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Theorem cosordlem 25591

Description: Lemma for cosord 25592. (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 10908	. . . . . . . 8 ⊢ 0 ∈ ℝ
3		pire 25520	. . . . . . . 8 ⊢ π ∈ ℝ
4	2, 3	elicc2i 13074	. . . . . . 7 ⊢ (𝐵 ∈ (0[,]π) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ π))
5	1, 4	sylib 217	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ π))
6	5	simp1d 1140	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
7	6	recnd 10934	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
8		cosord.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (0[,]π))
9	2, 3	elicc2i 13074	. . . . . . 7 ⊢ (𝐴 ∈ (0[,]π) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ π))
10	8, 9	sylib 217	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ π))
11	10	simp1d 1140	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
12	11	recnd 10934	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ)
13		subcos 15812	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((cos‘𝐴) − (cos‘𝐵)) = (2 · ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2)))))
14	7, 12, 13	syl2anc 583	. . 3 ⊢ (𝜑 → ((cos‘𝐴) − (cos‘𝐵)) = (2 · ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2)))))
15		2rp 12664	. . . 4 ⊢ 2 ∈ ℝ⁺
16	6, 11	readdcld 10935	. . . . . . . 8 ⊢ (𝜑 → (𝐵 + 𝐴) ∈ ℝ)
17	16	rehalfcld 12150	. . . . . . 7 ⊢ (𝜑 → ((𝐵 + 𝐴) / 2) ∈ ℝ)
18	17	resincld 15780	. . . . . 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 11063	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝐵)
23	6, 11, 22, 20	addgtge0d 11479	. . . . . . . . 9 ⊢ (𝜑 → 0 < (𝐵 + 𝐴))
24		2re 11977	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
25		2pos 12006	. . . . . . . . . 10 ⊢ 0 < 2
26		divgt0 11773	. . . . . . . . . 10 ⊢ ((((𝐵 + 𝐴) ∈ ℝ ∧ 0 < (𝐵 + 𝐴)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 < ((𝐵 + 𝐴) / 2))
27	24, 25, 26	mpanr12 701	. . . . . . . . 9 ⊢ (((𝐵 + 𝐴) ∈ ℝ ∧ 0 < (𝐵 + 𝐴)) → 0 < ((𝐵 + 𝐴) / 2))
28	16, 23, 27	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → 0 < ((𝐵 + 𝐴) / 2))
29	3	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → π ∈ ℝ)
30	11, 6, 6, 21	ltadd2dd 11064	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 + 𝐴) < (𝐵 + 𝐵))
31	7	2timesd 12146	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · 𝐵) = (𝐵 + 𝐵))
32	30, 31	breqtrrd 5098	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 + 𝐴) < (2 · 𝐵))
33	24	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
34	25	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 2)
35		ltdivmul 11780	. . . . . . . . . . 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 256	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 + 𝐴) / 2) < 𝐵)
38	5	simp3d 1142	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≤ π)
39	17, 6, 29, 37, 38	ltletrd 11065	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 + 𝐴) / 2) < π)
40		0xr 10953	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
41	3	rexri 10964	. . . . . . . . 9 ⊢ π ∈ ℝ^*
42		elioo2 13049	. . . . . . . . 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 688	. . . . . . . 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 25569	. . . . . . 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 12698	. . . . 5 ⊢ (𝜑 → (sin‘((𝐵 + 𝐴) / 2)) ∈ ℝ⁺)
48	6, 11	resubcld 11333	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
49	48	rehalfcld 12150	. . . . . . 7 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) ∈ ℝ)
50	49	resincld 15780	. . . . . 6 ⊢ (𝜑 → (sin‘((𝐵 − 𝐴) / 2)) ∈ ℝ)
51	11, 6	posdifd 11492	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
52	21, 51	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
53		divgt0 11773	. . . . . . . . . 10 ⊢ ((((𝐵 − 𝐴) ∈ ℝ ∧ 0 < (𝐵 − 𝐴)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 < ((𝐵 − 𝐴) / 2))
54	24, 25, 53	mpanr12 701	. . . . . . . . 9 ⊢ (((𝐵 − 𝐴) ∈ ℝ ∧ 0 < (𝐵 − 𝐴)) → 0 < ((𝐵 − 𝐴) / 2))
55	48, 52, 54	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → 0 < ((𝐵 − 𝐴) / 2))
56		rehalfcl 12129	. . . . . . . . . 10 ⊢ (π ∈ ℝ → (π / 2) ∈ ℝ)
57	3, 56	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → (π / 2) ∈ ℝ)
58	6, 11	subge02d 11497	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ 𝐴 ↔ (𝐵 − 𝐴) ≤ 𝐵))
59	20, 58	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ 𝐵)
60	48, 6, 29, 59, 38	letrd 11062	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ π)
61		lediv1 11770	. . . . . . . . . . 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 231	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) ≤ (π / 2))
64		pirp 25523	. . . . . . . . . 10 ⊢ π ∈ ℝ⁺
65		rphalflt 12688	. . . . . . . . . 10 ⊢ (π ∈ ℝ⁺ → (π / 2) < π)
66	64, 65	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → (π / 2) < π)
67	49, 57, 29, 63, 66	lelttrd 11063	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) < π)
68		elioo2 13049	. . . . . . . . 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 688	. . . . . . . 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 25569	. . . . . . 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 12698	. . . . 5 ⊢ (𝜑 → (sin‘((𝐵 − 𝐴) / 2)) ∈ ℝ⁺)
74	47, 73	rpmulcld 12717	. . . 4 ⊢ (𝜑 → ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2))) ∈ ℝ⁺)
75		rpmulcl 12682	. . . 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 586	. . 3 ⊢ (𝜑 → (2 · ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2)))) ∈ ℝ⁺)
77	14, 76	eqeltrd 2839	. 2 ⊢ (𝜑 → ((cos‘𝐴) − (cos‘𝐵)) ∈ ℝ⁺)
78	6	recoscld 15781	. . 3 ⊢ (𝜑 → (cos‘𝐵) ∈ ℝ)
79	11	recoscld 15781	. . 3 ⊢ (𝜑 → (cos‘𝐴) ∈ ℝ)
80		difrp 12697	. . 3 ⊢ (((cos‘𝐵) ∈ ℝ ∧ (cos‘𝐴) ∈ ℝ) → ((cos‘𝐵) < (cos‘𝐴) ↔ ((cos‘𝐴) − (cos‘𝐵)) ∈ ℝ⁺))
81	78, 79, 80	syl2anc 583	. 2 ⊢ (𝜑 → ((cos‘𝐵) < (cos‘𝐴) ↔ ((cos‘𝐴) − (cos‘𝐵)) ∈ ℝ⁺))
82	77, 81	mpbird 256	1 ⊢ (𝜑 → (cos‘𝐵) < (cos‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 + caddc 10805 · cmul 10807 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 − cmin 11135 / cdiv 11562 2c2 11958 ℝ⁺crp 12659 (,)cioo 13008 [,]cicc 13011 sincsin 15701 cosccos 15702 πcpi 15704

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936

This theorem is referenced by: cosord 25592 cos0pilt1 25593

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