0lem3  24355  mulogsum  24368  mulog2sumlem2  24371  pntlemr  24438  pntlemf  24441  quad3  30310  wallispilem4  37870  dirkertrigeqlem3  37902  dirkercncflem1  37905  fourierswlem  38034  dignn0flhalflem2  40048  dignn0ehalf  40049
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