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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 10967 | . . . . 5 ⊢ 2 ∈ ℝ | |
2 | remulcl 9900 | . . . . 5 ⊢ ((2 ∈ ℝ ∧ 𝑋 ∈ ℝ) → (2 · 𝑋) ∈ ℝ) | |
3 | 1, 2 | mpan 702 | . . . 4 ⊢ (𝑋 ∈ ℝ → (2 · 𝑋) ∈ ℝ) |
4 | 3 | 3ad2ant1 1075 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ (1 / 2)) → (2 · 𝑋) ∈ ℝ) |
5 | 0le2 10988 | . . . . 5 ⊢ 0 ≤ 2 | |
6 | mulge0 10425 | . . . . 5 ⊢ (((2 ∈ ℝ ∧ 0 ≤ 2) ∧ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋)) → 0 ≤ (2 · 𝑋)) | |
7 | 1, 5, 6 | mpanl12 714 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → 0 ≤ (2 · 𝑋)) |
8 | 7 | 3adant3 1074 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ (1 / 2)) → 0 ≤ (2 · 𝑋)) |
9 | 1re 9918 | . . . . . 6 ⊢ 1 ∈ ℝ | |
10 | 2pos 10989 | . . . . . . 7 ⊢ 0 < 2 | |
11 | 1, 10 | pm3.2i 470 | . . . . . 6 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
12 | lemuldiv2 10783 | . . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 1 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((2 · 𝑋) ≤ 1 ↔ 𝑋 ≤ (1 / 2))) | |
13 | 9, 11, 12 | mp3an23 1408 | . . . . 5 ⊢ (𝑋 ∈ ℝ → ((2 · 𝑋) ≤ 1 ↔ 𝑋 ≤ (1 / 2))) |
14 | 13 | biimpar 501 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝑋 ≤ (1 / 2)) → (2 · 𝑋) ≤ 1) |
15 | 14 | 3adant2 1073 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ (1 / 2)) → (2 · 𝑋) ≤ 1) |
16 | 4, 8, 15 | 3jca 1235 | . 2 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ (1 / 2)) → ((2 · 𝑋) ∈ ℝ ∧ 0 ≤ (2 · 𝑋) ∧ (2 · 𝑋) ≤ 1)) |
17 | 0re 9919 | . . 3 ⊢ 0 ∈ ℝ | |
18 | halfre 11123 | . . 3 ⊢ (1 / 2) ∈ ℝ | |
19 | 17, 18 | elicc2i 12110 | . 2 ⊢ (𝑋 ∈ (0[,](1 / 2)) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ (1 / 2))) |
20 | 17, 9 | elicc2i 12110 | . 2 ⊢ ((2 · 𝑋) ∈ (0[,]1) ↔ ((2 · 𝑋) ∈ ℝ ∧ 0 ≤ (2 · 𝑋) ∧ (2 · 𝑋) ≤ 1)) |
21 | 16, 19, 20 | 3imtr4i 280 | 1 ⊢ (𝑋 ∈ (0[,](1 / 2)) → (2 · 𝑋) ∈ (0[,]1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 · cmul 9820 < clt 9953 ≤ cle 9954 / cdiv 10563 2c2 10947 [,]cicc 12049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-2 10956 df-icc 12053 |
This theorem is referenced by: iihalf1cn 22539 phtpycc 22598 copco 22626 pcohtpylem 22627 pcopt 22630 pcopt2 22631 pcorevlem 22634 |
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