![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem37 | Structured version Visualization version GIF version |
Description: This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_(t0) = 0, and p > 0 on T - U. Z is used for t0, P is used for p, (πΊβπ) is used for p_(ti). (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
stoweidlem37.1 | β’ π = {β β π΄ β£ ((ββπ) = 0 β§ βπ‘ β π (0 β€ (ββπ‘) β§ (ββπ‘) β€ 1))} |
stoweidlem37.2 | β’ π = (π‘ β π β¦ ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ‘))) |
stoweidlem37.3 | β’ (π β π β β) |
stoweidlem37.4 | β’ (π β πΊ:(1...π)βΆπ) |
stoweidlem37.5 | β’ ((π β§ π β π΄) β π:πβΆβ) |
stoweidlem37.6 | β’ (π β π β π) |
Ref | Expression |
---|---|
stoweidlem37 | β’ (π β (πβπ) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stoweidlem37.6 | . . 3 β’ (π β π β π) | |
2 | stoweidlem37.1 | . . . 4 β’ π = {β β π΄ β£ ((ββπ) = 0 β§ βπ‘ β π (0 β€ (ββπ‘) β§ (ββπ‘) β€ 1))} | |
3 | stoweidlem37.2 | . . . 4 β’ π = (π‘ β π β¦ ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ‘))) | |
4 | stoweidlem37.3 | . . . 4 β’ (π β π β β) | |
5 | stoweidlem37.4 | . . . 4 β’ (π β πΊ:(1...π)βΆπ) | |
6 | stoweidlem37.5 | . . . 4 β’ ((π β§ π β π΄) β π:πβΆβ) | |
7 | 2, 3, 4, 5, 6 | stoweidlem30 44361 | . . 3 β’ ((π β§ π β π) β (πβπ) = ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ))) |
8 | 1, 7 | mpdan 686 | . 2 β’ (π β (πβπ) = ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ))) |
9 | 5 | ffvelcdmda 7039 | . . . . . . 7 β’ ((π β§ π β (1...π)) β (πΊβπ) β π) |
10 | fveq1 6845 | . . . . . . . . . 10 β’ (β = (πΊβπ) β (ββπ) = ((πΊβπ)βπ)) | |
11 | 10 | eqeq1d 2735 | . . . . . . . . 9 β’ (β = (πΊβπ) β ((ββπ) = 0 β ((πΊβπ)βπ) = 0)) |
12 | fveq1 6845 | . . . . . . . . . . . 12 β’ (β = (πΊβπ) β (ββπ‘) = ((πΊβπ)βπ‘)) | |
13 | 12 | breq2d 5121 | . . . . . . . . . . 11 β’ (β = (πΊβπ) β (0 β€ (ββπ‘) β 0 β€ ((πΊβπ)βπ‘))) |
14 | 12 | breq1d 5119 | . . . . . . . . . . 11 β’ (β = (πΊβπ) β ((ββπ‘) β€ 1 β ((πΊβπ)βπ‘) β€ 1)) |
15 | 13, 14 | anbi12d 632 | . . . . . . . . . 10 β’ (β = (πΊβπ) β ((0 β€ (ββπ‘) β§ (ββπ‘) β€ 1) β (0 β€ ((πΊβπ)βπ‘) β§ ((πΊβπ)βπ‘) β€ 1))) |
16 | 15 | ralbidv 3171 | . . . . . . . . 9 β’ (β = (πΊβπ) β (βπ‘ β π (0 β€ (ββπ‘) β§ (ββπ‘) β€ 1) β βπ‘ β π (0 β€ ((πΊβπ)βπ‘) β§ ((πΊβπ)βπ‘) β€ 1))) |
17 | 11, 16 | anbi12d 632 | . . . . . . . 8 β’ (β = (πΊβπ) β (((ββπ) = 0 β§ βπ‘ β π (0 β€ (ββπ‘) β§ (ββπ‘) β€ 1)) β (((πΊβπ)βπ) = 0 β§ βπ‘ β π (0 β€ ((πΊβπ)βπ‘) β§ ((πΊβπ)βπ‘) β€ 1)))) |
18 | 17, 2 | elrab2 3652 | . . . . . . 7 β’ ((πΊβπ) β π β ((πΊβπ) β π΄ β§ (((πΊβπ)βπ) = 0 β§ βπ‘ β π (0 β€ ((πΊβπ)βπ‘) β§ ((πΊβπ)βπ‘) β€ 1)))) |
19 | 9, 18 | sylib 217 | . . . . . 6 β’ ((π β§ π β (1...π)) β ((πΊβπ) β π΄ β§ (((πΊβπ)βπ) = 0 β§ βπ‘ β π (0 β€ ((πΊβπ)βπ‘) β§ ((πΊβπ)βπ‘) β€ 1)))) |
20 | 19 | simprld 771 | . . . . 5 β’ ((π β§ π β (1...π)) β ((πΊβπ)βπ) = 0) |
21 | 20 | sumeq2dv 15596 | . . . 4 β’ (π β Ξ£π β (1...π)((πΊβπ)βπ) = Ξ£π β (1...π)0) |
22 | fzfi 13886 | . . . . 5 β’ (1...π) β Fin | |
23 | olc 867 | . . . . 5 β’ ((1...π) β Fin β ((1...π) β (β€β₯β1) β¨ (1...π) β Fin)) | |
24 | sumz 15615 | . . . . 5 β’ (((1...π) β (β€β₯β1) β¨ (1...π) β Fin) β Ξ£π β (1...π)0 = 0) | |
25 | 22, 23, 24 | mp2b 10 | . . . 4 β’ Ξ£π β (1...π)0 = 0 |
26 | 21, 25 | eqtrdi 2789 | . . 3 β’ (π β Ξ£π β (1...π)((πΊβπ)βπ) = 0) |
27 | 26 | oveq2d 7377 | . 2 β’ (π β ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ)) = ((1 / π) Β· 0)) |
28 | 4 | nncnd 12177 | . . . 4 β’ (π β π β β) |
29 | 4 | nnne0d 12211 | . . . 4 β’ (π β π β 0) |
30 | 28, 29 | reccld 11932 | . . 3 β’ (π β (1 / π) β β) |
31 | 30 | mul01d 11362 | . 2 β’ (π β ((1 / π) Β· 0) = 0) |
32 | 8, 27, 31 | 3eqtrd 2777 | 1 β’ (π β (πβπ) = 0) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β¨ wo 846 = wceq 1542 β wcel 2107 βwral 3061 {crab 3406 β wss 3914 class class class wbr 5109 β¦ cmpt 5192 βΆwf 6496 βcfv 6500 (class class class)co 7361 Fincfn 8889 βcr 11058 0cc0 11059 1c1 11060 Β· cmul 11064 β€ cle 11198 / cdiv 11820 βcn 12161 β€β₯cuz 12771 ...cfz 13433 Ξ£csu 15579 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-z 12508 df-uz 12772 df-rp 12924 df-fz 13434 df-fzo 13577 df-seq 13916 df-exp 13977 df-hash 14240 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-clim 15379 df-sum 15580 |
This theorem is referenced by: stoweidlem44 44375 |
Copyright terms: Public domain | W3C validator |