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Mirrors > Home > MPE Home > Th. List > iihalf1 | Structured version Visualization version GIF version |
Description: Map the first half of II into II. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
iihalf1 | ⊢ (𝑋 ∈ (0[,](1 / 2)) → (2 · 𝑋) ∈ (0[,]1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 11512 | . . . . 5 ⊢ 2 ∈ ℝ | |
2 | remulcl 10418 | . . . . 5 ⊢ ((2 ∈ ℝ ∧ 𝑋 ∈ ℝ) → (2 · 𝑋) ∈ ℝ) | |
3 | 1, 2 | mpan 677 | . . . 4 ⊢ (𝑋 ∈ ℝ → (2 · 𝑋) ∈ ℝ) |
4 | 3 | 3ad2ant1 1113 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ (1 / 2)) → (2 · 𝑋) ∈ ℝ) |
5 | 0le2 11547 | . . . . 5 ⊢ 0 ≤ 2 | |
6 | mulge0 10957 | . . . . 5 ⊢ (((2 ∈ ℝ ∧ 0 ≤ 2) ∧ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋)) → 0 ≤ (2 · 𝑋)) | |
7 | 1, 5, 6 | mpanl12 689 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → 0 ≤ (2 · 𝑋)) |
8 | 7 | 3adant3 1112 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ (1 / 2)) → 0 ≤ (2 · 𝑋)) |
9 | 1re 10437 | . . . . . 6 ⊢ 1 ∈ ℝ | |
10 | 2pos 11548 | . . . . . . 7 ⊢ 0 < 2 | |
11 | 1, 10 | pm3.2i 463 | . . . . . 6 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
12 | lemuldiv2 11320 | . . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 1 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((2 · 𝑋) ≤ 1 ↔ 𝑋 ≤ (1 / 2))) | |
13 | 9, 11, 12 | mp3an23 1432 | . . . . 5 ⊢ (𝑋 ∈ ℝ → ((2 · 𝑋) ≤ 1 ↔ 𝑋 ≤ (1 / 2))) |
14 | 13 | biimpar 470 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝑋 ≤ (1 / 2)) → (2 · 𝑋) ≤ 1) |
15 | 14 | 3adant2 1111 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ (1 / 2)) → (2 · 𝑋) ≤ 1) |
16 | 4, 8, 15 | 3jca 1108 | . 2 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ (1 / 2)) → ((2 · 𝑋) ∈ ℝ ∧ 0 ≤ (2 · 𝑋) ∧ (2 · 𝑋) ≤ 1)) |
17 | 0re 10439 | . . 3 ⊢ 0 ∈ ℝ | |
18 | halfre 11659 | . . 3 ⊢ (1 / 2) ∈ ℝ | |
19 | 17, 18 | elicc2i 12616 | . 2 ⊢ (𝑋 ∈ (0[,](1 / 2)) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ (1 / 2))) |
20 | 17, 9 | elicc2i 12616 | . 2 ⊢ ((2 · 𝑋) ∈ (0[,]1) ↔ ((2 · 𝑋) ∈ ℝ ∧ 0 ≤ (2 · 𝑋) ∧ (2 · 𝑋) ≤ 1)) |
21 | 16, 19, 20 | 3imtr4i 284 | 1 ⊢ (𝑋 ∈ (0[,](1 / 2)) → (2 · 𝑋) ∈ (0[,]1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 ∈ wcel 2050 class class class wbr 4925 (class class class)co 6974 ℝcr 10332 0cc0 10333 1c1 10334 · cmul 10338 < clt 10472 ≤ cle 10473 / cdiv 11096 2c2 11493 [,]cicc 12555 |
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-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 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 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 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-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-po 5322 df-so 5323 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-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 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-2 11501 df-icc 12559 |
This theorem is referenced by: iihalf1cn 23251 phtpycc 23310 copco 23337 pcohtpylem 23338 pcopt 23341 pcopt2 23342 pcorevlem 23345 |
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