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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 484 | . . . . . . 7 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
3 | 2 | biantrurd 532 | . . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ (0[,)+∞)))) |
4 | i1ff 25725 | . . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | |
5 | 4 | ffvelcdmda 7104 | . . . . . . . 8 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) |
6 | 5 | biantrurd 532 | . . . . . . 7 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (0 ≤ (𝐹‘𝑥) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))) |
7 | elrege0 13491 | . . . . . . 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 480 | . . . . . . 7 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → 𝐹:ℝ⟶ℝ) |
10 | ffn 6737 | . . . . . . 7 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ) | |
11 | elpreima 7078 | . . . . . . 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 4554 | . . . 4 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) = if(𝑥 ∈ (◡𝐹 “ (0[,)+∞)), (𝐹‘𝑥), 0)) |
15 | 14 | mpteq2dva 5248 | . . 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 2787 | . 2 ⊢ (𝐹 ∈ dom ∫1 → 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ (0[,)+∞)), (𝐹‘𝑥), 0))) |
17 | i1fima 25727 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ (0[,)+∞)) ∈ dom vol) | |
18 | eqid 2735 | . . . 4 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ (0[,)+∞)), (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ (0[,)+∞)), (𝐹‘𝑥), 0)) | |
19 | 18 | i1fres 25755 | . . 3 ⊢ ((𝐹 ∈ dom ∫1 ∧ (◡𝐹 “ (0[,)+∞)) ∈ dom vol) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ (0[,)+∞)), (𝐹‘𝑥), 0)) ∈ dom ∫1) |
20 | 17, 19 | mpdan 687 | . 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 ◡ccnv 5688 dom cdm 5689 “ cima 5692 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 +∞cpnf 11290 ≤ cle 11294 [,)cico 13386 volcvol 25512 ∫1citg1 25664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-rest 17469 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-top 22916 df-topon 22933 df-bases 22969 df-cmp 23411 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 |
This theorem is referenced by: i1fposd 25757 i1fibl 25858 itg2addnclem 37658 ftc1anclem5 37684 |
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