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Theorem cosordlem 25222

Description: Lemma for cosord 25223. (Contributed by Mario Carneiro, 10-May-2014.)

**Hypotheses**
Ref	Expression
cosord.1	⊢ (𝜑 → 𝐴 ∈ (0[,]π))
cosord.2	⊢ (𝜑 → 𝐵 ∈ (0[,]π))
cosord.3	⊢ (𝜑 → 𝐴 < 𝐵)

**Assertion**
Ref	Expression
cosordlem	⊢ (𝜑 → (cos‘𝐵) < (cos‘𝐴))

**Proof of Theorem cosordlem**
Step	Hyp	Ref	Expression
1		cosord.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (0[,]π))
2		0re 10682	. . . . . . . 8 ⊢ 0 ∈ ℝ
3		pire 25151	. . . . . . . 8 ⊢ π ∈ ℝ
4	2, 3	elicc2i 12846	. . . . . . 7 ⊢ (𝐵 ∈ (0[,]π) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ π))
5	1, 4	sylib 221	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ π))
6	5	simp1d 1140	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
7	6	recnd 10708	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
8		cosord.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (0[,]π))
9	2, 3	elicc2i 12846	. . . . . . 7 ⊢ (𝐴 ∈ (0[,]π) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ π))
10	8, 9	sylib 221	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ π))
11	10	simp1d 1140	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
12	11	recnd 10708	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ)
13		subcos 15577	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((cos‘𝐴) − (cos‘𝐵)) = (2 · ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2)))))
14	7, 12, 13	syl2anc 588	. . 3 ⊢ (𝜑 → ((cos‘𝐴) − (cos‘𝐵)) = (2 · ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2)))))
15		2rp 12436	. . . 4 ⊢ 2 ∈ ℝ⁺
16	6, 11	readdcld 10709	. . . . . . . 8 ⊢ (𝜑 → (𝐵 + 𝐴) ∈ ℝ)
17	16	rehalfcld 11922	. . . . . . 7 ⊢ (𝜑 → ((𝐵 + 𝐴) / 2) ∈ ℝ)
18	17	resincld 15545	. . . . . 6 ⊢ (𝜑 → (sin‘((𝐵 + 𝐴) / 2)) ∈ ℝ)
19	2	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℝ)
20	10	simp2d 1141	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 𝐴)
21		cosord.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 < 𝐵)
22	19, 11, 6, 20, 21	lelttrd 10837	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝐵)
23	6, 11, 22, 20	addgtge0d 11253	. . . . . . . . 9 ⊢ (𝜑 → 0 < (𝐵 + 𝐴))
24		2re 11749	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
25		2pos 11778	. . . . . . . . . 10 ⊢ 0 < 2
26		divgt0 11547	. . . . . . . . . 10 ⊢ ((((𝐵 + 𝐴) ∈ ℝ ∧ 0 < (𝐵 + 𝐴)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 < ((𝐵 + 𝐴) / 2))
27	24, 25, 26	mpanr12 705	. . . . . . . . 9 ⊢ (((𝐵 + 𝐴) ∈ ℝ ∧ 0 < (𝐵 + 𝐴)) → 0 < ((𝐵 + 𝐴) / 2))
28	16, 23, 27	syl2anc 588	. . . . . . . 8 ⊢ (𝜑 → 0 < ((𝐵 + 𝐴) / 2))
29	3	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → π ∈ ℝ)
30	11, 6, 6, 21	ltadd2dd 10838	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 + 𝐴) < (𝐵 + 𝐵))
31	7	2timesd 11918	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · 𝐵) = (𝐵 + 𝐵))
32	30, 31	breqtrrd 5061	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 + 𝐴) < (2 · 𝐵))
33	24	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
34	25	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 2)
35		ltdivmul 11554	. . . . . . . . . . 11 ⊢ (((𝐵 + 𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((𝐵 + 𝐴) / 2) < 𝐵 ↔ (𝐵 + 𝐴) < (2 · 𝐵)))
36	16, 6, 33, 34, 35	syl112anc 1372	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐵 + 𝐴) / 2) < 𝐵 ↔ (𝐵 + 𝐴) < (2 · 𝐵)))
37	32, 36	mpbird 260	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 + 𝐴) / 2) < 𝐵)
38	5	simp3d 1142	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≤ π)
39	17, 6, 29, 37, 38	ltletrd 10839	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 + 𝐴) / 2) < π)
40		0xr 10727	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
41	3	rexri 10738	. . . . . . . . 9 ⊢ π ∈ ℝ^*
42		elioo2 12821	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^*) → (((𝐵 + 𝐴) / 2) ∈ (0(,)π) ↔ (((𝐵 + 𝐴) / 2) ∈ ℝ ∧ 0 < ((𝐵 + 𝐴) / 2) ∧ ((𝐵 + 𝐴) / 2) < π)))
43	40, 41, 42	mp2an 692	. . . . . . . 8 ⊢ (((𝐵 + 𝐴) / 2) ∈ (0(,)π) ↔ (((𝐵 + 𝐴) / 2) ∈ ℝ ∧ 0 < ((𝐵 + 𝐴) / 2) ∧ ((𝐵 + 𝐴) / 2) < π))
44	17, 28, 39, 43	syl3anbrc 1341	. . . . . . 7 ⊢ (𝜑 → ((𝐵 + 𝐴) / 2) ∈ (0(,)π))
45		sinq12gt0 25200	. . . . . . 7 ⊢ (((𝐵 + 𝐴) / 2) ∈ (0(,)π) → 0 < (sin‘((𝐵 + 𝐴) / 2)))
46	44, 45	syl 17	. . . . . 6 ⊢ (𝜑 → 0 < (sin‘((𝐵 + 𝐴) / 2)))
47	18, 46	elrpd 12470	. . . . 5 ⊢ (𝜑 → (sin‘((𝐵 + 𝐴) / 2)) ∈ ℝ⁺)
48	6, 11	resubcld 11107	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
49	48	rehalfcld 11922	. . . . . . 7 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) ∈ ℝ)
50	49	resincld 15545	. . . . . 6 ⊢ (𝜑 → (sin‘((𝐵 − 𝐴) / 2)) ∈ ℝ)
51	11, 6	posdifd 11266	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
52	21, 51	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
53		divgt0 11547	. . . . . . . . . 10 ⊢ ((((𝐵 − 𝐴) ∈ ℝ ∧ 0 < (𝐵 − 𝐴)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 < ((𝐵 − 𝐴) / 2))
54	24, 25, 53	mpanr12 705	. . . . . . . . 9 ⊢ (((𝐵 − 𝐴) ∈ ℝ ∧ 0 < (𝐵 − 𝐴)) → 0 < ((𝐵 − 𝐴) / 2))
55	48, 52, 54	syl2anc 588	. . . . . . . 8 ⊢ (𝜑 → 0 < ((𝐵 − 𝐴) / 2))
56		rehalfcl 11901	. . . . . . . . . 10 ⊢ (π ∈ ℝ → (π / 2) ∈ ℝ)
57	3, 56	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → (π / 2) ∈ ℝ)
58	6, 11	subge02d 11271	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ 𝐴 ↔ (𝐵 − 𝐴) ≤ 𝐵))
59	20, 58	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ 𝐵)
60	48, 6, 29, 59, 38	letrd 10836	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ π)
61		lediv1 11544	. . . . . . . . . . 11 ⊢ (((𝐵 − 𝐴) ∈ ℝ ∧ π ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((𝐵 − 𝐴) ≤ π ↔ ((𝐵 − 𝐴) / 2) ≤ (π / 2)))
62	48, 29, 33, 34, 61	syl112anc 1372	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝐴) ≤ π ↔ ((𝐵 − 𝐴) / 2) ≤ (π / 2)))
63	60, 62	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) ≤ (π / 2))
64		pirp 25154	. . . . . . . . . 10 ⊢ π ∈ ℝ⁺
65		rphalflt 12460	. . . . . . . . . 10 ⊢ (π ∈ ℝ⁺ → (π / 2) < π)
66	64, 65	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → (π / 2) < π)
67	49, 57, 29, 63, 66	lelttrd 10837	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) < π)
68		elioo2 12821	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^*) → (((𝐵 − 𝐴) / 2) ∈ (0(,)π) ↔ (((𝐵 − 𝐴) / 2) ∈ ℝ ∧ 0 < ((𝐵 − 𝐴) / 2) ∧ ((𝐵 − 𝐴) / 2) < π)))
69	40, 41, 68	mp2an 692	. . . . . . . 8 ⊢ (((𝐵 − 𝐴) / 2) ∈ (0(,)π) ↔ (((𝐵 − 𝐴) / 2) ∈ ℝ ∧ 0 < ((𝐵 − 𝐴) / 2) ∧ ((𝐵 − 𝐴) / 2) < π))
70	49, 55, 67, 69	syl3anbrc 1341	. . . . . . 7 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) ∈ (0(,)π))
71		sinq12gt0 25200	. . . . . . 7 ⊢ (((𝐵 − 𝐴) / 2) ∈ (0(,)π) → 0 < (sin‘((𝐵 − 𝐴) / 2)))
72	70, 71	syl 17	. . . . . 6 ⊢ (𝜑 → 0 < (sin‘((𝐵 − 𝐴) / 2)))
73	50, 72	elrpd 12470	. . . . 5 ⊢ (𝜑 → (sin‘((𝐵 − 𝐴) / 2)) ∈ ℝ⁺)
74	47, 73	rpmulcld 12489	. . . 4 ⊢ (𝜑 → ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2))) ∈ ℝ⁺)
75		rpmulcl 12454	. . . 4 ⊢ ((2 ∈ ℝ⁺ ∧ ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2))) ∈ ℝ⁺) → (2 · ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2)))) ∈ ℝ⁺)
76	15, 74, 75	sylancr 591	. . 3 ⊢ (𝜑 → (2 · ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2)))) ∈ ℝ⁺)
77	14, 76	eqeltrd 2853	. 2 ⊢ (𝜑 → ((cos‘𝐴) − (cos‘𝐵)) ∈ ℝ⁺)
78	6	recoscld 15546	. . 3 ⊢ (𝜑 → (cos‘𝐵) ∈ ℝ)
79	11	recoscld 15546	. . 3 ⊢ (𝜑 → (cos‘𝐴) ∈ ℝ)
80		difrp 12469	. . 3 ⊢ (((cos‘𝐵) ∈ ℝ ∧ (cos‘𝐴) ∈ ℝ) → ((cos‘𝐵) < (cos‘𝐴) ↔ ((cos‘𝐴) − (cos‘𝐵)) ∈ ℝ⁺))
81	78, 79, 80	syl2anc 588	. 2 ⊢ (𝜑 → ((cos‘𝐵) < (cos‘𝐴) ↔ ((cos‘𝐴) − (cos‘𝐵)) ∈ ℝ⁺))
82	77, 81	mpbird 260	1 ⊢ (𝜑 → (cos‘𝐵) < (cos‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 class class class wbr 5033 ‘cfv 6336 (class class class)co 7151 ℂcc 10574 ℝcr 10575 0cc0 10576 + caddc 10579 · cmul 10581 ℝ^*cxr 10713 < clt 10714 ≤ cle 10715 − cmin 10909 / cdiv 11336 2c2 11730 ℝ⁺crp 12431 (,)cioo 12780 [,]cicc 12783 sincsin 15466 cosccos 15467 πcpi 15469

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5157 ax-sep 5170 ax-nul 5177 ax-pow 5235 ax-pr 5299 ax-un 7460 ax-inf2 9138 ax-cnex 10632 ax-resscn 10633 ax-1cn 10634 ax-icn 10635 ax-addcl 10636 ax-addrcl 10637 ax-mulcl 10638 ax-mulrcl 10639 ax-mulcom 10640 ax-addass 10641 ax-mulass 10642 ax-distr 10643 ax-i2m1 10644 ax-1ne0 10645 ax-1rid 10646 ax-rnegex 10647 ax-rrecex 10648 ax-cnre 10649 ax-pre-lttri 10650 ax-pre-lttrn 10651 ax-pre-ltadd 10652 ax-pre-mulgt0 10653 ax-pre-sup 10654 ax-addf 10655 ax-mulf 10656

This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-nel 3057 df-ral 3076 df-rex 3077 df-reu 3078 df-rmo 3079 df-rab 3080 df-v 3412 df-sbc 3698 df-csb 3807 df-dif 3862 df-un 3864 df-in 3866 df-ss 3876 df-pss 3878 df-nul 4227 df-if 4422 df-pw 4497 df-sn 4524 df-pr 4526 df-tp 4528 df-op 4530 df-uni 4800 df-int 4840 df-iun 4886 df-iin 4887 df-br 5034 df-opab 5096 df-mpt 5114 df-tr 5140 df-id 5431 df-eprel 5436 df-po 5444 df-so 5445 df-fr 5484 df-se 5485 df-we 5486 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6127 df-ord 6173 df-on 6174 df-lim 6175 df-suc 6176 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-isom 6345 df-riota 7109 df-ov 7154 df-oprab 7155 df-mpo 7156 df-of 7406 df-om 7581 df-1st 7694 df-2nd 7695 df-supp 7837 df-wrecs 7958 df-recs 8019 df-rdg 8057 df-1o 8113 df-2o 8114 df-er 8300 df-map 8419 df-pm 8420 df-ixp 8481 df-en 8529 df-dom 8530 df-sdom 8531 df-fin 8532 df-fsupp 8868 df-fi 8909 df-sup 8940 df-inf 8941 df-oi 9008 df-card 9402 df-pnf 10716 df-mnf 10717 df-xr 10718 df-ltxr 10719 df-le 10720 df-sub 10911 df-neg 10912 df-div 11337 df-nn 11676 df-2 11738 df-3 11739 df-4 11740 df-5 11741 df-6 11742 df-7 11743 df-8 11744 df-9 11745 df-n0 11936 df-z 12022 df-dec 12139 df-uz 12284 df-q 12390 df-rp 12432 df-xneg 12549 df-xadd 12550 df-xmul 12551 df-ioo 12784 df-ioc 12785 df-ico 12786 df-icc 12787 df-fz 12941 df-fzo 13084 df-fl 13212 df-seq 13420 df-exp 13481 df-fac 13685 df-bc 13714 df-hash 13742 df-shft 14475 df-cj 14507 df-re 14508 df-im 14509 df-sqrt 14643 df-abs 14644 df-limsup 14877 df-clim 14894 df-rlim 14895 df-sum 15092 df-ef 15470 df-sin 15472 df-cos 15473 df-pi 15475 df-struct 16544 df-ndx 16545 df-slot 16546 df-base 16548 df-sets 16549 df-ress 16550 df-plusg 16637 df-mulr 16638 df-starv 16639 df-sca 16640 df-vsca 16641 df-ip 16642 df-tset 16643 df-ple 16644 df-ds 16646 df-unif 16647 df-hom 16648 df-cco 16649 df-rest 16755 df-topn 16756 df-0g 16774 df-gsum 16775 df-topgen 16776 df-pt 16777 df-prds 16780 df-xrs 16834 df-qtop 16839 df-imas 16840 df-xps 16842 df-mre 16916 df-mrc 16917 df-acs 16919 df-mgm 17919 df-sgrp 17968 df-mnd 17979 df-submnd 18024 df-mulg 18293 df-cntz 18515 df-cmn 18976 df-psmet 20159 df-xmet 20160 df-met 20161 df-bl 20162 df-mopn 20163 df-fbas 20164 df-fg 20165 df-cnfld 20168 df-top 21595 df-topon 21612 df-topsp 21634 df-bases 21647 df-cld 21720 df-ntr 21721 df-cls 21722 df-nei 21799 df-lp 21837 df-perf 21838 df-cn 21928 df-cnp 21929 df-haus 22016 df-tx 22263 df-hmeo 22456 df-fil 22547 df-fm 22639 df-flim 22640 df-flf 22641 df-xms 23023 df-ms 23024 df-tms 23025 df-cncf 23580 df-limc 24566 df-dv 24567

This theorem is referenced by: cosord 25223 cos0pilt1 25224

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