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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 13420 | . . . 4 ⊢ (0(,]1) ⊆ (0[,]1) | |
2 | 1 | sseli 3973 | . . 3 ⊢ (𝑋 ∈ (0(,]1) → 𝑋 ∈ (0[,]1)) |
3 | eqeq1 2730 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0)) | |
4 | fveq2 6885 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (log‘𝑥) = (log‘𝑋)) | |
5 | 4 | negeqd 11458 | . . . . 5 ⊢ (𝑥 = 𝑋 → -(log‘𝑥) = -(log‘𝑋)) |
6 | 3, 5 | ifbieq2d 4549 | . . . 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 11271 | . . . . 5 ⊢ +∞ ∈ V | |
9 | negex 11462 | . . . . 5 ⊢ -(log‘𝑋) ∈ V | |
10 | 8, 9 | ifex 4573 | . . . 4 ⊢ if(𝑋 = 0, +∞, -(log‘𝑋)) ∈ V |
11 | 6, 7, 10 | fvmpt 6992 | . . 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 11265 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
14 | 1re 11218 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
15 | elioc2 13393 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ) → (𝑋 ∈ (0(,]1) ↔ (𝑋 ∈ ℝ ∧ 0 < 𝑋 ∧ 𝑋 ≤ 1))) | |
16 | 13, 14, 15 | mp2an 689 | . . . . . 6 ⊢ (𝑋 ∈ (0(,]1) ↔ (𝑋 ∈ ℝ ∧ 0 < 𝑋 ∧ 𝑋 ≤ 1)) |
17 | 16 | simp2bi 1143 | . . . . 5 ⊢ (𝑋 ∈ (0(,]1) → 0 < 𝑋) |
18 | 17 | gt0ne0d 11782 | . . . 4 ⊢ (𝑋 ∈ (0(,]1) → 𝑋 ≠ 0) |
19 | 18 | neneqd 2939 | . . 3 ⊢ (𝑋 ∈ (0(,]1) → ¬ 𝑋 = 0) |
20 | 19 | iffalsed 4534 | . 2 ⊢ (𝑋 ∈ (0(,]1) → if(𝑋 = 0, +∞, -(log‘𝑋)) = -(log‘𝑋)) |
21 | 12, 20 | eqtrd 2766 | 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 4523 class class class wbr 5141 ↦ cmpt 5224 ‘cfv 6537 (class class class)co 7405 ℝcr 11111 0cc0 11112 1c1 11113 +∞cpnf 11249 ℝ*cxr 11251 < clt 11252 ≤ cle 11253 -cneg 11449 (,]cioc 13331 [,]cicc 13333 logclog 26443 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-i2m1 11180 ax-1ne0 11181 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-neg 11451 df-ioc 13335 df-icc 13337 |
This theorem is referenced by: xrge0iifiso 33445 xrge0iifhom 33447 |
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