Theorem abslem2 14079

Description: Lemma involving absolute values. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Assertion

Ref	Expression
abslem2	|- ( ( A e. CC /\ A =/= 0 ) -> ( ( ( * ` ( A / ( abs ` A ) ) ) x. A ) + ( ( A / ( abs ` A ) ) x. ( * ` A ) ) ) = ( 2 x. ( abs ` A ) ) )

Proof of Theorem abslem2

Step	Hyp	Ref	Expression
1		absvalsq 14020	. . . . . . . . 9 |- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( A x. ( * ` A ) ) )
2	1	adantr 481	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( abs ` A ) ^ 2 ) = ( A x. ( * ` A ) ) )
3		abscl 14018	. . . . . . . . . . 11 |- ( A e. CC -> ( abs ` A ) e. RR )
4	3	adantr 481	. . . . . . . . . 10 |- ( ( A e. CC /\ A =/= 0 ) -> ( abs ` A ) e. RR )
5	4	recnd 10068	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 ) -> ( abs ` A ) e. CC )
6	5	sqvald 13005	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( abs ` A ) ^ 2 ) = ( ( abs ` A ) x. ( abs ` A ) ) )
7	2, 6	eqtr3d 2658	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 ) -> ( A x. ( * ` A ) ) = ( ( abs ` A ) x. ( abs ` A ) ) )
8	7	oveq1d 6665	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A x. ( * ` A ) ) / ( abs ` A ) ) = ( ( ( abs ` A ) x. ( abs ` A ) ) / ( abs ` A ) ) )
9		simpl 473	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 ) -> A e. CC )
10	9	cjcld 13936	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 ) -> ( * ` A ) e. CC )
11		abs00 14029	. . . . . . . . 9 |- ( A e. CC -> ( ( abs ` A ) = 0 <-> A = 0 ) )
12	11	necon3bid 2838	. . . . . . . 8 |- ( A e. CC -> ( ( abs ` A ) =/= 0 <-> A =/= 0 ) )
13	12	biimpar 502	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 ) -> ( abs ` A ) =/= 0 )
14	9, 10, 5, 13	div23d 10838	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A x. ( * ` A ) ) / ( abs ` A ) ) = ( ( A / ( abs ` A ) ) x. ( * ` A ) ) )
15	5, 5, 13	divcan3d 10806	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( ( abs ` A ) x. ( abs ` A ) ) / ( abs ` A ) ) = ( abs ` A ) )
16	8, 14, 15	3eqtr3d 2664	. . . . 5 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( A / ( abs ` A ) ) x. ( * ` A ) ) = ( abs ` A ) )
17	16	fveq2d 6195	. . . 4 |- ( ( A e. CC /\ A =/= 0 ) -> ( * ` ( ( A / ( abs ` A ) ) x. ( * ` A ) ) ) = ( * ` ( abs ` A ) ) )
18	9, 5, 13	divcld 10801	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 ) -> ( A / ( abs ` A ) ) e. CC )
19	18, 10	cjmuld 13961	. . . . 5 |- ( ( A e. CC /\ A =/= 0 ) -> ( * ` ( ( A / ( abs ` A ) ) x. ( * ` A ) ) ) = ( ( * ` ( A / ( abs ` A ) ) ) x. ( * ` ( * ` A ) ) ) )
20	9	cjcjd 13939	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 ) -> ( * ` ( * ` A ) ) = A )
21	20	oveq2d 6666	. . . . 5 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( * ` ( A / ( abs ` A ) ) ) x. ( * ` ( * ` A ) ) ) = ( ( * ` ( A / ( abs ` A ) ) ) x. A ) )
22	19, 21	eqtrd 2656	. . . 4 |- ( ( A e. CC /\ A =/= 0 ) -> ( * ` ( ( A / ( abs ` A ) ) x. ( * ` A ) ) ) = ( ( * ` ( A / ( abs ` A ) ) ) x. A ) )
23	4	cjred 13966	. . . 4 |- ( ( A e. CC /\ A =/= 0 ) -> ( * ` ( abs ` A ) ) = ( abs ` A ) )
24	17, 22, 23	3eqtr3d 2664	. . 3 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( * ` ( A / ( abs ` A ) ) ) x. A ) = ( abs ` A ) )
25	24, 16	oveq12d 6668	. 2 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( ( * ` ( A / ( abs ` A ) ) ) x. A ) + ( ( A / ( abs ` A ) ) x. ( * ` A ) ) ) = ( ( abs ` A ) + ( abs ` A ) ) )
26	5	2timesd 11275	. 2 |- ( ( A e. CC /\ A =/= 0 ) -> ( 2 x. ( abs ` A ) ) = ( ( abs ` A ) + ( abs ` A ) ) )
27	25, 26	eqtr4d 2659	1 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( ( * ` ( A / ( abs ` A ) ) ) x. A ) + ( ( A / ( abs ` A ) ) x. ( * ` A ) ) ) = ( 2 x. ( abs ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   / cdiv 10684   2c2 11070   ^cexp 12860   *ccj 13836   abscabs 13974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976

This theorem is referenced by:  bcsiALT  28036