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Theorem divassap 8472
Description: An associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
Assertion
Ref Expression
divassap  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( A  x.  B )  /  C
)  =  ( A  x.  ( B  /  C ) ) )

Proof of Theorem divassap
StepHypRef Expression
1 recclap 8461 . . 3  |-  ( ( C  e.  CC  /\  C #  0 )  ->  (
1  /  C )  e.  CC )
2 mulass 7773 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  (
1  /  C )  e.  CC )  -> 
( ( A  x.  B )  x.  (
1  /  C ) )  =  ( A  x.  ( B  x.  ( 1  /  C
) ) ) )
31, 2syl3an3 1252 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( A  x.  B )  x.  (
1  /  C ) )  =  ( A  x.  ( B  x.  ( 1  /  C
) ) ) )
4 mulcl 7769 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
543adant3 1002 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( A  x.  B
)  e.  CC )
6 simp3l 1010 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  C  e.  CC )
7 simp3r 1011 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  C #  0 )
8 divrecap 8470 . . 3  |-  ( ( ( A  x.  B
)  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  (
( A  x.  B
)  /  C )  =  ( ( A  x.  B )  x.  ( 1  /  C
) ) )
95, 6, 7, 8syl3anc 1217 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( A  x.  B )  /  C
)  =  ( ( A  x.  B )  x.  ( 1  /  C ) ) )
10 simp2 983 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  B  e.  CC )
11 divrecap 8470 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( B  /  C )  =  ( B  x.  (
1  /  C ) ) )
1210, 6, 7, 11syl3anc 1217 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( B  /  C
)  =  ( B  x.  ( 1  /  C ) ) )
1312oveq2d 5796 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( A  x.  ( B  /  C ) )  =  ( A  x.  ( B  x.  (
1  /  C ) ) ) )
143, 9, 133eqtr4d 2183 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( A  x.  B )  /  C
)  =  ( A  x.  ( B  /  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963    = wceq 1332    e. wcel 1481   class class class wbr 3935  (class class class)co 5780   CCcc 7640   0cc0 7642   1c1 7643    x. cmul 7647   # cap 8365    / cdiv 8454
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458  ax-cnex 7733  ax-resscn 7734  ax-1cn 7735  ax-1re 7736  ax-icn 7737  ax-addcl 7738  ax-addrcl 7739  ax-mulcl 7740  ax-mulrcl 7741  ax-addcom 7742  ax-mulcom 7743  ax-addass 7744  ax-mulass 7745  ax-distr 7746  ax-i2m1 7747  ax-0lt1 7748  ax-1rid 7749  ax-0id 7750  ax-rnegex 7751  ax-precex 7752  ax-cnre 7753  ax-pre-ltirr 7754  ax-pre-ltwlin 7755  ax-pre-lttrn 7756  ax-pre-apti 7757  ax-pre-ltadd 7758  ax-pre-mulgt0 7759  ax-pre-mulext 7760
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-br 3936  df-opab 3996  df-id 4221  df-po 4224  df-iso 4225  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-iota 5094  df-fun 5131  df-fv 5137  df-riota 5736  df-ov 5783  df-oprab 5784  df-mpo 5785  df-pnf 7824  df-mnf 7825  df-xr 7826  df-ltxr 7827  df-le 7828  df-sub 7957  df-neg 7958  df-reap 8359  df-ap 8366  df-div 8455
This theorem is referenced by:  div23ap  8473  div32ap  8474  divmulassap  8477  divmulasscomap  8478  divassapzi  8544  divassapi  8550  divassapd  8608  zdivmul  9163  efi4p  11453
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