Theorem atanbndlem 24652

Description: Lemma for atanbnd 24653. (Contributed by Mario Carneiro, 5-Apr-2015.)

Assertion

Ref	Expression
atanbndlem	|- ( A e. RR+ -> (arctan ` A ) e. ( -u ( pi / 2 ) (,) ( pi / 2 ) ) )

Proof of Theorem atanbndlem

Step	Hyp	Ref	Expression
1		rpre 11839	. . 3 |- ( A e. RR+ -> A e. RR )
2		atanrecl 24638	. . 3 |- ( A e. RR -> (arctan ` A ) e. RR )
3	1, 2	syl 17	. 2 |- ( A e. RR+ -> (arctan ` A ) e. RR )
4		picn 24211	. . . 4 |- pi e. CC
5		2cn 11091	. . . 4 |- 2 e. CC
6		2ne0 11113	. . . 4 |- 2 =/= 0
7		divneg 10719	. . . 4 |- ( ( pi e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> -u ( pi / 2 ) = ( -u pi / 2 ) )
8	4, 5, 6, 7	mp3an 1424	. . 3 |- -u ( pi / 2 ) = ( -u pi / 2 )
9		ax-1cn 9994	. . . . . . . . . . . 12 |- 1 e. CC
10		ax-icn 9995	. . . . . . . . . . . . 13 |- _i e. CC
11	1	recnd 10068	. . . . . . . . . . . . 13 |- ( A e. RR+ -> A e. CC )
12		mulcl 10020	. . . . . . . . . . . . 13 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
13	10, 11, 12	sylancr 695	. . . . . . . . . . . 12 |- ( A e. RR+ -> ( _i x. A ) e. CC )
14		addcl 10018	. . . . . . . . . . . 12 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + ( _i x. A ) ) e. CC )
15	9, 13, 14	sylancr 695	. . . . . . . . . . 11 |- ( A e. RR+ -> ( 1 + ( _i x. A ) ) e. CC )
16		atanre 24612	. . . . . . . . . . . . . 14 |- ( A e. RR -> A e. dom arctan )
17	1, 16	syl 17	. . . . . . . . . . . . 13 |- ( A e. RR+ -> A e. dom arctan )
18		atandm2 24604	. . . . . . . . . . . . 13 |- ( A e. dom arctan <-> ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) )
19	17, 18	sylib 208	. . . . . . . . . . . 12 |- ( A e. RR+ -> ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) )
20	19	simp3d 1075	. . . . . . . . . . 11 |- ( A e. RR+ -> ( 1 + ( _i x. A ) ) =/= 0 )
21	15, 20	logcld 24317	. . . . . . . . . 10 |- ( A e. RR+ -> ( log ` ( 1 + ( _i x. A ) ) ) e. CC )
22		subcl 10280	. . . . . . . . . . . 12 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 - ( _i x. A ) ) e. CC )
23	9, 13, 22	sylancr 695	. . . . . . . . . . 11 |- ( A e. RR+ -> ( 1 - ( _i x. A ) ) e. CC )
24	19	simp2d 1074	. . . . . . . . . . 11 |- ( A e. RR+ -> ( 1 - ( _i x. A ) ) =/= 0 )
25	23, 24	logcld 24317	. . . . . . . . . 10 |- ( A e. RR+ -> ( log ` ( 1 - ( _i x. A ) ) ) e. CC )
26	21, 25	subcld 10392	. . . . . . . . 9 |- ( A e. RR+ -> ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) e. CC )
27		imre 13848	. . . . . . . . 9 |- ( ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) e. CC -> ( Im ` ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( Re ` ( -u _i x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) )
28	26, 27	syl 17	. . . . . . . 8 |- ( A e. RR+ -> ( Im ` ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( Re ` ( -u _i x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) )
29		atanval 24611	. . . . . . . . . . . . . 14 |- ( A e. dom arctan -> (arctan ` A ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) )
30	17, 29	syl 17	. . . . . . . . . . . . 13 |- ( A e. RR+ -> (arctan ` A ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) )
31	30	oveq2d 6666	. . . . . . . . . . . 12 |- ( A e. RR+ -> ( 2 x. (arctan ` A ) ) = ( 2 x. ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) )
32	10, 5, 6	divcan2i 10768	. . . . . . . . . . . . . 14 |- ( 2 x. ( _i / 2 ) ) = _i
33	32	oveq1i 6660	. . . . . . . . . . . . 13 |- ( ( 2 x. ( _i / 2 ) ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = ( _i x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) )
34		2re 11090	. . . . . . . . . . . . . . . 16 |- 2 e. RR
35	34	a1i 11	. . . . . . . . . . . . . . 15 |- ( A e. RR+ -> 2 e. RR )
36	35	recnd 10068	. . . . . . . . . . . . . 14 |- ( A e. RR+ -> 2 e. CC )
37		halfcl 11257	. . . . . . . . . . . . . . 15 |- ( _i e. CC -> ( _i / 2 ) e. CC )
38	10, 37	mp1i 13	. . . . . . . . . . . . . 14 |- ( A e. RR+ -> ( _i / 2 ) e. CC )
39	25, 21	subcld 10392	. . . . . . . . . . . . . 14 |- ( A e. RR+ -> ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) e. CC )
40	36, 38, 39	mulassd 10063	. . . . . . . . . . . . 13 |- ( A e. RR+ -> ( ( 2 x. ( _i / 2 ) ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = ( 2 x. ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) )
41	33, 40	syl5eqr 2670	. . . . . . . . . . . 12 |- ( A e. RR+ -> ( _i x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = ( 2 x. ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) )
42	31, 41	eqtr4d 2659	. . . . . . . . . . 11 |- ( A e. RR+ -> ( 2 x. (arctan ` A ) ) = ( _i x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) )
43	21, 25	negsubdi2d 10408	. . . . . . . . . . . 12 |- ( A e. RR+ -> -u ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) = ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) )
44	43	oveq2d 6666	. . . . . . . . . . 11 |- ( A e. RR+ -> ( _i x. -u ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( _i x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) )
45	42, 44	eqtr4d 2659	. . . . . . . . . 10 |- ( A e. RR+ -> ( 2 x. (arctan ` A ) ) = ( _i x. -u ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) )
46		mulneg12 10468	. . . . . . . . . . 11 |- ( ( _i e. CC /\ ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) e. CC ) -> ( -u _i x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( _i x. -u ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) )
47	10, 26, 46	sylancr 695	. . . . . . . . . 10 |- ( A e. RR+ -> ( -u _i x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( _i x. -u ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) )
48	45, 47	eqtr4d 2659	. . . . . . . . 9 |- ( A e. RR+ -> ( 2 x. (arctan ` A ) ) = ( -u _i x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) )
49	48	fveq2d 6195	. . . . . . . 8 |- ( A e. RR+ -> ( Re ` ( 2 x. (arctan ` A ) ) ) = ( Re ` ( -u _i x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) )
50		remulcl 10021	. . . . . . . . . 10 |- ( ( 2 e. RR /\ (arctan ` A ) e. RR ) -> ( 2 x. (arctan ` A ) ) e. RR )
51	34, 3, 50	sylancr 695	. . . . . . . . 9 |- ( A e. RR+ -> ( 2 x. (arctan ` A ) ) e. RR )
52	51	rered 13964	. . . . . . . 8 |- ( A e. RR+ -> ( Re ` ( 2 x. (arctan ` A ) ) ) = ( 2 x. (arctan ` A ) ) )
53	28, 49, 52	3eqtr2rd 2663	. . . . . . 7 |- ( A e. RR+ -> ( 2 x. (arctan ` A ) ) = ( Im ` ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) )
54		rpgt0 11844	. . . . . . . . 9 |- ( A e. RR+ -> 0 < A )
55	1	rered 13964	. . . . . . . . 9 |- ( A e. RR+ -> ( Re ` A ) = A )
56	54, 55	breqtrrd 4681	. . . . . . . 8 |- ( A e. RR+ -> 0 < ( Re ` A ) )
57		atanlogsublem 24642	. . . . . . . 8 |- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) e. ( -u pi (,) pi ) )
58	17, 56, 57	syl2anc 693	. . . . . . 7 |- ( A e. RR+ -> ( Im ` ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) e. ( -u pi (,) pi ) )
59	53, 58	eqeltrd 2701	. . . . . 6 |- ( A e. RR+ -> ( 2 x. (arctan ` A ) ) e. ( -u pi (,) pi ) )
60		eliooord 12233	. . . . . 6 |- ( ( 2 x. (arctan ` A ) ) e. ( -u pi (,) pi ) -> ( -u pi < ( 2 x. (arctan ` A ) ) /\ ( 2 x. (arctan ` A ) ) < pi ) )
61	59, 60	syl 17	. . . . 5 |- ( A e. RR+ -> ( -u pi < ( 2 x. (arctan ` A ) ) /\ ( 2 x. (arctan ` A ) ) < pi ) )
62	61	simpld 475	. . . 4 |- ( A e. RR+ -> -u pi < ( 2 x. (arctan ` A ) ) )
63		pire 24210	. . . . . . 7 |- pi e. RR
64	63	renegcli 10342	. . . . . 6 |- -u pi e. RR
65	64	a1i 11	. . . . 5 |- ( A e. RR+ -> -u pi e. RR )
66		2pos 11112	. . . . . 6 |- 0 < 2
67	66	a1i 11	. . . . 5 |- ( A e. RR+ -> 0 < 2 )
68		ltdivmul 10898	. . . . 5 |- ( ( -u pi e. RR /\ (arctan ` A ) e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( -u pi / 2 ) < (arctan ` A ) <-> -u pi < ( 2 x. (arctan ` A ) ) ) )
69	65, 3, 35, 67, 68	syl112anc 1330	. . . 4 |- ( A e. RR+ -> ( ( -u pi / 2 ) < (arctan ` A ) <-> -u pi < ( 2 x. (arctan ` A ) ) ) )
70	62, 69	mpbird 247	. . 3 |- ( A e. RR+ -> ( -u pi / 2 ) < (arctan ` A ) )
71	8, 70	syl5eqbr 4688	. 2 |- ( A e. RR+ -> -u ( pi / 2 ) < (arctan ` A ) )
72	61	simprd 479	. . 3 |- ( A e. RR+ -> ( 2 x. (arctan ` A ) ) < pi )
73	63	a1i 11	. . . 4 |- ( A e. RR+ -> pi e. RR )
74		ltmuldiv2 10897	. . . 4 |- ( ( (arctan ` A ) e. RR /\ pi e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( 2 x. (arctan ` A ) ) < pi <-> (arctan ` A ) < ( pi / 2 ) ) )
75	3, 73, 35, 67, 74	syl112anc 1330	. . 3 |- ( A e. RR+ -> ( ( 2 x. (arctan ` A ) ) < pi <-> (arctan ` A ) < ( pi / 2 ) ) )
76	72, 75	mpbid 222	. 2 |- ( A e. RR+ -> (arctan ` A ) < ( pi / 2 ) )
77		halfpire 24216	. . . . 5 |- ( pi / 2 ) e. RR
78	77	renegcli 10342	. . . 4 |- -u ( pi / 2 ) e. RR
79	78	rexri 10097	. . 3 |- -u ( pi / 2 ) e. RR*
80	77	rexri 10097	. . 3 |- ( pi / 2 ) e. RR*
81		elioo2 12216	. . 3 |- ( ( -u ( pi / 2 ) e. RR* /\ ( pi / 2 ) e. RR* ) -> ( (arctan ` A ) e. ( -u ( pi / 2 ) (,) ( pi / 2 ) ) <-> ( (arctan ` A ) e. RR /\ -u ( pi / 2 ) < (arctan ` A ) /\ (arctan ` A ) < ( pi / 2 ) ) ) )
82	79, 80, 81	mp2an 708	. 2 |- ( (arctan ` A ) e. ( -u ( pi / 2 ) (,) ( pi / 2 ) ) <-> ( (arctan ` A ) e. RR /\ -u ( pi / 2 ) < (arctan ` A ) /\ (arctan ` A ) < ( pi / 2 ) ) )
83	3, 71, 76, 82	syl3anbrc 1246	1 |- ( A e. RR+ -> (arctan ` A ) e. ( -u ( pi / 2 ) (,) ( pi / 2 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   RR*cxr 10073   < clt 10074   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   RR+crp 11832   (,)cioo 12175   Recre 13837   Imcim 13838   picpi 14797   logclog 24301  arctancatan 24591

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-int 4476  df-iun 4522  df-iin 4523  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-tan 14802  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-atan 24594

This theorem is referenced by:  atanbnd  24653