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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 2816 | . . . . 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 6891 | . . . . 5 β’ (π = π β (πβπ ) = (πβπ)) | |
4 | fveq2 6891 | . . . . . . 7 β’ (π = π β ((πΊβπ)βπ ) = ((πΊβπ)βπ)) | |
5 | 4 | sumeq2sdv 15668 | . . . . . 6 β’ (π = π β Ξ£π β (1...π)((πΊβπ)βπ ) = Ξ£π β (1...π)((πΊβπ)βπ)) |
6 | 5 | oveq2d 7430 | . . . . 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 2743 | . . . 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 344 | . . 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 6891 | . . . . . 6 β’ (π‘ = π β ((πΊβπ)βπ‘) = ((πΊβπ)βπ )) | |
11 | 10 | sumeq2sdv 15668 | . . . . 5 β’ (π‘ = π β Ξ£π β (1...π)((πΊβπ)βπ‘) = Ξ£π β (1...π)((πΊβπ)βπ )) |
12 | 11 | oveq2d 7430 | . . . 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 484 | . . . 4 β’ ((π β§ π β π) β π β π) | |
14 | stoweidlem30.3 | . . . . . . 7 β’ (π β π β β) | |
15 | 14 | nnrecred 12279 | . . . . . 6 β’ (π β (1 / π) β β) |
16 | 15 | adantr 480 | . . . . 5 β’ ((π β§ π β π) β (1 / π) β β) |
17 | fzfid 13956 | . . . . . 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 45316 | . . . . . . . 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 1140 | . . . . . . 7 β’ (((π β§ π β (1...π)) β§ π β π) β ((πΊβπ)βπ ) β β) |
23 | 22 | an32s 651 | . . . . . 6 β’ (((π β§ π β π) β§ π β (1...π)) β ((πΊβπ)βπ ) β β) |
24 | 17, 23 | fsumrecl 15698 | . . . . 5 β’ ((π β§ π β π) β Ξ£π β (1...π)((πΊβπ)βπ ) β β) |
25 | 16, 24 | remulcld 11260 | . . . 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 7022 | . . 3 β’ ((π β§ π β π) β (πβπ ) = ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ ))) |
27 | 8, 26 | vtoclg 3538 | . 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 670 | 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 395 = wceq 1534 β wcel 2099 βwral 3056 {crab 3427 class class class wbr 5142 β¦ cmpt 5225 βΆwf 6538 βcfv 6542 (class class class)co 7414 βcr 11123 0cc0 11124 1c1 11125 Β· cmul 11129 β€ cle 11265 / cdiv 11887 βcn 12228 ...cfz 13502 Ξ£csu 15650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-n0 12489 df-z 12575 df-uz 12839 df-rp 12993 df-fz 13503 df-fzo 13646 df-seq 13985 df-exp 14045 df-hash 14308 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-clim 15450 df-sum 15651 |
This theorem is referenced by: stoweidlem37 45338 stoweidlem38 45339 stoweidlem44 45345 |
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