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Theorem divdiv1 10146
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 9444 . . . . 5  |-  1  e.  CC
2 ax-1ne0 9455 . . . . 5  |-  1  =/=  0
31, 2pm3.2i 455 . . . 4  |-  ( 1  e.  CC  /\  1  =/=  0 )
4 divdivdiv 10136 . . . 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 684 . . 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 1183 . 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 10127 . . . . 5  |-  ( C  e.  CC  ->  ( C  /  1 )  =  C )
87oveq2d 6209 . . . 4  |-  ( C  e.  CC  ->  (
( A  /  B
)  /  ( C  /  1 ) )  =  ( ( A  /  B )  /  C ) )
98ad2antrl 727 . . 3  |-  ( ( ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  B )  /  ( C  /  1 ) )  =  ( ( A  /  B )  /  C ) )
1093adant1 1006 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  B )  /  ( C  /  1 ) )  =  ( ( A  /  B )  /  C ) )
11 mulid1 9487 . . . 4  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
1211oveq1d 6208 . . 3  |-  ( A  e.  CC  ->  (
( A  x.  1 )  /  ( B  x.  C ) )  =  ( A  / 
( B  x.  C
) ) )
13123ad2ant1 1009 . 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 2500 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644  (class class class)co 6193   CCcc 9384   0cc0 9386   1c1 9387    x. cmul 9391    / cdiv 10097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098
This theorem is referenced by:  recdiv2  10148  divdiv1d  10242  fldiv2  11810  sin01bnd  13580  pythagtriplem12  14004  pythagtriplem14  14006  pythagtriplem16  14008  coseq1  22110  efeq1  22111  ang180lem1  22331  atan1  22449  fsumdvdscom  22651  bposlem8  22756  rplogsumlem2  22860  dchrvmasum2lem  22871  dchrisum0lem2  22893  dchrisum0lem3  22894  mulogsum  22907  mulog2sumlem2  22910  pntlemr  22977  pntlemf  22980  quad3  27440  wallispilem4  30004
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