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Mirrors > Home > MPE Home > Th. List > i1fposd | Structured version Visualization version GIF version |
Description: Deduction form of i1fposd 25088. (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 2908 | . . . . . 6 β’ β²π₯0 | |
2 | nfcv 2908 | . . . . . 6 β’ β²π₯ β€ | |
3 | nffvmpt1 6858 | . . . . . 6 β’ β²π₯((π₯ β β β¦ π΄)βπ¦) | |
4 | 1, 2, 3 | nfbr 5157 | . . . . 5 β’ β²π₯0 β€ ((π₯ β β β¦ π΄)βπ¦) |
5 | 4, 3, 1 | nfif 4521 | . . . 4 β’ β²π₯if(0 β€ ((π₯ β β β¦ π΄)βπ¦), ((π₯ β β β¦ π΄)βπ¦), 0) |
6 | nfcv 2908 | . . . 4 β’ β²π¦if(0 β€ ((π₯ β β β¦ π΄)βπ₯), ((π₯ β β β¦ π΄)βπ₯), 0) | |
7 | fveq2 6847 | . . . . . 6 β’ (π¦ = π₯ β ((π₯ β β β¦ π΄)βπ¦) = ((π₯ β β β¦ π΄)βπ₯)) | |
8 | 7 | breq2d 5122 | . . . . 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 4515 | . . . 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 5221 | . . 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 486 | . . . . . . 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 25056 | . . . . . . . . 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 7066 | . . . . . . 7 β’ ((π β§ π₯ β β) β π΄ β β) |
16 | eqid 2737 | . . . . . . . 8 β’ (π₯ β β β¦ π΄) = (π₯ β β β¦ π΄) | |
17 | 16 | fvmpt2 6964 | . . . . . . 7 β’ ((π₯ β β β§ π΄ β β) β ((π₯ β β β¦ π΄)βπ₯) = π΄) |
18 | 11, 15, 17 | syl2anc 585 | . . . . . 6 β’ ((π β§ π₯ β β) β ((π₯ β β β¦ π΄)βπ₯) = π΄) |
19 | 18 | breq2d 5122 | . . . . 5 β’ ((π β§ π₯ β β) β (0 β€ ((π₯ β β β¦ π΄)βπ₯) β 0 β€ π΄)) |
20 | 19, 18 | ifbieq1d 4515 | . . . 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 5210 | . . 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 2789 | . 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 2737 | . . . 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 25087 | . . 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 2839 | 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 397 = wceq 1542 β wcel 2107 ifcif 4491 class class class wbr 5110 β¦ cmpt 5193 dom cdm 5638 βΆwf 6497 βcfv 6501 βcr 11057 0cc0 11058 β€ cle 11197 β«1citg1 24995 |
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 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
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 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9453 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-q 12881 df-rp 12923 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13275 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 df-rest 17311 df-topgen 17332 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-top 22259 df-topon 22276 df-bases 22312 df-cmp 22754 df-ovol 24844 df-vol 24845 df-mbf 24999 df-itg1 25000 |
This theorem is referenced by: i1fibl 25188 itgitg1 25189 |
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