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Mirrors > Home > MPE Home > Th. List > sinltx | Structured version Visualization version GIF version |
Description: The sine of a positive real number is less than its argument. (Contributed by Mario Carneiro, 29-Jul-2014.) |
Ref | Expression |
---|---|
sinltx | ⊢ (𝐴 ∈ ℝ+ → (sin‘𝐴) < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 12398 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | 1 | adantr 483 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → 𝐴 ∈ ℝ) |
3 | 2 | resincld 15496 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → (sin‘𝐴) ∈ ℝ) |
4 | 1red 10642 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → 1 ∈ ℝ) | |
5 | sinbnd 15533 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (-1 ≤ (sin‘𝐴) ∧ (sin‘𝐴) ≤ 1)) | |
6 | 5 | simprd 498 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ≤ 1) |
7 | 1, 6 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (sin‘𝐴) ≤ 1) |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → (sin‘𝐴) ≤ 1) |
9 | simpr 487 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → 1 < 𝐴) | |
10 | 3, 4, 2, 8, 9 | lelttrd 10798 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → (sin‘𝐴) < 𝐴) |
11 | df-3an 1085 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 1) ↔ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ 𝐴 ≤ 1)) | |
12 | 0xr 10688 | . . . . 5 ⊢ 0 ∈ ℝ* | |
13 | 1re 10641 | . . . . 5 ⊢ 1 ∈ ℝ | |
14 | elioc2 12800 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ) → (𝐴 ∈ (0(,]1) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 1))) | |
15 | 12, 13, 14 | mp2an 690 | . . . 4 ⊢ (𝐴 ∈ (0(,]1) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 1)) |
16 | elrp 12392 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
17 | 16 | anbi1i 625 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) ↔ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ 𝐴 ≤ 1)) |
18 | 11, 15, 17 | 3bitr4i 305 | . . 3 ⊢ (𝐴 ∈ (0(,]1) ↔ (𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1)) |
19 | sin01bnd 15538 | . . . 4 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴 − ((𝐴↑3) / 3)) < (sin‘𝐴) ∧ (sin‘𝐴) < 𝐴)) | |
20 | 19 | simprd 498 | . . 3 ⊢ (𝐴 ∈ (0(,]1) → (sin‘𝐴) < 𝐴) |
21 | 18, 20 | sylbir 237 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (sin‘𝐴) < 𝐴) |
22 | 1red 10642 | . 2 ⊢ (𝐴 ∈ ℝ+ → 1 ∈ ℝ) | |
23 | 10, 21, 22, 1 | ltlecasei 10748 | 1 ⊢ (𝐴 ∈ ℝ+ → (sin‘𝐴) < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 − cmin 10870 -cneg 10871 / cdiv 11297 3c3 11694 ℝ+crp 12390 (,]cioc 12740 ↑cexp 13430 sincsin 15417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-ioc 12744 df-ico 12745 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 |
This theorem is referenced by: pigt3 25103 basellem8 25665 |
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