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Theorem absmul 11709
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 11557 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  x.  B )
)  =  ( ( * `  A )  x.  ( * `  B ) ) )
21oveq2d 5773 . . . . 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 11611 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  A
)  e.  CC )
64cjcld 11611 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  B
)  e.  CC )
73, 4, 5, 6mul4d 8957 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  x.  (
( * `  A
)  x.  ( * `
 B ) ) )  =  ( ( A  x.  ( * `
 A ) )  x.  ( B  x.  ( * `  B
) ) ) )
82, 7eqtrd 2288 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  x.  (
* `  ( A  x.  B ) ) )  =  ( ( A  x.  ( * `  A ) )  x.  ( B  x.  (
* `  B )
) ) )
98fveq2d 5427 . . 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 11559 . . . . 5  |-  ( A  e.  CC  ->  ( A  x.  ( * `  A ) )  e.  RR )
11 cjmulge0 11561 . . . . 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 11559 . . . . 5  |-  ( B  e.  CC  ->  ( B  x.  ( * `  B ) )  e.  RR )
14 cjmulge0 11561 . . . . 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 11675 . . . 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 2288 . 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 8754 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
20 absval 11653 . . 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 11653 . . 3  |-  ( A  e.  CC  ->  ( abs `  A )  =  ( sqr `  ( A  x.  ( * `  A ) ) ) )
23 absval 11653 . . 3  |-  ( B  e.  CC  ->  ( abs `  B )  =  ( sqr `  ( B  x.  ( * `  B ) ) ) )
2422, 23oveqan12d 5776 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  A
)  x.  ( abs `  B ) )  =  ( ( sqr `  ( A  x.  ( * `  A ) ) )  x.  ( sqr `  ( B  x.  ( * `  B ) ) ) ) )
2518, 21, 243eqtr4d 2298 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 1619    e. wcel 1621   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   CCcc 8668   RRcr 8669   0cc0 8670    x. cmul 8675    <_ cle 8801   *ccj 11511   sqrcsqr 11648   abscabs 11649
This theorem is referenced by:  absdiv  11710  absexp  11719  absimle  11724  abstri  11744  absmuli  11817  absmuld  11866  ef01bndlem  12391  absmulgcd  12653  gcdmultiplez  12657  absabv  16356  iblabs  19110  pige3  19812  atantayl  20160  efrlim  20191  lgslem3  20464  mul2sq  20531  cnnv  21170  bcsiALT  21683  nmcfnexi  22556
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
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 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-sup 7127  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-n0 9898  df-z 9957  df-uz 10163  df-rp 10287  df-seq 10978  df-exp 11036  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651
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