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Theorem cosordlem 26415

Description: Lemma for cosord 26416. (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 11152	. . . . . . . 8 ⊢ 0 ∈ ℝ
3		pire 26342	. . . . . . . 8 ⊢ π ∈ ℝ
4	2, 3	elicc2i 13349	. . . . . . 7 ⊢ (𝐵 ∈ (0[,]π) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ π))
5	1, 4	sylib 218	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ π))
6	5	simp1d 1142	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
7	6	recnd 11178	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
8		cosord.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (0[,]π))
9	2, 3	elicc2i 13349	. . . . . . 7 ⊢ (𝐴 ∈ (0[,]π) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ π))
10	8, 9	sylib 218	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ π))
11	10	simp1d 1142	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
12	11	recnd 11178	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ)
13		subcos 16119	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((cos‘𝐴) − (cos‘𝐵)) = (2 · ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2)))))
14	7, 12, 13	syl2anc 584	. . 3 ⊢ (𝜑 → ((cos‘𝐴) − (cos‘𝐵)) = (2 · ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2)))))
15		2rp 12932	. . . 4 ⊢ 2 ∈ ℝ⁺
16	6, 11	readdcld 11179	. . . . . . . 8 ⊢ (𝜑 → (𝐵 + 𝐴) ∈ ℝ)
17	16	rehalfcld 12405	. . . . . . 7 ⊢ (𝜑 → ((𝐵 + 𝐴) / 2) ∈ ℝ)
18	17	resincld 16087	. . . . . 6 ⊢ (𝜑 → (sin‘((𝐵 + 𝐴) / 2)) ∈ ℝ)
19	2	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℝ)
20	10	simp2d 1143	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 𝐴)
21		cosord.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 < 𝐵)
22	19, 11, 6, 20, 21	lelttrd 11308	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝐵)
23	6, 11, 22, 20	addgtge0d 11728	. . . . . . . . 9 ⊢ (𝜑 → 0 < (𝐵 + 𝐴))
24		2re 12236	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
25		2pos 12265	. . . . . . . . . 10 ⊢ 0 < 2
26		divgt0 12027	. . . . . . . . . 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 584	. . . . . . . 8 ⊢ (𝜑 → 0 < ((𝐵 + 𝐴) / 2))
29	3	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → π ∈ ℝ)
30	11, 6, 6, 21	ltadd2dd 11309	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 + 𝐴) < (𝐵 + 𝐵))
31	7	2timesd 12401	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · 𝐵) = (𝐵 + 𝐵))
32	30, 31	breqtrrd 5130	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 + 𝐴) < (2 · 𝐵))
33	24	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
34	25	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 2)
35		ltdivmul 12034	. . . . . . . . . . 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 1376	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐵 + 𝐴) / 2) < 𝐵 ↔ (𝐵 + 𝐴) < (2 · 𝐵)))
37	32, 36	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 + 𝐴) / 2) < 𝐵)
38	5	simp3d 1144	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≤ π)
39	17, 6, 29, 37, 38	ltletrd 11310	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 + 𝐴) / 2) < π)
40		0xr 11197	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
41	3	rexri 11208	. . . . . . . . 9 ⊢ π ∈ ℝ^*
42		elioo2 13323	. . . . . . . . 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 1344	. . . . . . 7 ⊢ (𝜑 → ((𝐵 + 𝐴) / 2) ∈ (0(,)π))
45		sinq12gt0 26392	. . . . . . 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 12968	. . . . 5 ⊢ (𝜑 → (sin‘((𝐵 + 𝐴) / 2)) ∈ ℝ⁺)
48	6, 11	resubcld 11582	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
49	48	rehalfcld 12405	. . . . . . 7 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) ∈ ℝ)
50	49	resincld 16087	. . . . . 6 ⊢ (𝜑 → (sin‘((𝐵 − 𝐴) / 2)) ∈ ℝ)
51	11, 6	posdifd 11741	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
52	21, 51	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
53		divgt0 12027	. . . . . . . . . 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 584	. . . . . . . 8 ⊢ (𝜑 → 0 < ((𝐵 − 𝐴) / 2))
56		rehalfcl 12385	. . . . . . . . . 10 ⊢ (π ∈ ℝ → (π / 2) ∈ ℝ)
57	3, 56	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → (π / 2) ∈ ℝ)
58	6, 11	subge02d 11746	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ 𝐴 ↔ (𝐵 − 𝐴) ≤ 𝐵))
59	20, 58	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ 𝐵)
60	48, 6, 29, 59, 38	letrd 11307	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ π)
61		lediv1 12024	. . . . . . . . . . 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 1376	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝐴) ≤ π ↔ ((𝐵 − 𝐴) / 2) ≤ (π / 2)))
63	60, 62	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) ≤ (π / 2))
64		pirp 26346	. . . . . . . . . 10 ⊢ π ∈ ℝ⁺
65		rphalflt 12958	. . . . . . . . . 10 ⊢ (π ∈ ℝ⁺ → (π / 2) < π)
66	64, 65	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → (π / 2) < π)
67	49, 57, 29, 63, 66	lelttrd 11308	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) < π)
68		elioo2 13323	. . . . . . . . 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 1344	. . . . . . 7 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) ∈ (0(,)π))
71		sinq12gt0 26392	. . . . . . 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 12968	. . . . 5 ⊢ (𝜑 → (sin‘((𝐵 − 𝐴) / 2)) ∈ ℝ⁺)
74	47, 73	rpmulcld 12987	. . . 4 ⊢ (𝜑 → ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2))) ∈ ℝ⁺)
75		rpmulcl 12952	. . . 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 587	. . 3 ⊢ (𝜑 → (2 · ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2)))) ∈ ℝ⁺)
77	14, 76	eqeltrd 2828	. 2 ⊢ (𝜑 → ((cos‘𝐴) − (cos‘𝐵)) ∈ ℝ⁺)
78	6	recoscld 16088	. . 3 ⊢ (𝜑 → (cos‘𝐵) ∈ ℝ)
79	11	recoscld 16088	. . 3 ⊢ (𝜑 → (cos‘𝐴) ∈ ℝ)
80		difrp 12967	. . 3 ⊢ (((cos‘𝐵) ∈ ℝ ∧ (cos‘𝐴) ∈ ℝ) → ((cos‘𝐵) < (cos‘𝐴) ↔ ((cos‘𝐴) − (cos‘𝐵)) ∈ ℝ⁺))
81	78, 79, 80	syl2anc 584	. 2 ⊢ (𝜑 → ((cos‘𝐵) < (cos‘𝐴) ↔ ((cos‘𝐴) − (cos‘𝐵)) ∈ ℝ⁺))
82	77, 81	mpbird 257	1 ⊢ (𝜑 → (cos‘𝐵) < (cos‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 class class class wbr 5102 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 + caddc 11047 · cmul 11049 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 − cmin 11381 / cdiv 11811 2c2 12217 ℝ⁺crp 12927 (,)cioo 13282 [,]cicc 13285 sincsin 16005 cosccos 16006 πcpi 16008

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-ef 16009 df-sin 16011 df-cos 16012 df-pi 16014 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-haus 23178 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-limc 25743 df-dv 25744

This theorem is referenced by: cosord 26416 cos0pilt1 26417

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