sinq12gt0 - Metamath Proof Explorer

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Theorem sinq12gt0 25916

Description: The sine of a number strictly between 0 and π is positive. (Contributed by Paul Chapman, 15-Mar-2008.)

**Assertion**
Ref	Expression
sinq12gt0	⊢ (𝐴 ∈ (0(,)π) → 0 < (sin‘𝐴))

**Proof of Theorem sinq12gt0**
Step	Hyp	Ref	Expression
1		0xr 11226	. . 3 ⊢ 0 ∈ ℝ^*
2		pire 25867	. . . 4 ⊢ π ∈ ℝ
3	2	rexri 11237	. . 3 ⊢ π ∈ ℝ^*
4		elioo2 13330	. . 3 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^*) → (𝐴 ∈ (0(,)π) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π)))
5	1, 3, 4	mp2an 690	. 2 ⊢ (𝐴 ∈ (0(,)π) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π))
6		rehalfcl 12403	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ)
7	6	3ad2ant1 1133	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π) → (𝐴 / 2) ∈ ℝ)
8		halfpos2 12406	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ 0 < (𝐴 / 2)))
9	8	biimpa 477	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < (𝐴 / 2))
10	9	3adant3 1132	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π) → 0 < (𝐴 / 2))
11		2re 12251	. . . . . . . . 9 ⊢ 2 ∈ ℝ
12		2pos 12280	. . . . . . . . 9 ⊢ 0 < 2
13	11, 12	pm3.2i 471	. . . . . . . 8 ⊢ (2 ∈ ℝ ∧ 0 < 2)
14		ltdiv1 12043	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ π ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (𝐴 < π ↔ (𝐴 / 2) < (π / 2)))
15	2, 13, 14	mp3an23 1453	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 < π ↔ (𝐴 / 2) < (π / 2)))
16	15	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 < π ↔ (𝐴 / 2) < (π / 2)))
17	16	biimp3a 1469	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π) → (𝐴 / 2) < (π / 2))
18		sincosq1lem 25906	. . . . 5 ⊢ (((𝐴 / 2) ∈ ℝ ∧ 0 < (𝐴 / 2) ∧ (𝐴 / 2) < (π / 2)) → 0 < (sin‘(𝐴 / 2)))
19	7, 10, 17, 18	syl3anc 1371	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π) → 0 < (sin‘(𝐴 / 2)))
20		resubcl 11489	. . . . . . . . 9 ⊢ ((π ∈ ℝ ∧ 𝐴 ∈ ℝ) → (π − 𝐴) ∈ ℝ)
21	2, 20	mpan 688	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (π − 𝐴) ∈ ℝ)
22		rehalfcl 12403	. . . . . . . 8 ⊢ ((π − 𝐴) ∈ ℝ → ((π − 𝐴) / 2) ∈ ℝ)
23	21, 22	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((π − 𝐴) / 2) ∈ ℝ)
24	23	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π) → ((π − 𝐴) / 2) ∈ ℝ)
25		posdif 11672	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ π ∈ ℝ) → (𝐴 < π ↔ 0 < (π − 𝐴)))
26	2, 25	mpan2 689	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 < π ↔ 0 < (π − 𝐴)))
27		halfpos2 12406	. . . . . . . . . 10 ⊢ ((π − 𝐴) ∈ ℝ → (0 < (π − 𝐴) ↔ 0 < ((π − 𝐴) / 2)))
28	21, 27	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (0 < (π − 𝐴) ↔ 0 < ((π − 𝐴) / 2)))
29	26, 28	bitrd 278	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 < π ↔ 0 < ((π − 𝐴) / 2)))
30	29	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 < π ↔ 0 < ((π − 𝐴) / 2)))
31	30	biimp3a 1469	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π) → 0 < ((π − 𝐴) / 2))
32		ltsubpos 11671	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ π ∈ ℝ) → (0 < 𝐴 ↔ (π − 𝐴) < π))
33	2, 32	mpan2 689	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ (π − 𝐴) < π))
34		ltdiv1 12043	. . . . . . . . . . 11 ⊢ (((π − 𝐴) ∈ ℝ ∧ π ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((π − 𝐴) < π ↔ ((π − 𝐴) / 2) < (π / 2)))
35	2, 13, 34	mp3an23 1453	. . . . . . . . . 10 ⊢ ((π − 𝐴) ∈ ℝ → ((π − 𝐴) < π ↔ ((π − 𝐴) / 2) < (π / 2)))
36	21, 35	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → ((π − 𝐴) < π ↔ ((π − 𝐴) / 2) < (π / 2)))
37	33, 36	bitrd 278	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ ((π − 𝐴) / 2) < (π / 2)))
38	37	biimpa 477	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ((π − 𝐴) / 2) < (π / 2))
39	38	3adant3 1132	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π) → ((π − 𝐴) / 2) < (π / 2))
40		sincosq1lem 25906	. . . . . 6 ⊢ ((((π − 𝐴) / 2) ∈ ℝ ∧ 0 < ((π − 𝐴) / 2) ∧ ((π − 𝐴) / 2) < (π / 2)) → 0 < (sin‘((π − 𝐴) / 2)))
41	24, 31, 39, 40	syl3anc 1371	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π) → 0 < (sin‘((π − 𝐴) / 2)))
42		recn 11165	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
43		picn 25868	. . . . . . . . . 10 ⊢ π ∈ ℂ
44		2cnne0 12387	. . . . . . . . . 10 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
45		divsubdir 11873	. . . . . . . . . 10 ⊢ ((π ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((π − 𝐴) / 2) = ((π / 2) − (𝐴 / 2)))
46	43, 44, 45	mp3an13 1452	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((π − 𝐴) / 2) = ((π / 2) − (𝐴 / 2)))
47	42, 46	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((π − 𝐴) / 2) = ((π / 2) − (𝐴 / 2)))
48	47	fveq2d 6866	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (sin‘((π − 𝐴) / 2)) = (sin‘((π / 2) − (𝐴 / 2))))
49	6	recnd 11207	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℂ)
50		sinhalfpim 25902	. . . . . . . 8 ⊢ ((𝐴 / 2) ∈ ℂ → (sin‘((π / 2) − (𝐴 / 2))) = (cos‘(𝐴 / 2)))
51	49, 50	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (sin‘((π / 2) − (𝐴 / 2))) = (cos‘(𝐴 / 2)))
52	48, 51	eqtrd 2771	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (sin‘((π − 𝐴) / 2)) = (cos‘(𝐴 / 2)))
53	52	3ad2ant1 1133	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π) → (sin‘((π − 𝐴) / 2)) = (cos‘(𝐴 / 2)))
54	41, 53	breqtrd 5151	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π) → 0 < (cos‘(𝐴 / 2)))
55		resincl 16048	. . . . . . . 8 ⊢ ((𝐴 / 2) ∈ ℝ → (sin‘(𝐴 / 2)) ∈ ℝ)
56		recoscl 16049	. . . . . . . 8 ⊢ ((𝐴 / 2) ∈ ℝ → (cos‘(𝐴 / 2)) ∈ ℝ)
57	55, 56	jca 512	. . . . . . 7 ⊢ ((𝐴 / 2) ∈ ℝ → ((sin‘(𝐴 / 2)) ∈ ℝ ∧ (cos‘(𝐴 / 2)) ∈ ℝ))
58		axmulgt0 11253	. . . . . . 7 ⊢ (((sin‘(𝐴 / 2)) ∈ ℝ ∧ (cos‘(𝐴 / 2)) ∈ ℝ) → ((0 < (sin‘(𝐴 / 2)) ∧ 0 < (cos‘(𝐴 / 2))) → 0 < ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2)))))
59	6, 57, 58	3syl 18	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((0 < (sin‘(𝐴 / 2)) ∧ 0 < (cos‘(𝐴 / 2))) → 0 < ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2)))))
60		remulcl 11160	. . . . . . . . 9 ⊢ (((sin‘(𝐴 / 2)) ∈ ℝ ∧ (cos‘(𝐴 / 2)) ∈ ℝ) → ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2))) ∈ ℝ)
61	6, 57, 60	3syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2))) ∈ ℝ)
62		axmulgt0 11253	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2))) ∈ ℝ) → ((0 < 2 ∧ 0 < ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2)))) → 0 < (2 · ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2))))))
63	11, 61, 62	sylancr 587	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((0 < 2 ∧ 0 < ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2)))) → 0 < (2 · ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2))))))
64	12, 63	mpani 694	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (0 < ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2))) → 0 < (2 · ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2))))))
65	59, 64	syld 47	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((0 < (sin‘(𝐴 / 2)) ∧ 0 < (cos‘(𝐴 / 2))) → 0 < (2 · ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2))))))
66	65	3ad2ant1 1133	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π) → ((0 < (sin‘(𝐴 / 2)) ∧ 0 < (cos‘(𝐴 / 2))) → 0 < (2 · ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2))))))
67	19, 54, 66	mp2and 697	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π) → 0 < (2 · ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2)))))
68		2cn 12252	. . . . . . . 8 ⊢ 2 ∈ ℂ
69		2ne0 12281	. . . . . . . 8 ⊢ 2 ≠ 0
70		divcan2 11845	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (2 · (𝐴 / 2)) = 𝐴)
71	68, 69, 70	mp3an23 1453	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (2 · (𝐴 / 2)) = 𝐴)
72	42, 71	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (2 · (𝐴 / 2)) = 𝐴)
73	72	fveq2d 6866	. . . . 5 ⊢ (𝐴 ∈ ℝ → (sin‘(2 · (𝐴 / 2))) = (sin‘𝐴))
74		sin2t 16085	. . . . . 6 ⊢ ((𝐴 / 2) ∈ ℂ → (sin‘(2 · (𝐴 / 2))) = (2 · ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2)))))
75	49, 74	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℝ → (sin‘(2 · (𝐴 / 2))) = (2 · ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2)))))
76	73, 75	eqtr3d 2773	. . . 4 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) = (2 · ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2)))))
77	76	3ad2ant1 1133	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π) → (sin‘𝐴) = (2 · ((sin‘(𝐴 / 2)) · (cos‘(𝐴 / 2)))))
78	67, 77	breqtrrd 5153	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π) → 0 < (sin‘𝐴))
79	5, 78	sylbi 216	1 ⊢ (𝐴 ∈ (0(,)π) → 0 < (sin‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 class class class wbr 5125 ‘cfv 6516 (class class class)co 7377 ℂcc 11073 ℝcr 11074 0cc0 11075 · cmul 11080 ℝ^*cxr 11212 < clt 11213 − cmin 11409 / cdiv 11836 2c2 12232 (,)cioo 13289 sincsin 15972 cosccos 15973 πcpi 15975

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4886 df-int 4928 df-iun 4976 df-iin 4977 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-se 5609 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-of 7637 df-om 7823 df-1st 7941 df-2nd 7942 df-supp 8113 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8670 df-map 8789 df-pm 8790 df-ixp 8858 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-fsupp 9328 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9470 df-card 9899 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-div 11837 df-nn 12178 df-2 12240 df-3 12241 df-4 12242 df-5 12243 df-6 12244 df-7 12245 df-8 12246 df-9 12247 df-n0 12438 df-z 12524 df-dec 12643 df-uz 12788 df-q 12898 df-rp 12940 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13450 df-fzo 13593 df-fl 13722 df-seq 13932 df-exp 13993 df-fac 14199 df-bc 14228 df-hash 14256 df-shft 14979 df-cj 15011 df-re 15012 df-im 15013 df-sqrt 15147 df-abs 15148 df-limsup 15380 df-clim 15397 df-rlim 15398 df-sum 15598 df-ef 15976 df-sin 15978 df-cos 15979 df-pi 15981 df-struct 17045 df-sets 17062 df-slot 17080 df-ndx 17092 df-base 17110 df-ress 17139 df-plusg 17175 df-mulr 17176 df-starv 17177 df-sca 17178 df-vsca 17179 df-ip 17180 df-tset 17181 df-ple 17182 df-ds 17184 df-unif 17185 df-hom 17186 df-cco 17187 df-rest 17333 df-topn 17334 df-0g 17352 df-gsum 17353 df-topgen 17354 df-pt 17355 df-prds 17358 df-xrs 17413 df-qtop 17418 df-imas 17419 df-xps 17421 df-mre 17495 df-mrc 17496 df-acs 17498 df-mgm 18526 df-sgrp 18575 df-mnd 18586 df-submnd 18631 df-mulg 18902 df-cntz 19126 df-cmn 19593 df-psmet 20840 df-xmet 20841 df-met 20842 df-bl 20843 df-mopn 20844 df-fbas 20845 df-fg 20846 df-cnfld 20849 df-top 22295 df-topon 22312 df-topsp 22334 df-bases 22348 df-cld 22422 df-ntr 22423 df-cls 22424 df-nei 22501 df-lp 22539 df-perf 22540 df-cn 22630 df-cnp 22631 df-haus 22718 df-tx 22965 df-hmeo 23158 df-fil 23249 df-fm 23341 df-flim 23342 df-flf 23343 df-xms 23725 df-ms 23726 df-tms 23727 df-cncf 24293 df-limc 25282 df-dv 25283

This theorem is referenced by: sinq12ge0 25917 sinq34lt0t 25918 cosq14gt0 25919 sineq0 25932 cosordlem 25938 tan2h 36177 sineq0ALT 43374 wallispilem1 44459

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