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Mirrors > Home > MPE Home > Th. List > efif1olem1 | Structured version Visualization version GIF version |
Description: Lemma for efif1o 24843. (Contributed by Mario Carneiro, 13-May-2014.) |
Ref | Expression |
---|---|
efif1olem1.1 | ⊢ 𝐷 = (𝐴(,](𝐴 + (2 · π))) |
Ref | Expression |
---|---|
efif1olem1 | ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (abs‘(𝑥 − 𝑦)) < (2 · π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 760 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ 𝐷) | |
2 | efif1olem1.1 | . . . . . . 7 ⊢ 𝐷 = (𝐴(,](𝐴 + (2 · π))) | |
3 | 1, 2 | syl6eleq 2870 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ (𝐴(,](𝐴 + (2 · π)))) |
4 | rexr 10484 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
5 | simpl 475 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝐴 ∈ ℝ) | |
6 | 2re 11512 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
7 | pire 24759 | . . . . . . . . 9 ⊢ π ∈ ℝ | |
8 | 6, 7 | remulcli 10454 | . . . . . . . 8 ⊢ (2 · π) ∈ ℝ |
9 | readdcl 10416 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (2 · π) ∈ ℝ) → (𝐴 + (2 · π)) ∈ ℝ) | |
10 | 5, 8, 9 | sylancl 577 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝐴 + (2 · π)) ∈ ℝ) |
11 | elioc2 12613 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐴 + (2 · π)) ∈ ℝ) → (𝑦 ∈ (𝐴(,](𝐴 + (2 · π))) ↔ (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 ≤ (𝐴 + (2 · π))))) | |
12 | 4, 10, 11 | syl2an2r 672 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑦 ∈ (𝐴(,](𝐴 + (2 · π))) ↔ (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 ≤ (𝐴 + (2 · π))))) |
13 | 3, 12 | mpbid 224 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 ≤ (𝐴 + (2 · π)))) |
14 | 13 | simp1d 1122 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ ℝ) |
15 | simprl 758 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐷) | |
16 | 15, 2 | syl6eleq 2870 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ (𝐴(,](𝐴 + (2 · π)))) |
17 | elioc2 12613 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐴 + (2 · π)) ∈ ℝ) → (𝑥 ∈ (𝐴(,](𝐴 + (2 · π))) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ (𝐴 + (2 · π))))) | |
18 | 4, 10, 17 | syl2an2r 672 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 ∈ (𝐴(,](𝐴 + (2 · π))) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ (𝐴 + (2 · π))))) |
19 | 16, 18 | mpbid 224 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ (𝐴 + (2 · π)))) |
20 | 19 | simp1d 1122 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ ℝ) |
21 | readdcl 10416 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ (2 · π) ∈ ℝ) → (𝑥 + (2 · π)) ∈ ℝ) | |
22 | 20, 8, 21 | sylancl 577 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + (2 · π)) ∈ ℝ) |
23 | 13 | simp3d 1124 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ≤ (𝐴 + (2 · π))) |
24 | 8 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (2 · π) ∈ ℝ) |
25 | 19 | simp2d 1123 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝐴 < 𝑥) |
26 | 5, 20, 24, 25 | ltadd1dd 11050 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝐴 + (2 · π)) < (𝑥 + (2 · π))) |
27 | 14, 10, 22, 23, 26 | lelttrd 10596 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 < (𝑥 + (2 · π))) |
28 | 14, 24, 20 | ltsubaddd 11035 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → ((𝑦 − (2 · π)) < 𝑥 ↔ 𝑦 < (𝑥 + (2 · π)))) |
29 | 27, 28 | mpbird 249 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑦 − (2 · π)) < 𝑥) |
30 | readdcl 10416 | . . . 4 ⊢ ((𝑦 ∈ ℝ ∧ (2 · π) ∈ ℝ) → (𝑦 + (2 · π)) ∈ ℝ) | |
31 | 14, 8, 30 | sylancl 577 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑦 + (2 · π)) ∈ ℝ) |
32 | 19 | simp3d 1124 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ≤ (𝐴 + (2 · π))) |
33 | 13 | simp2d 1123 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝐴 < 𝑦) |
34 | 5, 14, 24, 33 | ltadd1dd 11050 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝐴 + (2 · π)) < (𝑦 + (2 · π))) |
35 | 20, 10, 31, 32, 34 | lelttrd 10596 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 < (𝑦 + (2 · π))) |
36 | 20, 14, 24 | absdifltd 14652 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → ((abs‘(𝑥 − 𝑦)) < (2 · π) ↔ ((𝑦 − (2 · π)) < 𝑥 ∧ 𝑥 < (𝑦 + (2 · π))))) |
37 | 29, 35, 36 | mpbir2and 700 | 1 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (abs‘(𝑥 − 𝑦)) < (2 · π)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 class class class wbr 4925 ‘cfv 6185 (class class class)co 6974 ℝcr 10332 + caddc 10336 · cmul 10338 ℝ*cxr 10471 < clt 10472 ≤ cle 10473 − cmin 10668 2c2 11493 (,]cioc 12553 abscabs 14452 πcpi 15278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 ax-addf 10412 ax-mulf 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-er 8087 df-map 8206 df-pm 8207 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-fi 8668 df-sup 8699 df-inf 8700 df-oi 8767 df-card 9160 df-cda 9386 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-q 12161 df-rp 12203 df-xneg 12322 df-xadd 12323 df-xmul 12324 df-ioo 12556 df-ioc 12557 df-ico 12558 df-icc 12559 df-fz 12707 df-fzo 12848 df-fl 12975 df-seq 13183 df-exp 13243 df-fac 13447 df-bc 13476 df-hash 13504 df-shft 14285 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-limsup 14687 df-clim 14704 df-rlim 14705 df-sum 14902 df-ef 15279 df-sin 15281 df-cos 15282 df-pi 15284 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-starv 16434 df-sca 16435 df-vsca 16436 df-ip 16437 df-tset 16438 df-ple 16439 df-ds 16441 df-unif 16442 df-hom 16443 df-cco 16444 df-rest 16550 df-topn 16551 df-0g 16569 df-gsum 16570 df-topgen 16571 df-pt 16572 df-prds 16575 df-xrs 16629 df-qtop 16634 df-imas 16635 df-xps 16637 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-mulg 18024 df-cntz 18230 df-cmn 18680 df-psmet 20251 df-xmet 20252 df-met 20253 df-bl 20254 df-mopn 20255 df-fbas 20256 df-fg 20257 df-cnfld 20260 df-top 21218 df-topon 21235 df-topsp 21257 df-bases 21270 df-cld 21343 df-ntr 21344 df-cls 21345 df-nei 21422 df-lp 21460 df-perf 21461 df-cn 21551 df-cnp 21552 df-haus 21639 df-tx 21886 df-hmeo 22079 df-fil 22170 df-fm 22262 df-flim 22263 df-flf 22264 df-xms 22645 df-ms 22646 df-tms 22647 df-cncf 23201 df-limc 24179 df-dv 24180 |
This theorem is referenced by: efif1o 24843 eff1o 24846 |
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