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Theorem cosordlem 26657

Description: Lemma for cosord 26658. (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 11206	. . . . . . . 8 ⊢ 0 ∈ ℝ
3		pire 26581	. . . . . . . 8 ⊢ π ∈ ℝ
4	2, 3	elicc2i 13435	. . . . . . 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 1158	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
7	6	recnd 11233	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
8		cosord.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (0[,]π))
9	2, 3	elicc2i 13435	. . . . . . 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 1158	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
12	11	recnd 11233	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ)
13		subcos 16227	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((cos‘𝐴) − (cos‘𝐵)) = (2 · ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2)))))
14	7, 12, 13	syl2anc 595	. . 3 ⊢ (𝜑 → ((cos‘𝐴) − (cos‘𝐵)) = (2 · ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2)))))
15		2rp 13017	. . . 4 ⊢ 2 ∈ ℝ⁺
16	6, 11	readdcld 11234	. . . . . . . 8 ⊢ (𝜑 → (𝐵 + 𝐴) ∈ ℝ)
17	16	rehalfcld 12487	. . . . . . 7 ⊢ (𝜑 → ((𝐵 + 𝐴) / 2) ∈ ℝ)
18	17	resincld 16195	. . . . . 6 ⊢ (𝜑 → (sin‘((𝐵 + 𝐴) / 2)) ∈ ℝ)
19	2	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℝ)
20	10	simp2d 1159	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 𝐴)
21		cosord.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 < 𝐵)
22	19, 11, 6, 20, 21	lelttrd 11364	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝐵)
23	6, 11, 22, 20	addgtge0d 11784	. . . . . . . . 9 ⊢ (𝜑 → 0 < (𝐵 + 𝐴))
24		2re 12311	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
25		2pos 12341	. . . . . . . . . 10 ⊢ 0 < 2
26		divgt0 12079	. . . . . . . . . 10 ⊢ ((((𝐵 + 𝐴) ∈ ℝ ∧ 0 < (𝐵 + 𝐴)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 < ((𝐵 + 𝐴) / 2))
27	24, 25, 26	mpanr12 717	. . . . . . . . 9 ⊢ (((𝐵 + 𝐴) ∈ ℝ ∧ 0 < (𝐵 + 𝐴)) → 0 < ((𝐵 + 𝐴) / 2))
28	16, 23, 27	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → 0 < ((𝐵 + 𝐴) / 2))
29	3	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → π ∈ ℝ)
30	11, 6, 6, 21	ltadd2dd 11365	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 + 𝐴) < (𝐵 + 𝐵))
31	7	2timesd 12483	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · 𝐵) = (𝐵 + 𝐵))
32	30, 31	breqtrrd 5140	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 + 𝐴) < (2 · 𝐵))
33	24	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
34	25	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 2)
35		ltdivmul 12086	. . . . . . . . . . 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 1399	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐵 + 𝐴) / 2) < 𝐵 ↔ (𝐵 + 𝐴) < (2 · 𝐵)))
37	32, 36	mpbird 260	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 + 𝐴) / 2) < 𝐵)
38	5	simp3d 1160	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≤ π)
39	17, 6, 29, 37, 38	ltletrd 11366	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 + 𝐴) / 2) < π)
40		0xr 11252	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
41	3	rexri 11263	. . . . . . . . 9 ⊢ π ∈ ℝ^*
42		elioo2 13409	. . . . . . . . 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 704	. . . . . . . 8 ⊢ (((𝐵 + 𝐴) / 2) ∈ (0(,)π) ↔ (((𝐵 + 𝐴) / 2) ∈ ℝ ∧ 0 < ((𝐵 + 𝐴) / 2) ∧ ((𝐵 + 𝐴) / 2) < π))
44	17, 28, 39, 43	syl3anbrc 1360	. . . . . . 7 ⊢ (𝜑 → ((𝐵 + 𝐴) / 2) ∈ (0(,)π))
45		sinq12gt0 26634	. . . . . . 7 ⊢ (((𝐵 + 𝐴) / 2) ∈ (0(,)π) → 0 < (sin‘((𝐵 + 𝐴) / 2)))
46	44, 45	syl 18	. . . . . 6 ⊢ (𝜑 → 0 < (sin‘((𝐵 + 𝐴) / 2)))
47	18, 46	elrpd 13053	. . . . 5 ⊢ (𝜑 → (sin‘((𝐵 + 𝐴) / 2)) ∈ ℝ⁺)
48	6, 11	resubcld 11638	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
49	48	rehalfcld 12487	. . . . . . 7 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) ∈ ℝ)
50	49	resincld 16195	. . . . . 6 ⊢ (𝜑 → (sin‘((𝐵 − 𝐴) / 2)) ∈ ℝ)
51	11, 6	posdifd 11797	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
52	21, 51	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
53		divgt0 12079	. . . . . . . . . 10 ⊢ ((((𝐵 − 𝐴) ∈ ℝ ∧ 0 < (𝐵 − 𝐴)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 < ((𝐵 − 𝐴) / 2))
54	24, 25, 53	mpanr12 717	. . . . . . . . 9 ⊢ (((𝐵 − 𝐴) ∈ ℝ ∧ 0 < (𝐵 − 𝐴)) → 0 < ((𝐵 − 𝐴) / 2))
55	48, 52, 54	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → 0 < ((𝐵 − 𝐴) / 2))
56		rehalfcl 12467	. . . . . . . . . 10 ⊢ (π ∈ ℝ → (π / 2) ∈ ℝ)
57	3, 56	mp1i 14	. . . . . . . . 9 ⊢ (𝜑 → (π / 2) ∈ ℝ)
58	6, 11	subge02d 11802	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ 𝐴 ↔ (𝐵 − 𝐴) ≤ 𝐵))
59	20, 58	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ 𝐵)
60	48, 6, 29, 59, 38	letrd 11363	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ π)
61		lediv1 12076	. . . . . . . . . . 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 1399	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝐴) ≤ π ↔ ((𝐵 − 𝐴) / 2) ≤ (π / 2)))
63	60, 62	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) ≤ (π / 2))
64		pirp 26588	. . . . . . . . . 10 ⊢ π ∈ ℝ⁺
65		rphalflt 13043	. . . . . . . . . 10 ⊢ (π ∈ ℝ⁺ → (π / 2) < π)
66	64, 65	mp1i 14	. . . . . . . . 9 ⊢ (𝜑 → (π / 2) < π)
67	49, 57, 29, 63, 66	lelttrd 11364	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) < π)
68		elioo2 13409	. . . . . . . . 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 704	. . . . . . . 8 ⊢ (((𝐵 − 𝐴) / 2) ∈ (0(,)π) ↔ (((𝐵 − 𝐴) / 2) ∈ ℝ ∧ 0 < ((𝐵 − 𝐴) / 2) ∧ ((𝐵 − 𝐴) / 2) < π))
70	49, 55, 67, 69	syl3anbrc 1360	. . . . . . 7 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) ∈ (0(,)π))
71		sinq12gt0 26634	. . . . . . 7 ⊢ (((𝐵 − 𝐴) / 2) ∈ (0(,)π) → 0 < (sin‘((𝐵 − 𝐴) / 2)))
72	70, 71	syl 18	. . . . . 6 ⊢ (𝜑 → 0 < (sin‘((𝐵 − 𝐴) / 2)))
73	50, 72	elrpd 13053	. . . . 5 ⊢ (𝜑 → (sin‘((𝐵 − 𝐴) / 2)) ∈ ℝ⁺)
74	47, 73	rpmulcld 13072	. . . 4 ⊢ (𝜑 → ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2))) ∈ ℝ⁺)
75		rpmulcl 13037	. . . 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 598	. . 3 ⊢ (𝜑 → (2 · ((sin‘((𝐵 + 𝐴) / 2)) · (sin‘((𝐵 − 𝐴) / 2)))) ∈ ℝ⁺)
77	14, 76	eqeltrd 2869	. 2 ⊢ (𝜑 → ((cos‘𝐴) − (cos‘𝐵)) ∈ ℝ⁺)
78	6	recoscld 16196	. . 3 ⊢ (𝜑 → (cos‘𝐵) ∈ ℝ)
79	11	recoscld 16196	. . 3 ⊢ (𝜑 → (cos‘𝐴) ∈ ℝ)
80		difrp 13052	. . 3 ⊢ (((cos‘𝐵) ∈ ℝ ∧ (cos‘𝐴) ∈ ℝ) → ((cos‘𝐵) < (cos‘𝐴) ↔ ((cos‘𝐴) − (cos‘𝐵)) ∈ ℝ⁺))
81	78, 79, 80	syl2anc 595	. 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 1101 = wceq 1567 ∈ wcel 2149 class class class wbr 5110 ‘cfv 6534 (class class class)co 7408 ℂcc 11094 ℝcr 11095 0cc0 11096 + caddc 11099 · cmul 11101 ℝ^*cxr 11238 < clt 11239 ≤ cle 11240 − cmin 11437 / cdiv 11867 2c2 12291 ℝ⁺crp 13012 (,)cioo 13368 [,]cicc 13371 sincsin 16113 cosccos 16114 πcpi 16116

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-inf2 9606 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-pre-sup 11174 ax-addf 11175

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6300 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7672 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-div 11868 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12501 df-z 12588 df-dec 12708 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13372 df-ioc 13373 df-ico 13374 df-icc 13375 df-fz 13532 df-fzo 13679 df-fl 13821 df-seq 14034 df-exp 14094 df-fac 14306 df-bc 14335 df-hash 14363 df-shft 15100 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-limsup 15518 df-clim 15535 df-rlim 15536 df-sum 15734 df-ef 16117 df-sin 16119 df-cos 16120 df-pi 16122 df-struct 17203 df-sets 17220 df-slot 17238 df-ndx 17250 df-base 17266 df-ress 17287 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-rest 17471 df-topn 17472 df-0g 17490 df-gsum 17491 df-topgen 17492 df-pt 17493 df-prds 17496 df-xrs 17552 df-qtop 17557 df-imas 17558 df-xps 17560 df-mre 17634 df-mrc 17635 df-acs 17637 df-mgm 18694 df-sgrp 18773 df-mnd 18789 df-submnd 18838 df-mulg 19130 df-cntz 19383 df-cmn 19848 df-psmet 21479 df-xmet 21480 df-met 21481 df-bl 21482 df-mopn 21483 df-fbas 21484 df-fg 21485 df-cnfld 21488 df-top 23016 df-topon 23033 df-topsp 23055 df-bases 23068 df-cld 23141 df-ntr 23142 df-cls 23143 df-nei 23220 df-lp 23258 df-perf 23259 df-cn 23349 df-cnp 23350 df-haus 23437 df-tx 23684 df-hmeo 23877 df-fil 23968 df-fm 24060 df-flim 24061 df-flf 24062 df-xms 24442 df-ms 24443 df-tms 24444 df-cncf 25002 df-limc 25990 df-dv 25991

This theorem is referenced by: cosord 26658 cos0pilt1 26659

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