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Theorem sinltx 16137

Description: The sine of a positive real number is less than its argument. (Contributed by Mario Carneiro, 29-Jul-2014.)

**Assertion**
Ref	Expression
sinltx	⊢ (𝐴 ∈ ℝ⁺ → (sin‘𝐴) < 𝐴)

**Proof of Theorem sinltx**
Step	Hyp	Ref	Expression
1		rpre 12985	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 < 𝐴) → 𝐴 ∈ ℝ)
3	2	resincld 16091	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 < 𝐴) → (sin‘𝐴) ∈ ℝ)
4		1red 11216	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 < 𝐴) → 1 ∈ ℝ)
5		sinbnd 16128	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (-1 ≤ (sin‘𝐴) ∧ (sin‘𝐴) ≤ 1))
6	5	simprd 495	. . . . 5 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ≤ 1)
7	1, 6	syl 17	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (sin‘𝐴) ≤ 1)
8	7	adantr 480	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 < 𝐴) → (sin‘𝐴) ≤ 1)
9		simpr 484	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 < 𝐴) → 1 < 𝐴)
10	3, 4, 2, 8, 9	lelttrd 11373	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 < 𝐴) → (sin‘𝐴) < 𝐴)
11		df-3an 1086	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 1) ↔ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ 𝐴 ≤ 1))
12		0xr 11262	. . . . 5 ⊢ 0 ∈ ℝ^*
13		1re 11215	. . . . 5 ⊢ 1 ∈ ℝ
14		elioc2 13390	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝐴 ∈ (0(,]1) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 1)))
15	12, 13, 14	mp2an 689	. . . 4 ⊢ (𝐴 ∈ (0(,]1) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 1))
16		elrp 12979	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
17	16	anbi1i 623	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) ↔ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ 𝐴 ≤ 1))
18	11, 15, 17	3bitr4i 303	. . 3 ⊢ (𝐴 ∈ (0(,]1) ↔ (𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1))
19		sin01bnd 16133	. . . 4 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴 − ((𝐴↑3) / 3)) < (sin‘𝐴) ∧ (sin‘𝐴) < 𝐴))
20	19	simprd 495	. . 3 ⊢ (𝐴 ∈ (0(,]1) → (sin‘𝐴) < 𝐴)
21	18, 20	sylbir 234	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (sin‘𝐴) < 𝐴)
22		1red 11216	. 2 ⊢ (𝐴 ∈ ℝ⁺ → 1 ∈ ℝ)
23	10, 21, 22, 1	ltlecasei 11323	1 ⊢ (𝐴 ∈ ℝ⁺ → (sin‘𝐴) < 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5141 ‘cfv 6536 (class class class)co 7404 ℝcr 11108 0cc0 11109 1c1 11110 ℝ^*cxr 11248 < clt 11249 ≤ cle 11250 − cmin 11445 -cneg 11446 / cdiv 11872 3c3 12269 ℝ⁺crp 12977 (,]cioc 13328 ↑cexp 14030 sincsin 16011

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-ioc 13332 df-ico 13333 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14031 df-fac 14237 df-bc 14266 df-hash 14294 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018

This theorem is referenced by: pigt3 26403 basellem8 26971

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