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Theorem absmul 12905
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 12753 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  x.  B )
)  =  ( ( * `  A )  x.  ( * `  B ) ) )
21oveq2d 6219 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  x.  (
* `  ( A  x.  B ) ) )  =  ( ( A  x.  B )  x.  ( ( * `  A )  x.  (
* `  B )
) ) )
3 simpl 457 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
4 simpr 461 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
53cjcld 12807 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  A
)  e.  CC )
64cjcld 12807 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  B
)  e.  CC )
73, 4, 5, 6mul4d 9696 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  x.  (
( * `  A
)  x.  ( * `
 B ) ) )  =  ( ( A  x.  ( * `
 A ) )  x.  ( B  x.  ( * `  B
) ) ) )
82, 7eqtrd 2495 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  x.  (
* `  ( A  x.  B ) ) )  =  ( ( A  x.  ( * `  A ) )  x.  ( B  x.  (
* `  B )
) ) )
98fveq2d 5806 . . 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 12755 . . . . 5  |-  ( A  e.  CC  ->  ( A  x.  ( * `  A ) )  e.  RR )
11 cjmulge0 12757 . . . . 5  |-  ( A  e.  CC  ->  0  <_  ( A  x.  (
* `  A )
) )
1210, 11jca 532 . . . 4  |-  ( A  e.  CC  ->  (
( A  x.  (
* `  A )
)  e.  RR  /\  0  <_  ( A  x.  ( * `  A
) ) ) )
13 cjmulrcl 12755 . . . . 5  |-  ( B  e.  CC  ->  ( B  x.  ( * `  B ) )  e.  RR )
14 cjmulge0 12757 . . . . 5  |-  ( B  e.  CC  ->  0  <_  ( B  x.  (
* `  B )
) )
1513, 14jca 532 . . . 4  |-  ( B  e.  CC  ->  (
( B  x.  (
* `  B )
)  e.  RR  /\  0  <_  ( B  x.  ( * `  B
) ) ) )
16 sqrmul 12871 . . . 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 477 . . 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 2495 . 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 9481 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
20 absval 12849 . . 3  |-  ( ( A  x.  B )  e.  CC  ->  ( abs `  ( A  x.  B ) )  =  ( sqr `  (
( A  x.  B
)  x.  ( * `
 ( A  x.  B ) ) ) ) )
2119, 20syl 16 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  x.  B )
)  =  ( sqr `  ( ( A  x.  B )  x.  (
* `  ( A  x.  B ) ) ) ) )
22 absval 12849 . . 3  |-  ( A  e.  CC  ->  ( abs `  A )  =  ( sqr `  ( A  x.  ( * `  A ) ) ) )
23 absval 12849 . . 3  |-  ( B  e.  CC  ->  ( abs `  B )  =  ( sqr `  ( B  x.  ( * `  B ) ) ) )
2422, 23oveqan12d 6222 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  A
)  x.  ( abs `  B ) )  =  ( ( sqr `  ( A  x.  ( * `  A ) ) )  x.  ( sqr `  ( B  x.  ( * `  B ) ) ) ) )
2518, 21, 243eqtr4d 2505 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  x.  B )
)  =  ( ( abs `  A )  x.  ( abs `  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   CCcc 9395   RRcr 9396   0cc0 9397    x. cmul 9402    <_ cle 9534   *ccj 12707   sqrcsqr 12844   abscabs 12845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7806  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-seq 11928  df-exp 11987  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847
This theorem is referenced by:  absdiv  12906  absexp  12915  absimle  12920  abstri  12940  absmuli  13013  absmuld  13062  ef01bndlem  13590  absmulgcd  13853  gcdmultiplez  13857  absabv  18005  iblabs  21449  pige3  22122  atantayl  22475  efrlim  22506  lgslem3  22780  mul2sq  22847  cnnv  24246  bcsiALT  24760  nmcfnexi  25634  cnzh  26567  rezh  26568  iblabsnc  28627  iblmulc2nc  28628  ftc1anclem6  28643  ftc1anclem7  28644  ftc1anclem8  28645
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