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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 486 | . . . . . . 7 β’ ((πΉ β dom β«1 β§ π₯ β β) β π₯ β β) | |
3 | 2 | biantrurd 534 | . . . . . 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 25043 | . . . . . . . . 9 β’ (πΉ β dom β«1 β πΉ:ββΆβ) | |
5 | 4 | ffvelcdmda 7036 | . . . . . . . 8 β’ ((πΉ β dom β«1 β§ π₯ β β) β (πΉβπ₯) β β) |
6 | 5 | biantrurd 534 | . . . . . . 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 13372 | . . . . . . 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 482 | . . . . . . 7 β’ ((πΉ β dom β«1 β§ π₯ β β) β πΉ:ββΆβ) |
10 | ffn 6669 | . . . . . . 7 β’ (πΉ:ββΆβ β πΉ Fn β) | |
11 | elpreima 7009 | . . . . . . 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 4510 | . . . 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 5206 | . . 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 2789 | . 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 25045 | . . 3 β’ (πΉ β dom β«1 β (β‘πΉ β (0[,)+β)) β dom vol) | |
18 | eqid 2737 | . . . 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 25073 | . . 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 2838 | 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 397 = wceq 1542 β wcel 2107 ifcif 4487 class class class wbr 5106 β¦ cmpt 5189 β‘ccnv 5633 dom cdm 5634 β cima 5637 Fn wfn 6492 βΆwf 6493 βcfv 6497 (class class class)co 7358 βcr 11051 0cc0 11052 +βcpnf 11187 β€ cle 11191 [,)cico 13267 volcvol 24830 β«1citg1 24982 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9578 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8649 df-map 8768 df-pm 8769 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fi 9348 df-sup 9379 df-inf 9380 df-oi 9447 df-dju 9838 df-card 9876 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-n0 12415 df-z 12501 df-uz 12765 df-q 12875 df-rp 12917 df-xneg 13034 df-xadd 13035 df-xmul 13036 df-ioo 13269 df-ico 13271 df-icc 13272 df-fz 13426 df-fzo 13569 df-fl 13698 df-seq 13908 df-exp 13969 df-hash 14232 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-clim 15371 df-sum 15572 df-rest 17305 df-topgen 17326 df-psmet 20791 df-xmet 20792 df-met 20793 df-bl 20794 df-mopn 20795 df-top 22246 df-topon 22263 df-bases 22299 df-cmp 22741 df-ovol 24831 df-vol 24832 df-mbf 24986 df-itg1 24987 |
This theorem is referenced by: i1fposd 25075 i1fibl 25175 itg2addnclem 36132 ftc1anclem5 36158 |
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