Theorem absmul 11857

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

Step	Hyp	Ref	Expression
1		cjmul 11672	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) ) )
2	1	oveq2d 6104	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) x. ( * ` ( A x. B ) ) ) = ( ( A x. B ) x. ( ( * ` A ) x. ( * ` B ) ) ) )
3		simpl 109	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
4		simpr 110	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
5	3	cjcld 11728	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( * ` A ) e. CC )
6	4	cjcld 11728	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( * ` B ) e. CC )
7	3, 4, 5, 6	mul4d 8488	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) x. ( ( * ` A ) x. ( * ` B ) ) ) = ( ( A x. ( * ` A ) ) x. ( B x. ( * ` B ) ) ) )
8	2, 7	eqtrd 2271	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) x. ( * ` ( A x. B ) ) ) = ( ( A x. ( * ` A ) ) x. ( B x. ( * ` B ) ) ) )
9	8	fveq2d 5702	. . 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 11674	. . . . 5 |- ( A e. CC -> ( A x. ( * ` A ) ) e. RR )
11		cjmulge0 11676	. . . . 5 |- ( A e. CC -> 0 <_ ( A x. ( * ` A ) ) )
12	10, 11	jca 306	. . . 4 |- ( A e. CC -> ( ( A x. ( * ` A ) ) e. RR /\ 0 <_ ( A x. ( * ` A ) ) ) )
13		cjmulrcl 11674	. . . . 5 |- ( B e. CC -> ( B x. ( * ` B ) ) e. RR )
14		cjmulge0 11676	. . . . 5 |- ( B e. CC -> 0 <_ ( B x. ( * ` B ) ) )
15	13, 14	jca 306	. . . 4 |- ( B e. CC -> ( ( B x. ( * ` B ) ) e. RR /\ 0 <_ ( B x. ( * ` B ) ) ) )
16		sqrtmul 11823	. . . 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 ) ) ) ) )
17	12, 15, 16	syl2an 289	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( sqr ` ( ( A x. ( * ` A ) ) x. ( B x. ( * ` B ) ) ) ) = ( ( sqr ` ( A x. ( * ` A ) ) ) x. ( sqr ` ( B x. ( * ` B ) ) ) ) )
18	9, 17	eqtrd 2271	. 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 8312	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
20		absval 11789	. . 3 |- ( ( A x. B ) e. CC -> ( abs ` ( A x. B ) ) = ( sqr ` ( ( A x. B ) x. ( * ` ( A x. B ) ) ) ) )
21	19, 20	syl 14	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A x. B ) ) = ( sqr ` ( ( A x. B ) x. ( * ` ( A x. B ) ) ) ) )
22		absval 11789	. . 3 |- ( A e. CC -> ( abs ` A ) = ( sqr ` ( A x. ( * ` A ) ) ) )
23		absval 11789	. . 3 |- ( B e. CC -> ( abs ` B ) = ( sqr ` ( B x. ( * ` B ) ) ) )
24	22, 23	oveqan12d 6107	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) x. ( abs ` B ) ) = ( ( sqr ` ( A x. ( * ` A ) ) ) x. ( sqr ` ( B x. ( * ` B ) ) ) ) )
25	18, 21, 24	3eqtr4d 2281	1 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A x. B ) ) = ( ( abs ` A ) x. ( abs ` B ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   CCcc 8183   RRcr 8184   0cc0 8185   x. cmul 8190   <_ cle 8367   *ccj 11626   sqrcsqrt 11784   abscabs 11785

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-rp 10072  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787

This theorem is used by:  absdivap  11858  absexp  11868  absimle  11873  abstri  11893  absmuli  11940  absmuld  11983  ef01bndlem  12548  absmulgcd  12819  gcdmultiplez  12823  lgslem3  16326  mul2sq  16440