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Theorem divdivap1 8497
Description: Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
Assertion
Ref Expression
divdivap1  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( ( A  /  B )  /  C )  =  ( A  /  ( B  x.  C ) ) )

Proof of Theorem divdivap1
StepHypRef Expression
1 ax-1cn 7727 . . . . 5  |-  1  e.  CC
2 1ap0 8366 . . . . 5  |-  1 #  0
31, 2pm3.2i 270 . . . 4  |-  ( 1  e.  CC  /\  1 #  0 )
4 divdivdivap 8487 . . . 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 434 . . 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 1176 . 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 8477 . . . . 5  |-  ( C  e.  CC  ->  ( C  /  1 )  =  C )
87oveq2d 5790 . . . 4  |-  ( C  e.  CC  ->  (
( A  /  B
)  /  ( C  /  1 ) )  =  ( ( A  /  B )  /  C ) )
98ad2antrl 481 . . 3  |-  ( ( ( B  e.  CC  /\  B #  0 )  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( ( A  /  B )  / 
( C  /  1
) )  =  ( ( A  /  B
)  /  C ) )
1093adant1 999 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( ( A  /  B )  / 
( C  /  1
) )  =  ( ( A  /  B
)  /  C ) )
11 mulid1 7777 . . . 4  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
1211oveq1d 5789 . . 3  |-  ( A  e.  CC  ->  (
( A  x.  1 )  /  ( B  x.  C ) )  =  ( A  / 
( B  x.  C
) ) )
13123ad2ant1 1002 . 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 2180 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 set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    = wceq 1331    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7632   0cc0 7634   1c1 7635    x. cmul 7639   # cap 8357    / cdiv 8446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7725  ax-resscn 7726  ax-1cn 7727  ax-1re 7728  ax-icn 7729  ax-addcl 7730  ax-addrcl 7731  ax-mulcl 7732  ax-mulrcl 7733  ax-addcom 7734  ax-mulcom 7735  ax-addass 7736  ax-mulass 7737  ax-distr 7738  ax-i2m1 7739  ax-0lt1 7740  ax-1rid 7741  ax-0id 7742  ax-rnegex 7743  ax-precex 7744  ax-cnre 7745  ax-pre-ltirr 7746  ax-pre-ltwlin 7747  ax-pre-lttrn 7748  ax-pre-apti 7749  ax-pre-ltadd 7750  ax-pre-mulgt0 7751  ax-pre-mulext 7752
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7816  df-mnf 7817  df-xr 7818  df-ltxr 7819  df-le 7820  df-sub 7949  df-neg 7950  df-reap 8351  df-ap 8358  df-div 8447
This theorem is referenced by:  recdivap2  8499  divdivap1d  8596  sin01bnd  11477
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