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Theorem div12ap 8571
Description: A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
Assertion
Ref Expression
div12ap  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( A  x.  ( B  /  C ) )  =  ( B  x.  ( A  /  C
) ) )

Proof of Theorem div12ap
StepHypRef Expression
1 divclap 8555 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( B  /  C )  e.  CC )
213expb 1186 . . . 4  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( B  /  C )  e.  CC )
3 mulcom 7863 . . . 4  |-  ( ( A  e.  CC  /\  ( B  /  C
)  e.  CC )  ->  ( A  x.  ( B  /  C
) )  =  ( ( B  /  C
)  x.  A ) )
42, 3sylan2 284 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) )  ->  ( A  x.  ( B  /  C
) )  =  ( ( B  /  C
)  x.  A ) )
543impb 1181 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( A  x.  ( B  /  C ) )  =  ( ( B  /  C )  x.  A ) )
6 div13ap 8570 . . 3  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C #  0 )  /\  A  e.  CC )  ->  ( ( B  /  C )  x.  A
)  =  ( ( A  /  C )  x.  B ) )
763comr 1193 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( B  /  C )  x.  A
)  =  ( ( A  /  C )  x.  B ) )
8 divclap 8555 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( A  /  C )  e.  CC )
983expb 1186 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( A  /  C )  e.  CC )
10 mulcom 7863 . . . . 5  |-  ( ( ( A  /  C
)  e.  CC  /\  B  e.  CC )  ->  ( ( A  /  C )  x.  B
)  =  ( B  x.  ( A  /  C ) ) )
119, 10sylan 281 . . . 4  |-  ( ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  /\  B  e.  CC )  ->  ( ( A  /  C )  x.  B )  =  ( B  x.  ( A  /  C ) ) )
12113impa 1177 . . 3  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 )  /\  B  e.  CC )  ->  ( ( A  /  C )  x.  B
)  =  ( B  x.  ( A  /  C ) ) )
13123com23 1191 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( A  /  C )  x.  B
)  =  ( B  x.  ( A  /  C ) ) )
145, 7, 133eqtrd 2194 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( A  x.  ( B  /  C ) )  =  ( B  x.  ( A  /  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963    = wceq 1335    e. wcel 2128   class class class wbr 3967  (class class class)co 5826   CCcc 7732   0cc0 7734    x. cmul 7739   # cap 8460    / cdiv 8549
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-mulrcl 7833  ax-addcom 7834  ax-mulcom 7835  ax-addass 7836  ax-mulass 7837  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-1rid 7841  ax-0id 7842  ax-rnegex 7843  ax-precex 7844  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-apti 7849  ax-pre-ltadd 7850  ax-pre-mulgt0 7851  ax-pre-mulext 7852
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-id 4255  df-po 4258  df-iso 4259  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-iota 5137  df-fun 5174  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-reap 8454  df-ap 8461  df-div 8550
This theorem is referenced by:  div2negap  8612  div12apd  8704  efival  11640  cos01bnd  11666  cos01gt0  11670  sincosq4sgn  13220
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