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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 10438 | . . . . 5 ⊢ 1 ∈ ℝ | |
2 | resubcl 10750 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 𝑋 ∈ ℝ) → (1 − 𝑋) ∈ ℝ) | |
3 | 1, 2 | mpan 678 | . . . 4 ⊢ (𝑋 ∈ ℝ → (1 − 𝑋) ∈ ℝ) |
4 | 3 | 3ad2ant1 1114 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → (1 − 𝑋) ∈ ℝ) |
5 | simp3 1119 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → 𝑋 ≤ 1) | |
6 | simp1 1117 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → 𝑋 ∈ ℝ) | |
7 | subge0 10953 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 𝑋 ∈ ℝ) → (0 ≤ (1 − 𝑋) ↔ 𝑋 ≤ 1)) | |
8 | 1, 6, 7 | sylancr 579 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → (0 ≤ (1 − 𝑋) ↔ 𝑋 ≤ 1)) |
9 | 5, 8 | mpbird 249 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → 0 ≤ (1 − 𝑋)) |
10 | simp2 1118 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → 0 ≤ 𝑋) | |
11 | subge02 10956 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 𝑋 ∈ ℝ) → (0 ≤ 𝑋 ↔ (1 − 𝑋) ≤ 1)) | |
12 | 1, 6, 11 | sylancr 579 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → (0 ≤ 𝑋 ↔ (1 − 𝑋) ≤ 1)) |
13 | 10, 12 | mpbid 224 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → (1 − 𝑋) ≤ 1) |
14 | 4, 9, 13 | 3jca 1109 | . 2 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) → ((1 − 𝑋) ∈ ℝ ∧ 0 ≤ (1 − 𝑋) ∧ (1 − 𝑋) ≤ 1)) |
15 | elicc01 12669 | . 2 ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1)) | |
16 | elicc01 12669 | . 2 ⊢ ((1 − 𝑋) ∈ (0[,]1) ↔ ((1 − 𝑋) ∈ ℝ ∧ 0 ≤ (1 − 𝑋) ∧ (1 − 𝑋) ≤ 1)) | |
17 | 14, 15, 16 | 3imtr4i 284 | 1 ⊢ (𝑋 ∈ (0[,]1) → (1 − 𝑋) ∈ (0[,]1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1069 ∈ wcel 2051 class class class wbr 4926 (class class class)co 6975 ℝcr 10333 0cc0 10334 1c1 10335 ≤ cle 10474 − cmin 10669 [,]cicc 12556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-po 5323 df-so 5324 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-icc 12560 |
This theorem is referenced by: iirevcn 23253 icccvx 23273 phtpycom 23311 pcorev2 23351 pi1xfrcnv 23380 dvlipcn 24310 efcvx 24756 logccv 24963 leibpi 25238 cvxcl 25280 resconn 32111 |
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