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Mirrors > Home > MPE Home > Th. List > i1fpos | Structured version Visualization version GIF version |
Description: The positive part of a simple function is simple. (Contributed by Mario Carneiro, 28-Jun-2014.) |
Ref | Expression |
---|---|
i1fpos.1 | ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) |
Ref | Expression |
---|---|
i1fpos | ⊢ (𝐹 ∈ dom ∫1 → 𝐺 ∈ dom ∫1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | i1fpos.1 | . . 3 ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) | |
2 | simpr 486 | . . . . . . 7 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
3 | 2 | biantrurd 534 | . . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ (0[,)+∞)))) |
4 | i1ff 24992 | . . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | |
5 | 4 | ffvelcdmda 7032 | . . . . . . . 8 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) |
6 | 5 | biantrurd 534 | . . . . . . 7 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (0 ≤ (𝐹‘𝑥) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))) |
7 | elrege0 13326 | . . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥))) | |
8 | 6, 7 | bitr4di 289 | . . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (0 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑥) ∈ (0[,)+∞))) |
9 | 4 | adantr 482 | . . . . . . 7 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → 𝐹:ℝ⟶ℝ) |
10 | ffn 6666 | . . . . . . 7 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ) | |
11 | elpreima 7006 | . . . . . . 7 ⊢ (𝐹 Fn ℝ → (𝑥 ∈ (◡𝐹 “ (0[,)+∞)) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ (0[,)+∞)))) | |
12 | 9, 10, 11 | 3syl 18 | . . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (◡𝐹 “ (0[,)+∞)) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ (0[,)+∞)))) |
13 | 3, 8, 12 | 3bitr4d 311 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (0 ≤ (𝐹‘𝑥) ↔ 𝑥 ∈ (◡𝐹 “ (0[,)+∞)))) |
14 | 13 | ifbid 4508 | . . . 4 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) = if(𝑥 ∈ (◡𝐹 “ (0[,)+∞)), (𝐹‘𝑥), 0)) |
15 | 14 | mpteq2dva 5204 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ (0[,)+∞)), (𝐹‘𝑥), 0))) |
16 | 1, 15 | eqtrid 2790 | . 2 ⊢ (𝐹 ∈ dom ∫1 → 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ (0[,)+∞)), (𝐹‘𝑥), 0))) |
17 | i1fima 24994 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ (0[,)+∞)) ∈ dom vol) | |
18 | eqid 2738 | . . . 4 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ (0[,)+∞)), (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ (0[,)+∞)), (𝐹‘𝑥), 0)) | |
19 | 18 | i1fres 25022 | . . 3 ⊢ ((𝐹 ∈ dom ∫1 ∧ (◡𝐹 “ (0[,)+∞)) ∈ dom vol) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ (0[,)+∞)), (𝐹‘𝑥), 0)) ∈ dom ∫1) |
20 | 17, 19 | mpdan 686 | . 2 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ (0[,)+∞)), (𝐹‘𝑥), 0)) ∈ dom ∫1) |
21 | 16, 20 | eqeltrd 2839 | 1 ⊢ (𝐹 ∈ dom ∫1 → 𝐺 ∈ dom ∫1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ifcif 4485 class class class wbr 5104 ↦ cmpt 5187 ◡ccnv 5631 dom cdm 5632 “ cima 5635 Fn wfn 6489 ⟶wf 6490 ‘cfv 6494 (class class class)co 7352 ℝcr 11009 0cc0 11010 +∞cpnf 11145 ≤ cle 11149 [,)cico 13221 volcvol 24779 ∫1citg1 24931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-inf2 9536 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 ax-pre-sup 11088 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-se 5588 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7610 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8607 df-map 8726 df-pm 8727 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-fi 9306 df-sup 9337 df-inf 9338 df-oi 9405 df-dju 9796 df-card 9834 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-div 11772 df-nn 12113 df-2 12175 df-3 12176 df-n0 12373 df-z 12459 df-uz 12723 df-q 12829 df-rp 12871 df-xneg 12988 df-xadd 12989 df-xmul 12990 df-ioo 13223 df-ico 13225 df-icc 13226 df-fz 13380 df-fzo 13523 df-fl 13652 df-seq 13862 df-exp 13923 df-hash 14185 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-abs 15081 df-clim 15330 df-sum 15531 df-rest 17264 df-topgen 17285 df-psmet 20741 df-xmet 20742 df-met 20743 df-bl 20744 df-mopn 20745 df-top 22195 df-topon 22212 df-bases 22248 df-cmp 22690 df-ovol 24780 df-vol 24781 df-mbf 24935 df-itg1 24936 |
This theorem is referenced by: i1fposd 25024 i1fibl 25124 itg2addnclem 36061 ftc1anclem5 36087 |
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