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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 | β’ πΊ = (π₯ β β β¦ if(0 β€ (πΉβπ₯), (πΉβπ₯), 0)) |
Ref | Expression |
---|---|
i1fpos | β’ (πΉ β dom β«1 β πΊ β dom β«1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | i1fpos.1 | . . 3 β’ πΊ = (π₯ β β β¦ if(0 β€ (πΉβπ₯), (πΉβπ₯), 0)) | |
2 | simpr 484 | . . . . . . 7 β’ ((πΉ β dom β«1 β§ π₯ β β) β π₯ β β) | |
3 | 2 | biantrurd 532 | . . . . . 6 β’ ((πΉ β dom β«1 β§ π₯ β β) β ((πΉβπ₯) β (0[,)+β) β (π₯ β β β§ (πΉβπ₯) β (0[,)+β)))) |
4 | i1ff 25618 | . . . . . . . . 9 β’ (πΉ β dom β«1 β πΉ:ββΆβ) | |
5 | 4 | ffvelcdmda 7094 | . . . . . . . 8 β’ ((πΉ β dom β«1 β§ π₯ β β) β (πΉβπ₯) β β) |
6 | 5 | biantrurd 532 | . . . . . . 7 β’ ((πΉ β dom β«1 β§ π₯ β β) β (0 β€ (πΉβπ₯) β ((πΉβπ₯) β β β§ 0 β€ (πΉβπ₯)))) |
7 | elrege0 13464 | . . . . . . 7 β’ ((πΉβπ₯) β (0[,)+β) β ((πΉβπ₯) β β β§ 0 β€ (πΉβπ₯))) | |
8 | 6, 7 | bitr4di 289 | . . . . . 6 β’ ((πΉ β dom β«1 β§ π₯ β β) β (0 β€ (πΉβπ₯) β (πΉβπ₯) β (0[,)+β))) |
9 | 4 | adantr 480 | . . . . . . 7 β’ ((πΉ β dom β«1 β§ π₯ β β) β πΉ:ββΆβ) |
10 | ffn 6722 | . . . . . . 7 β’ (πΉ:ββΆβ β πΉ Fn β) | |
11 | elpreima 7067 | . . . . . . 7 β’ (πΉ Fn β β (π₯ β (β‘πΉ β (0[,)+β)) β (π₯ β β β§ (πΉβπ₯) β (0[,)+β)))) | |
12 | 9, 10, 11 | 3syl 18 | . . . . . 6 β’ ((πΉ β dom β«1 β§ π₯ β β) β (π₯ β (β‘πΉ β (0[,)+β)) β (π₯ β β β§ (πΉβπ₯) β (0[,)+β)))) |
13 | 3, 8, 12 | 3bitr4d 311 | . . . . 5 β’ ((πΉ β dom β«1 β§ π₯ β β) β (0 β€ (πΉβπ₯) β π₯ β (β‘πΉ β (0[,)+β)))) |
14 | 13 | ifbid 4552 | . . . 4 β’ ((πΉ β dom β«1 β§ π₯ β β) β if(0 β€ (πΉβπ₯), (πΉβπ₯), 0) = if(π₯ β (β‘πΉ β (0[,)+β)), (πΉβπ₯), 0)) |
15 | 14 | mpteq2dva 5248 | . . 3 β’ (πΉ β dom β«1 β (π₯ β β β¦ if(0 β€ (πΉβπ₯), (πΉβπ₯), 0)) = (π₯ β β β¦ if(π₯ β (β‘πΉ β (0[,)+β)), (πΉβπ₯), 0))) |
16 | 1, 15 | eqtrid 2780 | . 2 β’ (πΉ β dom β«1 β πΊ = (π₯ β β β¦ if(π₯ β (β‘πΉ β (0[,)+β)), (πΉβπ₯), 0))) |
17 | i1fima 25620 | . . 3 β’ (πΉ β dom β«1 β (β‘πΉ β (0[,)+β)) β dom vol) | |
18 | eqid 2728 | . . . 4 β’ (π₯ β β β¦ if(π₯ β (β‘πΉ β (0[,)+β)), (πΉβπ₯), 0)) = (π₯ β β β¦ if(π₯ β (β‘πΉ β (0[,)+β)), (πΉβπ₯), 0)) | |
19 | 18 | i1fres 25648 | . . 3 β’ ((πΉ β dom β«1 β§ (β‘πΉ β (0[,)+β)) β dom vol) β (π₯ β β β¦ if(π₯ β (β‘πΉ β (0[,)+β)), (πΉβπ₯), 0)) β dom β«1) |
20 | 17, 19 | mpdan 686 | . 2 β’ (πΉ β dom β«1 β (π₯ β β β¦ if(π₯ β (β‘πΉ β (0[,)+β)), (πΉβπ₯), 0)) β dom β«1) |
21 | 16, 20 | eqeltrd 2829 | 1 β’ (πΉ β dom β«1 β πΊ β dom β«1) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 ifcif 4529 class class class wbr 5148 β¦ cmpt 5231 β‘ccnv 5677 dom cdm 5678 β cima 5681 Fn wfn 6543 βΆwf 6544 βcfv 6548 (class class class)co 7420 βcr 11138 0cc0 11139 +βcpnf 11276 β€ cle 11280 [,)cico 13359 volcvol 25405 β«1citg1 25557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-dju 9925 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-sum 15666 df-rest 17404 df-topgen 17425 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-top 22809 df-topon 22826 df-bases 22862 df-cmp 23304 df-ovol 25406 df-vol 25407 df-mbf 25561 df-itg1 25562 |
This theorem is referenced by: i1fposd 25650 i1fibl 25750 itg2addnclem 37144 ftc1anclem5 37170 |
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