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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 13446 | . . . 4 ⊢ (0(,]1) ⊆ (0[,]1) | |
2 | 1 | sseli 3968 | . . 3 ⊢ (𝑋 ∈ (0(,]1) → 𝑋 ∈ (0[,]1)) |
3 | eqeq1 2729 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0)) | |
4 | fveq2 6892 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (log‘𝑥) = (log‘𝑋)) | |
5 | 4 | negeqd 11484 | . . . . 5 ⊢ (𝑥 = 𝑋 → -(log‘𝑥) = -(log‘𝑋)) |
6 | 3, 5 | ifbieq2d 4550 | . . . 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 11297 | . . . . 5 ⊢ +∞ ∈ V | |
9 | negex 11488 | . . . . 5 ⊢ -(log‘𝑋) ∈ V | |
10 | 8, 9 | ifex 4574 | . . . 4 ⊢ if(𝑋 = 0, +∞, -(log‘𝑋)) ∈ V |
11 | 6, 7, 10 | fvmpt 7000 | . . 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 11291 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
14 | 1re 11244 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
15 | elioc2 13419 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ) → (𝑋 ∈ (0(,]1) ↔ (𝑋 ∈ ℝ ∧ 0 < 𝑋 ∧ 𝑋 ≤ 1))) | |
16 | 13, 14, 15 | mp2an 690 | . . . . . 6 ⊢ (𝑋 ∈ (0(,]1) ↔ (𝑋 ∈ ℝ ∧ 0 < 𝑋 ∧ 𝑋 ≤ 1)) |
17 | 16 | simp2bi 1143 | . . . . 5 ⊢ (𝑋 ∈ (0(,]1) → 0 < 𝑋) |
18 | 17 | gt0ne0d 11808 | . . . 4 ⊢ (𝑋 ∈ (0(,]1) → 𝑋 ≠ 0) |
19 | 18 | neneqd 2935 | . . 3 ⊢ (𝑋 ∈ (0(,]1) → ¬ 𝑋 = 0) |
20 | 19 | iffalsed 4535 | . 2 ⊢ (𝑋 ∈ (0(,]1) → if(𝑋 = 0, +∞, -(log‘𝑋)) = -(log‘𝑋)) |
21 | 12, 20 | eqtrd 2765 | 1 ⊢ (𝑋 ∈ (0(,]1) → (𝐹‘𝑋) = -(log‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ifcif 4524 class class class wbr 5143 ↦ cmpt 5226 ‘cfv 6543 (class class class)co 7416 ℝcr 11137 0cc0 11138 1c1 11139 +∞cpnf 11275 ℝ*cxr 11277 < clt 11278 ≤ cle 11279 -cneg 11475 (,]cioc 13357 [,]cicc 13359 logclog 26506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-i2m1 11206 ax-1ne0 11207 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-neg 11477 df-ioc 13361 df-icc 13363 |
This theorem is referenced by: xrge0iifiso 33593 xrge0iifhom 33595 |
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