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Mirrors > Home > MPE Home > Th. List > i1fposd | Structured version Visualization version GIF version |
Description: Deduction form of i1fposd 25657. (Contributed by Mario Carneiro, 6-Aug-2014.) |
Ref | Expression |
---|---|
i1fposd.1 | β’ (π β (π₯ β β β¦ π΄) β dom β«1) |
Ref | Expression |
---|---|
i1fposd | β’ (π β (π₯ β β β¦ if(0 β€ π΄, π΄, 0)) β dom β«1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2899 | . . . . . 6 β’ β²π₯0 | |
2 | nfcv 2899 | . . . . . 6 β’ β²π₯ β€ | |
3 | nffvmpt1 6913 | . . . . . 6 β’ β²π₯((π₯ β β β¦ π΄)βπ¦) | |
4 | 1, 2, 3 | nfbr 5199 | . . . . 5 β’ β²π₯0 β€ ((π₯ β β β¦ π΄)βπ¦) |
5 | 4, 3, 1 | nfif 4562 | . . . 4 β’ β²π₯if(0 β€ ((π₯ β β β¦ π΄)βπ¦), ((π₯ β β β¦ π΄)βπ¦), 0) |
6 | nfcv 2899 | . . . 4 β’ β²π¦if(0 β€ ((π₯ β β β¦ π΄)βπ₯), ((π₯ β β β¦ π΄)βπ₯), 0) | |
7 | fveq2 6902 | . . . . . 6 β’ (π¦ = π₯ β ((π₯ β β β¦ π΄)βπ¦) = ((π₯ β β β¦ π΄)βπ₯)) | |
8 | 7 | breq2d 5164 | . . . . 5 β’ (π¦ = π₯ β (0 β€ ((π₯ β β β¦ π΄)βπ¦) β 0 β€ ((π₯ β β β¦ π΄)βπ₯))) |
9 | 8, 7 | ifbieq1d 4556 | . . . 4 β’ (π¦ = π₯ β if(0 β€ ((π₯ β β β¦ π΄)βπ¦), ((π₯ β β β¦ π΄)βπ¦), 0) = if(0 β€ ((π₯ β β β¦ π΄)βπ₯), ((π₯ β β β¦ π΄)βπ₯), 0)) |
10 | 5, 6, 9 | cbvmpt 5263 | . . 3 β’ (π¦ β β β¦ if(0 β€ ((π₯ β β β¦ π΄)βπ¦), ((π₯ β β β¦ π΄)βπ¦), 0)) = (π₯ β β β¦ if(0 β€ ((π₯ β β β¦ π΄)βπ₯), ((π₯ β β β¦ π΄)βπ₯), 0)) |
11 | simpr 483 | . . . . . . 7 β’ ((π β§ π₯ β β) β π₯ β β) | |
12 | i1fposd.1 | . . . . . . . . 9 β’ (π β (π₯ β β β¦ π΄) β dom β«1) | |
13 | i1ff 25625 | . . . . . . . . 9 β’ ((π₯ β β β¦ π΄) β dom β«1 β (π₯ β β β¦ π΄):ββΆβ) | |
14 | 12, 13 | syl 17 | . . . . . . . 8 β’ (π β (π₯ β β β¦ π΄):ββΆβ) |
15 | 14 | fvmptelcdm 7128 | . . . . . . 7 β’ ((π β§ π₯ β β) β π΄ β β) |
16 | eqid 2728 | . . . . . . . 8 β’ (π₯ β β β¦ π΄) = (π₯ β β β¦ π΄) | |
17 | 16 | fvmpt2 7021 | . . . . . . 7 β’ ((π₯ β β β§ π΄ β β) β ((π₯ β β β¦ π΄)βπ₯) = π΄) |
18 | 11, 15, 17 | syl2anc 582 | . . . . . 6 β’ ((π β§ π₯ β β) β ((π₯ β β β¦ π΄)βπ₯) = π΄) |
19 | 18 | breq2d 5164 | . . . . 5 β’ ((π β§ π₯ β β) β (0 β€ ((π₯ β β β¦ π΄)βπ₯) β 0 β€ π΄)) |
20 | 19, 18 | ifbieq1d 4556 | . . . 4 β’ ((π β§ π₯ β β) β if(0 β€ ((π₯ β β β¦ π΄)βπ₯), ((π₯ β β β¦ π΄)βπ₯), 0) = if(0 β€ π΄, π΄, 0)) |
21 | 20 | mpteq2dva 5252 | . . 3 β’ (π β (π₯ β β β¦ if(0 β€ ((π₯ β β β¦ π΄)βπ₯), ((π₯ β β β¦ π΄)βπ₯), 0)) = (π₯ β β β¦ if(0 β€ π΄, π΄, 0))) |
22 | 10, 21 | eqtrid 2780 | . 2 β’ (π β (π¦ β β β¦ if(0 β€ ((π₯ β β β¦ π΄)βπ¦), ((π₯ β β β¦ π΄)βπ¦), 0)) = (π₯ β β β¦ if(0 β€ π΄, π΄, 0))) |
23 | eqid 2728 | . . . 4 β’ (π¦ β β β¦ if(0 β€ ((π₯ β β β¦ π΄)βπ¦), ((π₯ β β β¦ π΄)βπ¦), 0)) = (π¦ β β β¦ if(0 β€ ((π₯ β β β¦ π΄)βπ¦), ((π₯ β β β¦ π΄)βπ¦), 0)) | |
24 | 23 | i1fpos 25656 | . . 3 β’ ((π₯ β β β¦ π΄) β dom β«1 β (π¦ β β β¦ if(0 β€ ((π₯ β β β¦ π΄)βπ¦), ((π₯ β β β¦ π΄)βπ¦), 0)) β dom β«1) |
25 | 12, 24 | syl 17 | . 2 β’ (π β (π¦ β β β¦ if(0 β€ ((π₯ β β β¦ π΄)βπ¦), ((π₯ β β β¦ π΄)βπ¦), 0)) β dom β«1) |
26 | 22, 25 | eqeltrrd 2830 | 1 β’ (π β (π₯ β β β¦ if(0 β€ π΄, π΄, 0)) β dom β«1) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 ifcif 4532 class class class wbr 5152 β¦ cmpt 5235 dom cdm 5682 βΆwf 6549 βcfv 6553 βcr 11145 0cc0 11146 β€ cle 11287 β«1citg1 25564 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-sum 15673 df-rest 17411 df-topgen 17432 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-top 22816 df-topon 22833 df-bases 22869 df-cmp 23311 df-ovol 25413 df-vol 25414 df-mbf 25568 df-itg1 25569 |
This theorem is referenced by: i1fibl 25757 itgitg1 25758 |
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