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Mirrors > Home > MPE Home > Th. List > iirev | Structured version Visualization version GIF version |
Description: Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
iirev | ⊢ (𝑋 ∈ (0[,]1) → (1 − 𝑋) ∈ (0[,]1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 10635 | . . . . 5 ⊢ 1 ∈ ℝ | |
2 | resubcl 10944 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 𝑋 ∈ ℝ) → (1 − 𝑋) ∈ ℝ) | |
3 | 1, 2 | mpan 688 | . . . 4 ⊢ (𝑋 ∈ ℝ → (1 − 𝑋) ∈ ℝ) |
4 | 3 | 3ad2ant1 1129 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → (1 − 𝑋) ∈ ℝ) |
5 | simp3 1134 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → 𝑋 ≤ 1) | |
6 | simp1 1132 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → 𝑋 ∈ ℝ) | |
7 | subge0 11147 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 𝑋 ∈ ℝ) → (0 ≤ (1 − 𝑋) ↔ 𝑋 ≤ 1)) | |
8 | 1, 6, 7 | sylancr 589 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → (0 ≤ (1 − 𝑋) ↔ 𝑋 ≤ 1)) |
9 | 5, 8 | mpbird 259 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → 0 ≤ (1 − 𝑋)) |
10 | simp2 1133 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → 0 ≤ 𝑋) | |
11 | subge02 11150 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 𝑋 ∈ ℝ) → (0 ≤ 𝑋 ↔ (1 − 𝑋) ≤ 1)) | |
12 | 1, 6, 11 | sylancr 589 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → (0 ≤ 𝑋 ↔ (1 − 𝑋) ≤ 1)) |
13 | 10, 12 | mpbid 234 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → (1 − 𝑋) ≤ 1) |
14 | 4, 9, 13 | 3jca 1124 | . 2 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → ((1 − 𝑋) ∈ ℝ ∧ 0 ≤ (1 − 𝑋) ∧ (1 − 𝑋) ≤ 1)) |
15 | elicc01 12848 | . 2 ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1)) | |
16 | elicc01 12848 | . 2 ⊢ ((1 − 𝑋) ∈ (0[,]1) ↔ ((1 − 𝑋) ∈ ℝ ∧ 0 ≤ (1 − 𝑋) ∧ (1 − 𝑋) ≤ 1)) | |
17 | 14, 15, 16 | 3imtr4i 294 | 1 ⊢ (𝑋 ∈ (0[,]1) → (1 − 𝑋) ∈ (0[,]1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 ∈ wcel 2110 class class class wbr 5058 (class class class)co 7150 ℝcr 10530 0cc0 10531 1c1 10532 ≤ cle 10670 − cmin 10864 [,]cicc 12735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-icc 12739 |
This theorem is referenced by: iirevcn 23528 icccvx 23548 phtpycom 23586 pcorev2 23626 pi1xfrcnv 23655 dvlipcn 24585 efcvx 25031 logccv 25240 leibpi 25514 cvxcl 25556 resconn 32488 |
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