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 12813 | . . . 4 ⊢ (0(,]1) ⊆ (0[,]1) | |
2 | 1 | sseli 3960 | . . 3 ⊢ (𝑋 ∈ (0(,]1) → 𝑋 ∈ (0[,]1)) |
3 | eqeq1 2822 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0)) | |
4 | fveq2 6663 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (log‘𝑥) = (log‘𝑋)) | |
5 | 4 | negeqd 10868 | . . . . 5 ⊢ (𝑥 = 𝑋 → -(log‘𝑥) = -(log‘𝑋)) |
6 | 3, 5 | ifbieq2d 4488 | . . . 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 10682 | . . . . 5 ⊢ +∞ ∈ V | |
9 | negex 10872 | . . . . 5 ⊢ -(log‘𝑋) ∈ V | |
10 | 8, 9 | ifex 4511 | . . . 4 ⊢ if(𝑋 = 0, +∞, -(log‘𝑋)) ∈ V |
11 | 6, 7, 10 | fvmpt 6761 | . . 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 10676 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
14 | 1re 10629 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
15 | elioc2 12787 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ) → (𝑋 ∈ (0(,]1) ↔ (𝑋 ∈ ℝ ∧ 0 < 𝑋 ∧ 𝑋 ≤ 1))) | |
16 | 13, 14, 15 | mp2an 688 | . . . . . 6 ⊢ (𝑋 ∈ (0(,]1) ↔ (𝑋 ∈ ℝ ∧ 0 < 𝑋 ∧ 𝑋 ≤ 1)) |
17 | 16 | simp2bi 1138 | . . . . 5 ⊢ (𝑋 ∈ (0(,]1) → 0 < 𝑋) |
18 | 17 | gt0ne0d 11192 | . . . 4 ⊢ (𝑋 ∈ (0(,]1) → 𝑋 ≠ 0) |
19 | 18 | neneqd 3018 | . . 3 ⊢ (𝑋 ∈ (0(,]1) → ¬ 𝑋 = 0) |
20 | 19 | iffalsed 4474 | . 2 ⊢ (𝑋 ∈ (0(,]1) → if(𝑋 = 0, +∞, -(log‘𝑋)) = -(log‘𝑋)) |
21 | 12, 20 | eqtrd 2853 | 1 ⊢ (𝑋 ∈ (0(,]1) → (𝐹‘𝑋) = -(log‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ifcif 4463 class class class wbr 5057 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 0cc0 10525 1c1 10526 +∞cpnf 10660 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 -cneg 10859 (,]cioc 12727 [,]cicc 12729 logclog 25065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-i2m1 10593 ax-1ne0 10594 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-neg 10861 df-ioc 12731 df-icc 12733 |
This theorem is referenced by: xrge0iifiso 31077 xrge0iifhom 31079 |
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