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Mirrors > Home > MPE Home > Th. List > divelunit | Structured version Visualization version GIF version |
Description: A condition for a ratio to be a member of the closed unit interval. (Contributed by Scott Fenton, 11-Jun-2013.) |
Ref | Expression |
---|---|
divelunit | ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ∈ (0[,]1) ↔ 𝐴 ≤ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elicc01 12693 | . . 3 ⊢ ((𝐴 / 𝐵) ∈ (0[,]1) ↔ ((𝐴 / 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝐵) ∧ (𝐴 / 𝐵) ≤ 1)) | |
2 | df-3an 1080 | . . 3 ⊢ (((𝐴 / 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝐵) ∧ (𝐴 / 𝐵) ≤ 1) ↔ (((𝐴 / 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝐵)) ∧ (𝐴 / 𝐵) ≤ 1)) | |
3 | 1, 2 | bitri 276 | . 2 ⊢ ((𝐴 / 𝐵) ∈ (0[,]1) ↔ (((𝐴 / 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝐵)) ∧ (𝐴 / 𝐵) ≤ 1)) |
4 | 1re 10476 | . . . . 5 ⊢ 1 ∈ ℝ | |
5 | ledivmul 11353 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ≤ 1 ↔ 𝐴 ≤ (𝐵 · 1))) | |
6 | 4, 5 | mp3an2 1439 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ≤ 1 ↔ 𝐴 ≤ (𝐵 · 1))) |
7 | 6 | adantlr 711 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ≤ 1 ↔ 𝐴 ≤ (𝐵 · 1))) |
8 | simpll 763 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 𝐴 ∈ ℝ) | |
9 | simprl 767 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 𝐵 ∈ ℝ) | |
10 | gt0ne0 10942 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 0 < 𝐵) → 𝐵 ≠ 0) | |
11 | 10 | adantl 482 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 𝐵 ≠ 0) |
12 | 8, 9, 11 | redivcld 11305 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 / 𝐵) ∈ ℝ) |
13 | divge0 11346 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 ≤ (𝐴 / 𝐵)) | |
14 | 12, 13 | jca 512 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝐵))) |
15 | 14 | biantrurd 533 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ≤ 1 ↔ (((𝐴 / 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝐵)) ∧ (𝐴 / 𝐵) ≤ 1))) |
16 | recn 10462 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
17 | 16 | ad2antrl 724 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 𝐵 ∈ ℂ) |
18 | 17 | mulid1d 10493 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐵 · 1) = 𝐵) |
19 | 18 | breq2d 4968 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 ≤ (𝐵 · 1) ↔ 𝐴 ≤ 𝐵)) |
20 | 7, 15, 19 | 3bitr3d 310 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((((𝐴 / 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝐵)) ∧ (𝐴 / 𝐵) ≤ 1) ↔ 𝐴 ≤ 𝐵)) |
21 | 3, 20 | syl5bb 284 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ∈ (0[,]1) ↔ 𝐴 ≤ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1078 ∈ wcel 2079 ≠ wne 2982 class class class wbr 4956 (class class class)co 7007 ℂcc 10370 ℝcr 10371 0cc0 10372 1c1 10373 · cmul 10377 < clt 10510 ≤ cle 10511 / cdiv 11134 [,]cicc 12580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-po 5354 df-so 5355 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-icc 12584 |
This theorem is referenced by: brbtwn2 26362 axsegconlem7 26380 axcontlem2 26422 axcontlem4 26424 axcontlem7 26427 axcontlem8 26428 |
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