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Theorem absdiv 11827
Description: Absolute value distributes over division. (Contributed by NM, 27-Apr-2005.)
Assertion
Ref Expression
absdiv  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( abs `  ( A  /  B ) )  =  ( ( abs `  A
)  /  ( abs `  B ) ) )

Proof of Theorem absdiv
StepHypRef Expression
1 divcl 9475 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  e.  CC )
2 abscl 11810 . . . . 5  |-  ( ( A  /  B )  e.  CC  ->  ( abs `  ( A  /  B ) )  e.  RR )
31, 2syl 15 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( abs `  ( A  /  B ) )  e.  RR )
43recnd 8906 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( abs `  ( A  /  B ) )  e.  CC )
5 absrpcl 11820 . . . . 5  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( abs `  B
)  e.  RR+ )
653adant1 973 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( abs `  B )  e.  RR+ )
76rpcnd 10439 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( abs `  B )  e.  CC )
86rpne0d 10442 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( abs `  B )  =/=  0 )
94, 7, 8divcan4d 9587 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( ( abs `  ( A  /  B ) )  x.  ( abs `  B
) )  /  ( abs `  B ) )  =  ( abs `  ( A  /  B ) ) )
10 simp2 956 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  B  e.  CC )
11 absmul 11826 . . . . 5  |-  ( ( ( A  /  B
)  e.  CC  /\  B  e.  CC )  ->  ( abs `  (
( A  /  B
)  x.  B ) )  =  ( ( abs `  ( A  /  B ) )  x.  ( abs `  B
) ) )
121, 10, 11syl2anc 642 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( abs `  ( ( A  /  B )  x.  B ) )  =  ( ( abs `  ( A  /  B ) )  x.  ( abs `  B
) ) )
13 divcan1 9478 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  B
)  x.  B )  =  A )
1413fveq2d 5567 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( abs `  ( ( A  /  B )  x.  B ) )  =  ( abs `  A
) )
1512, 14eqtr3d 2350 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( abs `  ( A  /  B ) )  x.  ( abs `  B
) )  =  ( abs `  A ) )
1615oveq1d 5915 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( ( abs `  ( A  /  B ) )  x.  ( abs `  B
) )  /  ( abs `  B ) )  =  ( ( abs `  A )  /  ( abs `  B ) ) )
179, 16eqtr3d 2350 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( abs `  ( A  /  B ) )  =  ( ( abs `  A
)  /  ( abs `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   ` cfv 5292  (class class class)co 5900   CCcc 8780   RRcr 8781   0cc0 8782    x. cmul 8787    / cdiv 9468   RR+crp 10401   abscabs 11766
This theorem is referenced by:  absexpz  11837  abs1m  11866  absdivzi  11937  absdivd  11984  efif1olem4  19960  log2cnv  20293  qqhnm  23569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-seq 11094  df-exp 11152  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768
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