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Mirrors > Home > MPE Home > Th. List > efif1olem1 | Structured version Visualization version GIF version |
Description: Lemma for efif1o 26525. (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 771 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ 𝐷) | |
2 | efif1olem1.1 | . . . . . . 7 ⊢ 𝐷 = (𝐴(,](𝐴 + (2 · π))) | |
3 | 1, 2 | eleqtrdi 2835 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ (𝐴(,](𝐴 + (2 · π)))) |
4 | rexr 11292 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
5 | simpl 481 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝐴 ∈ ℝ) | |
6 | 2re 12319 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
7 | pire 26438 | . . . . . . . . 9 ⊢ π ∈ ℝ | |
8 | 6, 7 | remulcli 11262 | . . . . . . . 8 ⊢ (2 · π) ∈ ℝ |
9 | readdcl 11223 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (2 · π) ∈ ℝ) → (𝐴 + (2 · π)) ∈ ℝ) | |
10 | 5, 8, 9 | sylancl 584 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝐴 + (2 · π)) ∈ ℝ) |
11 | elioc2 13422 | . . . . . . 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 683 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑦 ∈ (𝐴(,](𝐴 + (2 · π))) ↔ (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 ≤ (𝐴 + (2 · π))))) |
13 | 3, 12 | mpbid 231 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 ≤ (𝐴 + (2 · π)))) |
14 | 13 | simp1d 1139 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ ℝ) |
15 | simprl 769 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐷) | |
16 | 15, 2 | eleqtrdi 2835 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ (𝐴(,](𝐴 + (2 · π)))) |
17 | elioc2 13422 | . . . . . . . 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 683 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 ∈ (𝐴(,](𝐴 + (2 · π))) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ (𝐴 + (2 · π))))) |
19 | 16, 18 | mpbid 231 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ (𝐴 + (2 · π)))) |
20 | 19 | simp1d 1139 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ ℝ) |
21 | readdcl 11223 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ (2 · π) ∈ ℝ) → (𝑥 + (2 · π)) ∈ ℝ) | |
22 | 20, 8, 21 | sylancl 584 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + (2 · π)) ∈ ℝ) |
23 | 13 | simp3d 1141 | . . . 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 1140 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝐴 < 𝑥) |
26 | 5, 20, 24, 25 | ltadd1dd 11857 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝐴 + (2 · π)) < (𝑥 + (2 · π))) |
27 | 14, 10, 22, 23, 26 | lelttrd 11404 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 < (𝑥 + (2 · π))) |
28 | 14, 24, 20 | ltsubaddd 11842 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → ((𝑦 − (2 · π)) < 𝑥 ↔ 𝑦 < (𝑥 + (2 · π)))) |
29 | 27, 28 | mpbird 256 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑦 − (2 · π)) < 𝑥) |
30 | readdcl 11223 | . . . 4 ⊢ ((𝑦 ∈ ℝ ∧ (2 · π) ∈ ℝ) → (𝑦 + (2 · π)) ∈ ℝ) | |
31 | 14, 8, 30 | sylancl 584 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑦 + (2 · π)) ∈ ℝ) |
32 | 19 | simp3d 1141 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ≤ (𝐴 + (2 · π))) |
33 | 13 | simp2d 1140 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝐴 < 𝑦) |
34 | 5, 14, 24, 33 | ltadd1dd 11857 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝐴 + (2 · π)) < (𝑦 + (2 · π))) |
35 | 20, 10, 31, 32, 34 | lelttrd 11404 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 < (𝑦 + (2 · π))) |
36 | 20, 14, 24 | absdifltd 15416 | . 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 711 | 1 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (abs‘(𝑥 − 𝑦)) < (2 · π)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 ℝcr 11139 + caddc 11143 · cmul 11145 ℝ*cxr 11279 < clt 11280 ≤ cle 11281 − cmin 11476 2c2 12300 (,]cioc 13360 abscabs 15217 πcpi 16046 |
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 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-ioc 13364 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-seq 14003 df-exp 14063 df-fac 14269 df-bc 14298 df-hash 14326 df-shft 15050 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-limsup 15451 df-clim 15468 df-rlim 15469 df-sum 15669 df-ef 16047 df-sin 16049 df-cos 16050 df-pi 16052 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-hom 17260 df-cco 17261 df-rest 17407 df-topn 17408 df-0g 17426 df-gsum 17427 df-topgen 17428 df-pt 17429 df-prds 17432 df-xrs 17487 df-qtop 17492 df-imas 17493 df-xps 17495 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-mulg 19032 df-cntz 19280 df-cmn 19749 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24270 df-ms 24271 df-tms 24272 df-cncf 24842 df-limc 25839 df-dv 25840 |
This theorem is referenced by: efif1o 26525 eff1o 26528 |
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