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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem30 | Structured version Visualization version GIF version |
Description: This lemma is used to prove the existence of a function p as in Lemma 1 [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 |
---|---|
stoweidlem30.1 | β’ π = {β β π΄ β£ ((ββπ) = 0 β§ βπ‘ β π (0 β€ (ββπ‘) β§ (ββπ‘) β€ 1))} |
stoweidlem30.2 | β’ π = (π‘ β π β¦ ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ‘))) |
stoweidlem30.3 | β’ (π β π β β) |
stoweidlem30.4 | β’ (π β πΊ:(1...π)βΆπ) |
stoweidlem30.5 | β’ ((π β§ π β π΄) β π:πβΆβ) |
Ref | Expression |
---|---|
stoweidlem30 | β’ ((π β§ π β π) β (πβπ) = ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2813 | . . . . 5 β’ (π = π β (π β π β π β π)) | |
2 | 1 | anbi2d 628 | . . . 4 β’ (π = π β ((π β§ π β π) β (π β§ π β π))) |
3 | fveq2 6890 | . . . . 5 β’ (π = π β (πβπ ) = (πβπ)) | |
4 | fveq2 6890 | . . . . . . 7 β’ (π = π β ((πΊβπ)βπ ) = ((πΊβπ)βπ)) | |
5 | 4 | sumeq2sdv 15677 | . . . . . 6 β’ (π = π β Ξ£π β (1...π)((πΊβπ)βπ ) = Ξ£π β (1...π)((πΊβπ)βπ)) |
6 | 5 | oveq2d 7429 | . . . . 5 β’ (π = π β ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ )) = ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ))) |
7 | 3, 6 | eqeq12d 2741 | . . . 4 β’ (π = π β ((πβπ ) = ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ )) β (πβπ) = ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ)))) |
8 | 2, 7 | imbi12d 343 | . . 3 β’ (π = π β (((π β§ π β π) β (πβπ ) = ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ ))) β ((π β§ π β π) β (πβπ) = ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ))))) |
9 | stoweidlem30.2 | . . . 4 β’ π = (π‘ β π β¦ ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ‘))) | |
10 | fveq2 6890 | . . . . . 6 β’ (π‘ = π β ((πΊβπ)βπ‘) = ((πΊβπ)βπ )) | |
11 | 10 | sumeq2sdv 15677 | . . . . 5 β’ (π‘ = π β Ξ£π β (1...π)((πΊβπ)βπ‘) = Ξ£π β (1...π)((πΊβπ)βπ )) |
12 | 11 | oveq2d 7429 | . . . 4 β’ (π‘ = π β ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ‘)) = ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ ))) |
13 | simpr 483 | . . . 4 β’ ((π β§ π β π) β π β π) | |
14 | stoweidlem30.3 | . . . . . . 7 β’ (π β π β β) | |
15 | 14 | nnrecred 12288 | . . . . . 6 β’ (π β (1 / π) β β) |
16 | 15 | adantr 479 | . . . . 5 β’ ((π β§ π β π) β (1 / π) β β) |
17 | fzfid 13965 | . . . . . 6 β’ ((π β§ π β π) β (1...π) β Fin) | |
18 | stoweidlem30.1 | . . . . . . . . 9 β’ π = {β β π΄ β£ ((ββπ) = 0 β§ βπ‘ β π (0 β€ (ββπ‘) β§ (ββπ‘) β€ 1))} | |
19 | stoweidlem30.4 | . . . . . . . . 9 β’ (π β πΊ:(1...π)βΆπ) | |
20 | stoweidlem30.5 | . . . . . . . . 9 β’ ((π β§ π β π΄) β π:πβΆβ) | |
21 | 18, 19, 20 | stoweidlem15 45462 | . . . . . . . 8 β’ (((π β§ π β (1...π)) β§ π β π) β (((πΊβπ)βπ ) β β β§ 0 β€ ((πΊβπ)βπ ) β§ ((πΊβπ)βπ ) β€ 1)) |
22 | 21 | simp1d 1139 | . . . . . . 7 β’ (((π β§ π β (1...π)) β§ π β π) β ((πΊβπ)βπ ) β β) |
23 | 22 | an32s 650 | . . . . . 6 β’ (((π β§ π β π) β§ π β (1...π)) β ((πΊβπ)βπ ) β β) |
24 | 17, 23 | fsumrecl 15707 | . . . . 5 β’ ((π β§ π β π) β Ξ£π β (1...π)((πΊβπ)βπ ) β β) |
25 | 16, 24 | remulcld 11269 | . . . 4 β’ ((π β§ π β π) β ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ )) β β) |
26 | 9, 12, 13, 25 | fvmptd3 7021 | . . 3 β’ ((π β§ π β π) β (πβπ ) = ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ ))) |
27 | 8, 26 | vtoclg 3533 | . 2 β’ (π β π β ((π β§ π β π) β (πβπ) = ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ)))) |
28 | 27 | anabsi7 669 | 1 β’ ((π β§ π β π) β (πβπ) = ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3051 {crab 3419 class class class wbr 5144 β¦ cmpt 5227 βΆwf 6539 βcfv 6543 (class class class)co 7413 βcr 11132 0cc0 11133 1c1 11134 Β· cmul 11138 β€ cle 11274 / cdiv 11896 βcn 12237 ...cfz 13511 Ξ£csu 15659 |
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 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-fz 13512 df-fzo 13655 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-sum 15660 |
This theorem is referenced by: stoweidlem37 45484 stoweidlem38 45485 stoweidlem44 45491 |
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