Theorem asinlem2 24596

Description: The argument to the logarithm in df-asin 24592 has the property that replacing A with -u A in the expression gives the reciprocal. (Contributed by Mario Carneiro, 1-Apr-2015.)

Assertion

Ref	Expression
asinlem2	|- ( A e. CC -> ( ( ( _i x. A ) + ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) x. ( ( _i x. -u A ) + ( sqr ` ( 1 - ( -u A ^ 2 ) ) ) ) ) = 1 )

Proof of Theorem asinlem2

Step	Hyp	Ref	Expression
1		ax-icn 9995	. . . . 5 |- _i e. CC
2		mulcl 10020	. . . . 5 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
3	1, 2	mpan 706	. . . 4 |- ( A e. CC -> ( _i x. A ) e. CC )
4		ax-1cn 9994	. . . . . 6 |- 1 e. CC
5		sqcl 12925	. . . . . 6 |- ( A e. CC -> ( A ^ 2 ) e. CC )
6		subcl 10280	. . . . . 6 |- ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( 1 - ( A ^ 2 ) ) e. CC )
7	4, 5, 6	sylancr 695	. . . . 5 |- ( A e. CC -> ( 1 - ( A ^ 2 ) ) e. CC )
8	7	sqrtcld 14176	. . . 4 |- ( A e. CC -> ( sqr ` ( 1 - ( A ^ 2 ) ) ) e. CC )
9	3, 8	addcomd 10238	. . 3 |- ( A e. CC -> ( ( _i x. A ) + ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) = ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) + ( _i x. A ) ) )
10		mulneg2 10467	. . . . . 6 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. -u A ) = -u ( _i x. A ) )
11	1, 10	mpan 706	. . . . 5 |- ( A e. CC -> ( _i x. -u A ) = -u ( _i x. A ) )
12		sqneg 12923	. . . . . . 7 |- ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) )
13	12	oveq2d 6666	. . . . . 6 |- ( A e. CC -> ( 1 - ( -u A ^ 2 ) ) = ( 1 - ( A ^ 2 ) ) )
14	13	fveq2d 6195	. . . . 5 |- ( A e. CC -> ( sqr ` ( 1 - ( -u A ^ 2 ) ) ) = ( sqr ` ( 1 - ( A ^ 2 ) ) ) )
15	11, 14	oveq12d 6668	. . . 4 |- ( A e. CC -> ( ( _i x. -u A ) + ( sqr ` ( 1 - ( -u A ^ 2 ) ) ) ) = ( -u ( _i x. A ) + ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) )
16	3	negcld 10379	. . . . 5 |- ( A e. CC -> -u ( _i x. A ) e. CC )
17	16, 8	addcomd 10238	. . . 4 |- ( A e. CC -> ( -u ( _i x. A ) + ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) = ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) + -u ( _i x. A ) ) )
18	8, 3	negsubd 10398	. . . 4 |- ( A e. CC -> ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) + -u ( _i x. A ) ) = ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) )
19	15, 17, 18	3eqtrd 2660	. . 3 |- ( A e. CC -> ( ( _i x. -u A ) + ( sqr ` ( 1 - ( -u A ^ 2 ) ) ) ) = ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) )
20	9, 19	oveq12d 6668	. 2 |- ( A e. CC -> ( ( ( _i x. A ) + ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) x. ( ( _i x. -u A ) + ( sqr ` ( 1 - ( -u A ^ 2 ) ) ) ) ) = ( ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) + ( _i x. A ) ) x. ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) ) )
21	7	sqsqrtd 14178	. . . 4 |- ( A e. CC -> ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) = ( 1 - ( A ^ 2 ) ) )
22		sqmul 12926	. . . . . 6 |- ( ( _i e. CC /\ A e. CC ) -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
23	1, 22	mpan 706	. . . . 5 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
24		i2 12965	. . . . . . 7 |- ( _i ^ 2 ) = -u 1
25	24	oveq1i 6660	. . . . . 6 |- ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = ( -u 1 x. ( A ^ 2 ) )
26	5	mulm1d 10482	. . . . . 6 |- ( A e. CC -> ( -u 1 x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
27	25, 26	syl5eq 2668	. . . . 5 |- ( A e. CC -> ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
28	23, 27	eqtrd 2656	. . . 4 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = -u ( A ^ 2 ) )
29	21, 28	oveq12d 6668	. . 3 |- ( A e. CC -> ( ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( ( 1 - ( A ^ 2 ) ) - -u ( A ^ 2 ) ) )
30		subsq 12972	. . . 4 |- ( ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) e. CC /\ ( _i x. A ) e. CC ) -> ( ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) + ( _i x. A ) ) x. ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) ) )
31	8, 3, 30	syl2anc 693	. . 3 |- ( A e. CC -> ( ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) + ( _i x. A ) ) x. ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) ) )
32	7, 5	subnegd 10399	. . 3 |- ( A e. CC -> ( ( 1 - ( A ^ 2 ) ) - -u ( A ^ 2 ) ) = ( ( 1 - ( A ^ 2 ) ) + ( A ^ 2 ) ) )
33	29, 31, 32	3eqtr3d 2664	. 2 |- ( A e. CC -> ( ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) + ( _i x. A ) ) x. ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) ) = ( ( 1 - ( A ^ 2 ) ) + ( A ^ 2 ) ) )
34		npcan 10290	. . 3 |- ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( ( 1 - ( A ^ 2 ) ) + ( A ^ 2 ) ) = 1 )
35	4, 5, 34	sylancr 695	. 2 |- ( A e. CC -> ( ( 1 - ( A ^ 2 ) ) + ( A ^ 2 ) ) = 1 )
36	20, 33, 35	3eqtrd 2660	1 |- ( A e. CC -> ( ( ( _i x. A ) + ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) x. ( ( _i x. -u A ) + ( sqr ` ( 1 - ( -u A ^ 2 ) ) ) ) ) = 1 )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   2c2 11070   ^cexp 12860   sqrcsqrt 13973

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:  asinlem3  24598  asinneg  24613