fourierdlem44 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem44 39044

Description: A condition for having (sin‘(𝐴 / 2)) non zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Assertion**
Ref	Expression
fourierdlem44	⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 ≠ 0) → (sin‘(𝐴 / 2)) ≠ 0)

**Proof of Theorem fourierdlem44**
Step	Hyp	Ref	Expression
1		0xr 9965	. . . . . 6 ⊢ 0 ∈ ℝ^*
2	1	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 0 < 𝐴) → 0 ∈ ℝ^*)
3		2re 10967	. . . . . . . 8 ⊢ 2 ∈ ℝ
4		pire 24014	. . . . . . . 8 ⊢ π ∈ ℝ
5	3, 4	remulcli 9933	. . . . . . 7 ⊢ (2 · π) ∈ ℝ
6	5	rexri 9976	. . . . . 6 ⊢ (2 · π) ∈ ℝ^*
7	6	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 0 < 𝐴) → (2 · π) ∈ ℝ^*)
8	4	renegcli 10221	. . . . . . . 8 ⊢ -π ∈ ℝ
9	8	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (-π[,]π) → -π ∈ ℝ)
10	4	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (-π[,]π) → π ∈ ℝ)
11		id 22	. . . . . . 7 ⊢ (𝐴 ∈ (-π[,]π) → 𝐴 ∈ (-π[,]π))
12		eliccre 38575	. . . . . . 7 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ ∧ 𝐴 ∈ (-π[,]π)) → 𝐴 ∈ ℝ)
13	9, 10, 11, 12	syl3anc 1318	. . . . . 6 ⊢ (𝐴 ∈ (-π[,]π) → 𝐴 ∈ ℝ)
14	13	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 0 < 𝐴) → 𝐴 ∈ ℝ)
15		simpr 476	. . . . 5 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 0 < 𝐴) → 0 < 𝐴)
16	5	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (-π[,]π) → (2 · π) ∈ ℝ)
17	9	rexrd 9968	. . . . . . . 8 ⊢ (𝐴 ∈ (-π[,]π) → -π ∈ ℝ^*)
18	10	rexrd 9968	. . . . . . . 8 ⊢ (𝐴 ∈ (-π[,]π) → π ∈ ℝ^*)
19		iccleub 12100	. . . . . . . 8 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ 𝐴 ∈ (-π[,]π)) → 𝐴 ≤ π)
20	17, 18, 11, 19	syl3anc 1318	. . . . . . 7 ⊢ (𝐴 ∈ (-π[,]π) → 𝐴 ≤ π)
21		pirp 24017	. . . . . . . . 9 ⊢ π ∈ ℝ⁺
22		2timesgt 38441	. . . . . . . . 9 ⊢ (π ∈ ℝ⁺ → π < (2 · π))
23	21, 22	ax-mp 5	. . . . . . . 8 ⊢ π < (2 · π)
24	23	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (-π[,]π) → π < (2 · π))
25	13, 10, 16, 20, 24	lelttrd 10074	. . . . . 6 ⊢ (𝐴 ∈ (-π[,]π) → 𝐴 < (2 · π))
26	25	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 0 < 𝐴) → 𝐴 < (2 · π))
27	2, 7, 14, 15, 26	eliood 38567	. . . 4 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 0 < 𝐴) → 𝐴 ∈ (0(,)(2 · π)))
28	27	adantlr 747	. . 3 ⊢ (((𝐴 ∈ (-π[,]π) ∧ 𝐴 ≠ 0) ∧ 0 < 𝐴) → 𝐴 ∈ (0(,)(2 · π)))
29		sinaover2ne0 38751	. . 3 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (sin‘(𝐴 / 2)) ≠ 0)
30	28, 29	syl 17	. 2 ⊢ (((𝐴 ∈ (-π[,]π) ∧ 𝐴 ≠ 0) ∧ 0 < 𝐴) → (sin‘(𝐴 / 2)) ≠ 0)
31		simpll 786	. . 3 ⊢ (((𝐴 ∈ (-π[,]π) ∧ 𝐴 ≠ 0) ∧ ¬ 0 < 𝐴) → 𝐴 ∈ (-π[,]π))
32	31, 13	syl 17	. . . 4 ⊢ (((𝐴 ∈ (-π[,]π) ∧ 𝐴 ≠ 0) ∧ ¬ 0 < 𝐴) → 𝐴 ∈ ℝ)
33		0red 9920	. . . 4 ⊢ (((𝐴 ∈ (-π[,]π) ∧ 𝐴 ≠ 0) ∧ ¬ 0 < 𝐴) → 0 ∈ ℝ)
34		simplr 788	. . . 4 ⊢ (((𝐴 ∈ (-π[,]π) ∧ 𝐴 ≠ 0) ∧ ¬ 0 < 𝐴) → 𝐴 ≠ 0)
35		simpr 476	. . . 4 ⊢ (((𝐴 ∈ (-π[,]π) ∧ 𝐴 ≠ 0) ∧ ¬ 0 < 𝐴) → ¬ 0 < 𝐴)
36	32, 33, 34, 35	lttri5d 38454	. . 3 ⊢ (((𝐴 ∈ (-π[,]π) ∧ 𝐴 ≠ 0) ∧ ¬ 0 < 𝐴) → 𝐴 < 0)
37	13	recnd 9947	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (-π[,]π) → 𝐴 ∈ ℂ)
38	37	halfcld 11154	. . . . . . . . . 10 ⊢ (𝐴 ∈ (-π[,]π) → (𝐴 / 2) ∈ ℂ)
39		sinneg 14715	. . . . . . . . . 10 ⊢ ((𝐴 / 2) ∈ ℂ → (sin‘-(𝐴 / 2)) = -(sin‘(𝐴 / 2)))
40	38, 39	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ (-π[,]π) → (sin‘-(𝐴 / 2)) = -(sin‘(𝐴 / 2)))
41		2cnd 10970	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (-π[,]π) → 2 ∈ ℂ)
42		2ne0 10990	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
43	42	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (-π[,]π) → 2 ≠ 0)
44	37, 41, 43	divnegd 10693	. . . . . . . . . 10 ⊢ (𝐴 ∈ (-π[,]π) → -(𝐴 / 2) = (-𝐴 / 2))
45	44	fveq2d 6107	. . . . . . . . 9 ⊢ (𝐴 ∈ (-π[,]π) → (sin‘-(𝐴 / 2)) = (sin‘(-𝐴 / 2)))
46	40, 45	eqtr3d 2646	. . . . . . . 8 ⊢ (𝐴 ∈ (-π[,]π) → -(sin‘(𝐴 / 2)) = (sin‘(-𝐴 / 2)))
47	46	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → -(sin‘(𝐴 / 2)) = (sin‘(-𝐴 / 2)))
48	1	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → 0 ∈ ℝ^*)
49	6	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → (2 · π) ∈ ℝ^*)
50	13	renegcld 10336	. . . . . . . . . 10 ⊢ (𝐴 ∈ (-π[,]π) → -𝐴 ∈ ℝ)
51	50	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → -𝐴 ∈ ℝ)
52		simpr 476	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → 𝐴 < 0)
53	13	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → 𝐴 ∈ ℝ)
54	53	lt0neg1d 10476	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → (𝐴 < 0 ↔ 0 < -𝐴))
55	52, 54	mpbid 221	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → 0 < -𝐴)
56	5	renegcli 10221	. . . . . . . . . . . . 13 ⊢ -(2 · π) ∈ ℝ
57	56	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → -(2 · π) ∈ ℝ)
58	8	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → -π ∈ ℝ)
59	4, 5	ltnegi 10451	. . . . . . . . . . . . . 14 ⊢ (π < (2 · π) ↔ -(2 · π) < -π)
60	23, 59	mpbi 219	. . . . . . . . . . . . 13 ⊢ -(2 · π) < -π
61	60	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → -(2 · π) < -π)
62		iccgelb 12101	. . . . . . . . . . . . . 14 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^* ∧ 𝐴 ∈ (-π[,]π)) → -π ≤ 𝐴)
63	17, 18, 11, 62	syl3anc 1318	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (-π[,]π) → -π ≤ 𝐴)
64	63	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → -π ≤ 𝐴)
65	57, 58, 53, 61, 64	ltletrd 10076	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → -(2 · π) < 𝐴)
66	57, 53	ltnegd 10484	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → (-(2 · π) < 𝐴 ↔ -𝐴 < --(2 · π)))
67	65, 66	mpbid 221	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → -𝐴 < --(2 · π))
68	16	recnd 9947	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (-π[,]π) → (2 · π) ∈ ℂ)
69	68	negnegd 10262	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (-π[,]π) → --(2 · π) = (2 · π))
70	69	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → --(2 · π) = (2 · π))
71	67, 70	breqtrd 4609	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → -𝐴 < (2 · π))
72	48, 49, 51, 55, 71	eliood 38567	. . . . . . . 8 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → -𝐴 ∈ (0(,)(2 · π)))
73		sinaover2ne0 38751	. . . . . . . 8 ⊢ (-𝐴 ∈ (0(,)(2 · π)) → (sin‘(-𝐴 / 2)) ≠ 0)
74	72, 73	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → (sin‘(-𝐴 / 2)) ≠ 0)
75	47, 74	eqnetrd 2849	. . . . . 6 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → -(sin‘(𝐴 / 2)) ≠ 0)
76	75	neneqd 2787	. . . . 5 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → ¬ -(sin‘(𝐴 / 2)) = 0)
77	38	sincld 14699	. . . . . . 7 ⊢ (𝐴 ∈ (-π[,]π) → (sin‘(𝐴 / 2)) ∈ ℂ)
78	77	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → (sin‘(𝐴 / 2)) ∈ ℂ)
79	78	negeq0d 10263	. . . . 5 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → ((sin‘(𝐴 / 2)) = 0 ↔ -(sin‘(𝐴 / 2)) = 0))
80	76, 79	mtbird 314	. . . 4 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → ¬ (sin‘(𝐴 / 2)) = 0)
81	80	neqned 2789	. . 3 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 < 0) → (sin‘(𝐴 / 2)) ≠ 0)
82	31, 36, 81	syl2anc 691	. 2 ⊢ (((𝐴 ∈ (-π[,]π) ∧ 𝐴 ≠ 0) ∧ ¬ 0 < 𝐴) → (sin‘(𝐴 / 2)) ≠ 0)
83	30, 82	pm2.61dan 828	1 ⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 ≠ 0) → (sin‘(𝐴 / 2)) ≠ 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 · cmul 9820 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 -cneg 10146 / cdiv 10563 2c2 10947 ℝ⁺crp 11708 (,)cioo 12046 [,]cicc 12049 sincsin 14633 πcpi 14636

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437

This theorem is referenced by: fourierdlem56 39055 fourierdlem57 39056 fourierdlem58 39057 fourierdlem62 39061 fourierdlem66 39065 fourierdlem68 39067 fourierdlem72 39071 fourierdlem76 39075 fourierdlem78 39077 fourierdlem80 39079 fourierdlem103 39102 fourierdlem104 39103

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