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Theorem absmul 11773
Description: Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)
Assertion
Ref Expression
absmul  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  x.  B )
)  =  ( ( abs `  A )  x.  ( abs `  B
) ) )

Proof of Theorem absmul
StepHypRef Expression
1 cjmul 11621 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  x.  B )
)  =  ( ( * `  A )  x.  ( * `  B ) ) )
21oveq2d 5835 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  x.  (
* `  ( A  x.  B ) ) )  =  ( ( A  x.  B )  x.  ( ( * `  A )  x.  (
* `  B )
) ) )
3 simpl 445 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
4 simpr 449 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
53cjcld 11675 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  A
)  e.  CC )
64cjcld 11675 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  B
)  e.  CC )
73, 4, 5, 6mul4d 9019 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  x.  (
( * `  A
)  x.  ( * `
 B ) ) )  =  ( ( A  x.  ( * `
 A ) )  x.  ( B  x.  ( * `  B
) ) ) )
82, 7eqtrd 2316 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  x.  (
* `  ( A  x.  B ) ) )  =  ( ( A  x.  ( * `  A ) )  x.  ( B  x.  (
* `  B )
) ) )
98fveq2d 5489 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( sqr `  (
( A  x.  B
)  x.  ( * `
 ( A  x.  B ) ) ) )  =  ( sqr `  ( ( A  x.  ( * `  A
) )  x.  ( B  x.  ( * `  B ) ) ) ) )
10 cjmulrcl 11623 . . . . 5  |-  ( A  e.  CC  ->  ( A  x.  ( * `  A ) )  e.  RR )
11 cjmulge0 11625 . . . . 5  |-  ( A  e.  CC  ->  0  <_  ( A  x.  (
* `  A )
) )
1210, 11jca 520 . . . 4  |-  ( A  e.  CC  ->  (
( A  x.  (
* `  A )
)  e.  RR  /\  0  <_  ( A  x.  ( * `  A
) ) ) )
13 cjmulrcl 11623 . . . . 5  |-  ( B  e.  CC  ->  ( B  x.  ( * `  B ) )  e.  RR )
14 cjmulge0 11625 . . . . 5  |-  ( B  e.  CC  ->  0  <_  ( B  x.  (
* `  B )
) )
1513, 14jca 520 . . . 4  |-  ( B  e.  CC  ->  (
( B  x.  (
* `  B )
)  e.  RR  /\  0  <_  ( B  x.  ( * `  B
) ) ) )
16 sqrmul 11739 . . . 4  |-  ( ( ( ( A  x.  ( * `  A
) )  e.  RR  /\  0  <_  ( A  x.  ( * `  A
) ) )  /\  ( ( B  x.  ( * `  B
) )  e.  RR  /\  0  <_  ( B  x.  ( * `  B
) ) ) )  ->  ( sqr `  (
( A  x.  (
* `  A )
)  x.  ( B  x.  ( * `  B ) ) ) )  =  ( ( sqr `  ( A  x.  ( * `  A ) ) )  x.  ( sqr `  ( B  x.  ( * `  B ) ) ) ) )
1712, 15, 16syl2an 465 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( sqr `  (
( A  x.  (
* `  A )
)  x.  ( B  x.  ( * `  B ) ) ) )  =  ( ( sqr `  ( A  x.  ( * `  A ) ) )  x.  ( sqr `  ( B  x.  ( * `  B ) ) ) ) )
189, 17eqtrd 2316 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( sqr `  (
( A  x.  B
)  x.  ( * `
 ( A  x.  B ) ) ) )  =  ( ( sqr `  ( A  x.  ( * `  A ) ) )  x.  ( sqr `  ( B  x.  ( * `  B ) ) ) ) )
19 mulcl 8816 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
20 absval 11717 . . 3  |-  ( ( A  x.  B )  e.  CC  ->  ( abs `  ( A  x.  B ) )  =  ( sqr `  (
( A  x.  B
)  x.  ( * `
 ( A  x.  B ) ) ) ) )
2119, 20syl 17 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  x.  B )
)  =  ( sqr `  ( ( A  x.  B )  x.  (
* `  ( A  x.  B ) ) ) ) )
22 absval 11717 . . 3  |-  ( A  e.  CC  ->  ( abs `  A )  =  ( sqr `  ( A  x.  ( * `  A ) ) ) )
23 absval 11717 . . 3  |-  ( B  e.  CC  ->  ( abs `  B )  =  ( sqr `  ( B  x.  ( * `  B ) ) ) )
2422, 23oveqan12d 5838 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  A
)  x.  ( abs `  B ) )  =  ( ( sqr `  ( A  x.  ( * `  A ) ) )  x.  ( sqr `  ( B  x.  ( * `  B ) ) ) ) )
2518, 21, 243eqtr4d 2326 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  x.  B )
)  =  ( ( abs `  A )  x.  ( abs `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1628    e. wcel 1688   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   CCcc 8730   RRcr 8731   0cc0 8732    x. cmul 8737    <_ cle 8863   *ccj 11575   sqrcsqr 11712   abscabs 11713
This theorem is referenced by:  absdiv  11774  absexp  11783  absimle  11788  abstri  11808  absmuli  11881  absmuld  11930  ef01bndlem  12458  absmulgcd  12720  gcdmultiplez  12724  absabv  16423  iblabs  19177  pige3  19879  atantayl  20227  efrlim  20258  lgslem3  20531  mul2sq  20598  cnnv  21237  bcsiALT  21750  nmcfnexi  22623
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-sup 7189  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-seq 11041  df-exp 11099  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715
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