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Theorem absmul 12099
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 11947 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  x.  B )
)  =  ( ( * `  A )  x.  ( * `  B ) ) )
21oveq2d 6097 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  x.  (
* `  ( A  x.  B ) ) )  =  ( ( A  x.  B )  x.  ( ( * `  A )  x.  (
* `  B )
) ) )
3 simpl 444 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
4 simpr 448 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
53cjcld 12001 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  A
)  e.  CC )
64cjcld 12001 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  B
)  e.  CC )
73, 4, 5, 6mul4d 9278 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  x.  (
( * `  A
)  x.  ( * `
 B ) ) )  =  ( ( A  x.  ( * `
 A ) )  x.  ( B  x.  ( * `  B
) ) ) )
82, 7eqtrd 2468 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  x.  (
* `  ( A  x.  B ) ) )  =  ( ( A  x.  ( * `  A ) )  x.  ( B  x.  (
* `  B )
) ) )
98fveq2d 5732 . . 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 11949 . . . . 5  |-  ( A  e.  CC  ->  ( A  x.  ( * `  A ) )  e.  RR )
11 cjmulge0 11951 . . . . 5  |-  ( A  e.  CC  ->  0  <_  ( A  x.  (
* `  A )
) )
1210, 11jca 519 . . . 4  |-  ( A  e.  CC  ->  (
( A  x.  (
* `  A )
)  e.  RR  /\  0  <_  ( A  x.  ( * `  A
) ) ) )
13 cjmulrcl 11949 . . . . 5  |-  ( B  e.  CC  ->  ( B  x.  ( * `  B ) )  e.  RR )
14 cjmulge0 11951 . . . . 5  |-  ( B  e.  CC  ->  0  <_  ( B  x.  (
* `  B )
) )
1513, 14jca 519 . . . 4  |-  ( B  e.  CC  ->  (
( B  x.  (
* `  B )
)  e.  RR  /\  0  <_  ( B  x.  ( * `  B
) ) ) )
16 sqrmul 12065 . . . 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 464 . . 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 2468 . 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 9074 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
20 absval 12043 . . 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 12043 . . 3  |-  ( A  e.  CC  ->  ( abs `  A )  =  ( sqr `  ( A  x.  ( * `  A ) ) ) )
23 absval 12043 . . 3  |-  ( B  e.  CC  ->  ( abs `  B )  =  ( sqr `  ( B  x.  ( * `  B ) ) ) )
2422, 23oveqan12d 6100 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  A
)  x.  ( abs `  B ) )  =  ( ( sqr `  ( A  x.  ( * `  A ) ) )  x.  ( sqr `  ( B  x.  ( * `  B ) ) ) ) )
2518, 21, 243eqtr4d 2478 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 4    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990    x. cmul 8995    <_ cle 9121   *ccj 11901   sqrcsqr 12038   abscabs 12039
This theorem is referenced by:  absdiv  12100  absexp  12109  absimle  12114  abstri  12134  absmuli  12207  absmuld  12256  ef01bndlem  12785  absmulgcd  13047  gcdmultiplez  13051  absabv  16756  iblabs  19720  pige3  20425  atantayl  20777  efrlim  20808  lgslem3  21082  mul2sq  21149  cnnv  22168  bcsiALT  22681  nmcfnexi  23554  cnzh  24354  rezh  24355  iblabsnc  26269  iblmulc2nc  26270  ftc1anclem6  26285  ftc1anclem7  26286  ftc1anclem8  26287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041
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