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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 13506 | . . 3 ⊢ ((𝐴 / 𝐵) ∈ (0[,]1) ↔ ((𝐴 / 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝐵) ∧ (𝐴 / 𝐵) ≤ 1)) | |
| 2 | df-3an 1089 | . . 3 ⊢ (((𝐴 / 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝐵) ∧ (𝐴 / 𝐵) ≤ 1) ↔ (((𝐴 / 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝐵)) ∧ (𝐴 / 𝐵) ≤ 1)) | |
| 3 | 1, 2 | bitri 275 | . 2 ⊢ ((𝐴 / 𝐵) ∈ (0[,]1) ↔ (((𝐴 / 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝐵)) ∧ (𝐴 / 𝐵) ≤ 1)) |
| 4 | 1re 11261 | . . . . 5 ⊢ 1 ∈ ℝ | |
| 5 | ledivmul 12144 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ≤ 1 ↔ 𝐴 ≤ (𝐵 · 1))) | |
| 6 | 4, 5 | mp3an2 1451 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ≤ 1 ↔ 𝐴 ≤ (𝐵 · 1))) |
| 7 | 6 | adantlr 715 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ≤ 1 ↔ 𝐴 ≤ (𝐵 · 1))) |
| 8 | simpll 767 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 𝐴 ∈ ℝ) | |
| 9 | simprl 771 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 𝐵 ∈ ℝ) | |
| 10 | gt0ne0 11728 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 0 < 𝐵) → 𝐵 ≠ 0) | |
| 11 | 10 | adantl 481 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 𝐵 ≠ 0) |
| 12 | 8, 9, 11 | redivcld 12095 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 / 𝐵) ∈ ℝ) |
| 13 | divge0 12137 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 ≤ (𝐴 / 𝐵)) | |
| 14 | 12, 13 | jca 511 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝐵))) |
| 15 | 14 | biantrurd 532 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ≤ 1 ↔ (((𝐴 / 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝐵)) ∧ (𝐴 / 𝐵) ≤ 1))) |
| 16 | recn 11245 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
| 17 | 16 | ad2antrl 728 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 𝐵 ∈ ℂ) |
| 18 | 17 | mulridd 11278 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐵 · 1) = 𝐵) |
| 19 | 18 | breq2d 5155 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 ≤ (𝐵 · 1) ↔ 𝐴 ≤ 𝐵)) |
| 20 | 7, 15, 19 | 3bitr3d 309 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((((𝐴 / 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝐵)) ∧ (𝐴 / 𝐵) ≤ 1) ↔ 𝐴 ≤ 𝐵)) |
| 21 | 3, 20 | bitrid 283 | 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 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 < clt 11295 ≤ cle 11296 / cdiv 11920 [,]cicc 13390 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-icc 13394 |
| This theorem is referenced by: brbtwn2 28920 axsegconlem7 28938 axcontlem2 28980 axcontlem4 28982 axcontlem7 28985 axcontlem8 28986 |
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