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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0iifcv | Structured version Visualization version GIF version |
Description: The defined function's value in the real. (Contributed by Thierry Arnoux, 1-Apr-2017.) |
Ref | Expression |
---|---|
xrge0iifhmeo.1 | ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) |
Ref | Expression |
---|---|
xrge0iifcv | ⊢ (𝑋 ∈ (0(,]1) → (𝐹‘𝑋) = -(log‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iocssicc 12454 | . . . 4 ⊢ (0(,]1) ⊆ (0[,]1) | |
2 | 1 | sseli 3740 | . . 3 ⊢ (𝑋 ∈ (0(,]1) → 𝑋 ∈ (0[,]1)) |
3 | eqeq1 2764 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0)) | |
4 | fveq2 6352 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (log‘𝑥) = (log‘𝑋)) | |
5 | 4 | negeqd 10467 | . . . . 5 ⊢ (𝑥 = 𝑋 → -(log‘𝑥) = -(log‘𝑋)) |
6 | 3, 5 | ifbieq2d 4255 | . . . 4 ⊢ (𝑥 = 𝑋 → if(𝑥 = 0, +∞, -(log‘𝑥)) = if(𝑋 = 0, +∞, -(log‘𝑋))) |
7 | xrge0iifhmeo.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) | |
8 | pnfex 10285 | . . . . 5 ⊢ +∞ ∈ V | |
9 | negex 10471 | . . . . 5 ⊢ -(log‘𝑋) ∈ V | |
10 | 8, 9 | ifex 4300 | . . . 4 ⊢ if(𝑋 = 0, +∞, -(log‘𝑋)) ∈ V |
11 | 6, 7, 10 | fvmpt 6444 | . . 3 ⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) = if(𝑋 = 0, +∞, -(log‘𝑋))) |
12 | 2, 11 | syl 17 | . 2 ⊢ (𝑋 ∈ (0(,]1) → (𝐹‘𝑋) = if(𝑋 = 0, +∞, -(log‘𝑋))) |
13 | 0xr 10278 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
14 | 1re 10231 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
15 | elioc2 12429 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ) → (𝑋 ∈ (0(,]1) ↔ (𝑋 ∈ ℝ ∧ 0 < 𝑋 ∧ 𝑋 ≤ 1))) | |
16 | 13, 14, 15 | mp2an 710 | . . . . . 6 ⊢ (𝑋 ∈ (0(,]1) ↔ (𝑋 ∈ ℝ ∧ 0 < 𝑋 ∧ 𝑋 ≤ 1)) |
17 | 16 | simp2bi 1141 | . . . . 5 ⊢ (𝑋 ∈ (0(,]1) → 0 < 𝑋) |
18 | 17 | gt0ne0d 10784 | . . . 4 ⊢ (𝑋 ∈ (0(,]1) → 𝑋 ≠ 0) |
19 | 18 | neneqd 2937 | . . 3 ⊢ (𝑋 ∈ (0(,]1) → ¬ 𝑋 = 0) |
20 | 19 | iffalsed 4241 | . 2 ⊢ (𝑋 ∈ (0(,]1) → if(𝑋 = 0, +∞, -(log‘𝑋)) = -(log‘𝑋)) |
21 | 12, 20 | eqtrd 2794 | 1 ⊢ (𝑋 ∈ (0(,]1) → (𝐹‘𝑋) = -(log‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ifcif 4230 class class class wbr 4804 ↦ cmpt 4881 ‘cfv 6049 (class class class)co 6813 ℝcr 10127 0cc0 10128 1c1 10129 +∞cpnf 10263 ℝ*cxr 10265 < clt 10266 ≤ cle 10267 -cneg 10459 (,]cioc 12369 [,]cicc 12371 logclog 24500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-i2m1 10196 ax-1ne0 10197 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-neg 10461 df-ioc 12373 df-icc 12375 |
This theorem is referenced by: xrge0iifiso 30290 xrge0iifhom 30292 |
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