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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 2822 | . . . . 5 β’ (π = π β (π β π β π β π)) | |
2 | 1 | anbi2d 630 | . . . 4 β’ (π = π β ((π β§ π β π) β (π β§ π β π))) |
3 | fveq2 6843 | . . . . 5 β’ (π = π β (πβπ ) = (πβπ)) | |
4 | fveq2 6843 | . . . . . . 7 β’ (π = π β ((πΊβπ)βπ ) = ((πΊβπ)βπ)) | |
5 | 4 | sumeq2sdv 15594 | . . . . . 6 β’ (π = π β Ξ£π β (1...π)((πΊβπ)βπ ) = Ξ£π β (1...π)((πΊβπ)βπ)) |
6 | 5 | oveq2d 7374 | . . . . 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 2749 | . . . 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 345 | . . 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 6843 | . . . . . 6 β’ (π‘ = π β ((πΊβπ)βπ‘) = ((πΊβπ)βπ )) | |
11 | 10 | sumeq2sdv 15594 | . . . . 5 β’ (π‘ = π β Ξ£π β (1...π)((πΊβπ)βπ‘) = Ξ£π β (1...π)((πΊβπ)βπ )) |
12 | 11 | oveq2d 7374 | . . . 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 486 | . . . 4 β’ ((π β§ π β π) β π β π) | |
14 | stoweidlem30.3 | . . . . . . 7 β’ (π β π β β) | |
15 | 14 | nnrecred 12209 | . . . . . 6 β’ (π β (1 / π) β β) |
16 | 15 | adantr 482 | . . . . 5 β’ ((π β§ π β π) β (1 / π) β β) |
17 | fzfid 13884 | . . . . . 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 44342 | . . . . . . . 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 1143 | . . . . . . 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 15624 | . . . . 5 β’ ((π β§ π β π) β Ξ£π β (1...π)((πΊβπ)βπ ) β β) |
25 | 16, 24 | remulcld 11190 | . . . 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 6972 | . . 3 β’ ((π β§ π β π) β (πβπ ) = ((1 / π) Β· Ξ£π β (1...π)((πΊβπ)βπ ))) |
27 | 8, 26 | vtoclg 3524 | . 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 397 = wceq 1542 β wcel 2107 βwral 3061 {crab 3406 class class class wbr 5106 β¦ cmpt 5189 βΆwf 6493 βcfv 6497 (class class class)co 7358 βcr 11055 0cc0 11056 1c1 11057 Β· cmul 11061 β€ cle 11195 / cdiv 11817 βcn 12158 ...cfz 13430 Ξ£csu 15576 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
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 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-fz 13431 df-fzo 13574 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-sum 15577 |
This theorem is referenced by: stoweidlem37 44364 stoweidlem38 44365 stoweidlem44 44371 |
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