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Mirrors > Home > ILE Home > Th. List > icccntr | GIF version |
Description: Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
icccntr.1 | ⊢ (𝐴 / 𝑅) = 𝐶 |
icccntr.2 | ⊢ (𝐵 / 𝑅) = 𝐷 |
Ref | Expression |
---|---|
icccntr | ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 / 𝑅) ∈ (𝐶[,]𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 102 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+) → 𝑋 ∈ ℝ) | |
2 | rerpdivcl 8613 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+) → (𝑋 / 𝑅) ∈ ℝ) | |
3 | 1, 2 | 2thd 164 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+) → (𝑋 ∈ ℝ ↔ (𝑋 / 𝑅) ∈ ℝ)) |
4 | 3 | adantl 262 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ∈ ℝ ↔ (𝑋 / 𝑅) ∈ ℝ)) |
5 | elrp 8585 | . . . . . . 7 ⊢ (𝑅 ∈ ℝ+ ↔ (𝑅 ∈ ℝ ∧ 0 < 𝑅)) | |
6 | lediv1 7835 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑋 ∈ ℝ ∧ (𝑅 ∈ ℝ ∧ 0 < 𝑅)) → (𝐴 ≤ 𝑋 ↔ (𝐴 / 𝑅) ≤ (𝑋 / 𝑅))) | |
7 | 5, 6 | syl3an3b 1173 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+) → (𝐴 ≤ 𝑋 ↔ (𝐴 / 𝑅) ≤ (𝑋 / 𝑅))) |
8 | 7 | 3expb 1105 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝐴 ≤ 𝑋 ↔ (𝐴 / 𝑅) ≤ (𝑋 / 𝑅))) |
9 | 8 | adantlr 446 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝐴 ≤ 𝑋 ↔ (𝐴 / 𝑅) ≤ (𝑋 / 𝑅))) |
10 | icccntr.1 | . . . . 5 ⊢ (𝐴 / 𝑅) = 𝐶 | |
11 | 10 | breq1i 3771 | . . . 4 ⊢ ((𝐴 / 𝑅) ≤ (𝑋 / 𝑅) ↔ 𝐶 ≤ (𝑋 / 𝑅)) |
12 | 9, 11 | syl6bb 185 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝐴 ≤ 𝑋 ↔ 𝐶 ≤ (𝑋 / 𝑅))) |
13 | lediv1 7835 | . . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝑅 ∈ ℝ ∧ 0 < 𝑅)) → (𝑋 ≤ 𝐵 ↔ (𝑋 / 𝑅) ≤ (𝐵 / 𝑅))) | |
14 | 5, 13 | syl3an3b 1173 | . . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑅 ∈ ℝ+) → (𝑋 ≤ 𝐵 ↔ (𝑋 / 𝑅) ≤ (𝐵 / 𝑅))) |
15 | 14 | 3expb 1105 | . . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ≤ 𝐵 ↔ (𝑋 / 𝑅) ≤ (𝐵 / 𝑅))) |
16 | 15 | an12s 499 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ≤ 𝐵 ↔ (𝑋 / 𝑅) ≤ (𝐵 / 𝑅))) |
17 | 16 | adantll 445 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ≤ 𝐵 ↔ (𝑋 / 𝑅) ≤ (𝐵 / 𝑅))) |
18 | icccntr.2 | . . . . 5 ⊢ (𝐵 / 𝑅) = 𝐷 | |
19 | 18 | breq2i 3772 | . . . 4 ⊢ ((𝑋 / 𝑅) ≤ (𝐵 / 𝑅) ↔ (𝑋 / 𝑅) ≤ 𝐷) |
20 | 17, 19 | syl6bb 185 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ≤ 𝐵 ↔ (𝑋 / 𝑅) ≤ 𝐷)) |
21 | 4, 12, 20 | 3anbi123d 1207 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → ((𝑋 ∈ ℝ ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵) ↔ ((𝑋 / 𝑅) ∈ ℝ ∧ 𝐶 ≤ (𝑋 / 𝑅) ∧ (𝑋 / 𝑅) ≤ 𝐷))) |
22 | elicc2 8807 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 ∈ ℝ ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵))) | |
23 | 22 | adantr 261 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 ∈ ℝ ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵))) |
24 | rerpdivcl 8613 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑅 ∈ ℝ+) → (𝐴 / 𝑅) ∈ ℝ) | |
25 | 10, 24 | syl5eqelr 2125 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑅 ∈ ℝ+) → 𝐶 ∈ ℝ) |
26 | rerpdivcl 8613 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝑅 ∈ ℝ+) → (𝐵 / 𝑅) ∈ ℝ) | |
27 | 18, 26 | syl5eqelr 2125 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝑅 ∈ ℝ+) → 𝐷 ∈ ℝ) |
28 | elicc2 8807 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝑋 / 𝑅) ∈ (𝐶[,]𝐷) ↔ ((𝑋 / 𝑅) ∈ ℝ ∧ 𝐶 ≤ (𝑋 / 𝑅) ∧ (𝑋 / 𝑅) ≤ 𝐷))) | |
29 | 25, 27, 28 | syl2an 273 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑅 ∈ ℝ+) ∧ (𝐵 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → ((𝑋 / 𝑅) ∈ (𝐶[,]𝐷) ↔ ((𝑋 / 𝑅) ∈ ℝ ∧ 𝐶 ≤ (𝑋 / 𝑅) ∧ (𝑋 / 𝑅) ≤ 𝐷))) |
30 | 29 | anandirs 527 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑅 ∈ ℝ+) → ((𝑋 / 𝑅) ∈ (𝐶[,]𝐷) ↔ ((𝑋 / 𝑅) ∈ ℝ ∧ 𝐶 ≤ (𝑋 / 𝑅) ∧ (𝑋 / 𝑅) ≤ 𝐷))) |
31 | 30 | adantrl 447 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → ((𝑋 / 𝑅) ∈ (𝐶[,]𝐷) ↔ ((𝑋 / 𝑅) ∈ ℝ ∧ 𝐶 ≤ (𝑋 / 𝑅) ∧ (𝑋 / 𝑅) ≤ 𝐷))) |
32 | 21, 23, 31 | 3bitr4d 209 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 / 𝑅) ∈ (𝐶[,]𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 885 = wceq 1243 ∈ wcel 1393 class class class wbr 3764 (class class class)co 5512 ℝcr 6888 0cc0 6889 < clt 7060 ≤ cle 7061 / cdiv 7651 ℝ+crp 8583 [,]cicc 8760 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 ax-pre-mulext 7002 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 df-div 7652 df-rp 8584 df-icc 8764 |
This theorem is referenced by: icccntri 8869 |
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