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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem58 | Structured version Visualization version GIF version |
Description: This theorem proves Lemma 2 in [BrosowskiDeutsh] p. 91. Here D is used to represent the set A of Lemma 2, because here the variable A is used for the subalgebra of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
stoweidlem58.1 | β’ β²π‘π· |
stoweidlem58.2 | β’ β²π‘π |
stoweidlem58.3 | β’ β²π‘π |
stoweidlem58.4 | β’ πΎ = (topGenβran (,)) |
stoweidlem58.5 | β’ π = βͺ π½ |
stoweidlem58.6 | β’ πΆ = (π½ Cn πΎ) |
stoweidlem58.7 | β’ (π β π½ β Comp) |
stoweidlem58.8 | β’ (π β π΄ β πΆ) |
stoweidlem58.9 | β’ ((π β§ π β π΄ β§ π β π΄) β (π‘ β π β¦ ((πβπ‘) + (πβπ‘))) β π΄) |
stoweidlem58.10 | β’ ((π β§ π β π΄ β§ π β π΄) β (π‘ β π β¦ ((πβπ‘) Β· (πβπ‘))) β π΄) |
stoweidlem58.11 | β’ ((π β§ π β β) β (π‘ β π β¦ π) β π΄) |
stoweidlem58.12 | β’ ((π β§ (π β π β§ π‘ β π β§ π β π‘)) β βπ β π΄ (πβπ) β (πβπ‘)) |
stoweidlem58.13 | β’ (π β π΅ β (Clsdβπ½)) |
stoweidlem58.14 | β’ (π β π· β (Clsdβπ½)) |
stoweidlem58.15 | β’ (π β (π΅ β© π·) = β ) |
stoweidlem58.16 | β’ π = (π β π΅) |
stoweidlem58.17 | β’ (π β πΈ β β+) |
stoweidlem58.18 | β’ (π β πΈ < (1 / 3)) |
Ref | Expression |
---|---|
stoweidlem58 | β’ (π β βπ₯ β π΄ (βπ‘ β π (0 β€ (π₯βπ‘) β§ (π₯βπ‘) β€ 1) β§ βπ‘ β π· (π₯βπ‘) < πΈ β§ βπ‘ β π΅ (1 β πΈ) < (π₯βπ‘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stoweidlem58.1 | . . 3 β’ β²π‘π· | |
2 | stoweidlem58.3 | . . . 4 β’ β²π‘π | |
3 | 1 | nfeq1 2919 | . . . 4 β’ β²π‘ π· = β |
4 | 2, 3 | nfan 1903 | . . 3 β’ β²π‘(π β§ π· = β ) |
5 | eqid 2733 | . . 3 β’ (π‘ β π β¦ 1) = (π‘ β π β¦ 1) | |
6 | stoweidlem58.5 | . . 3 β’ π = βͺ π½ | |
7 | stoweidlem58.11 | . . . 4 β’ ((π β§ π β β) β (π‘ β π β¦ π) β π΄) | |
8 | 7 | adantlr 714 | . . 3 β’ (((π β§ π· = β ) β§ π β β) β (π‘ β π β¦ π) β π΄) |
9 | stoweidlem58.13 | . . . 4 β’ (π β π΅ β (Clsdβπ½)) | |
10 | 9 | adantr 482 | . . 3 β’ ((π β§ π· = β ) β π΅ β (Clsdβπ½)) |
11 | stoweidlem58.17 | . . . 4 β’ (π β πΈ β β+) | |
12 | 11 | adantr 482 | . . 3 β’ ((π β§ π· = β ) β πΈ β β+) |
13 | simpr 486 | . . 3 β’ ((π β§ π· = β ) β π· = β ) | |
14 | 1, 4, 5, 6, 8, 10, 12, 13 | stoweidlem18 44345 | . 2 β’ ((π β§ π· = β ) β βπ₯ β π΄ (βπ‘ β π (0 β€ (π₯βπ‘) β§ (π₯βπ‘) β€ 1) β§ βπ‘ β π· (π₯βπ‘) < πΈ β§ βπ‘ β π΅ (1 β πΈ) < (π₯βπ‘))) |
15 | stoweidlem58.2 | . . 3 β’ β²π‘π | |
16 | nfcv 2904 | . . . . 5 β’ β²π‘β | |
17 | 1, 16 | nfne 3042 | . . . 4 β’ β²π‘ π· β β |
18 | 2, 17 | nfan 1903 | . . 3 β’ β²π‘(π β§ π· β β ) |
19 | eqid 2733 | . . 3 β’ {β β π΄ β£ βπ‘ β π (0 β€ (ββπ‘) β§ (ββπ‘) β€ 1)} = {β β π΄ β£ βπ‘ β π (0 β€ (ββπ‘) β§ (ββπ‘) β€ 1)} | |
20 | eqid 2733 | . . 3 β’ {π€ β π½ β£ βπ β β+ ββ β π΄ (βπ‘ β π (0 β€ (ββπ‘) β§ (ββπ‘) β€ 1) β§ βπ‘ β π€ (ββπ‘) < π β§ βπ‘ β (π β π)(1 β π) < (ββπ‘))} = {π€ β π½ β£ βπ β β+ ββ β π΄ (βπ‘ β π (0 β€ (ββπ‘) β§ (ββπ‘) β€ 1) β§ βπ‘ β π€ (ββπ‘) < π β§ βπ‘ β (π β π)(1 β π) < (ββπ‘))} | |
21 | stoweidlem58.4 | . . 3 β’ πΎ = (topGenβran (,)) | |
22 | stoweidlem58.6 | . . 3 β’ πΆ = (π½ Cn πΎ) | |
23 | stoweidlem58.16 | . . 3 β’ π = (π β π΅) | |
24 | stoweidlem58.7 | . . . 4 β’ (π β π½ β Comp) | |
25 | 24 | adantr 482 | . . 3 β’ ((π β§ π· β β ) β π½ β Comp) |
26 | stoweidlem58.8 | . . . 4 β’ (π β π΄ β πΆ) | |
27 | 26 | adantr 482 | . . 3 β’ ((π β§ π· β β ) β π΄ β πΆ) |
28 | stoweidlem58.9 | . . . 4 β’ ((π β§ π β π΄ β§ π β π΄) β (π‘ β π β¦ ((πβπ‘) + (πβπ‘))) β π΄) | |
29 | 28 | 3adant1r 1178 | . . 3 β’ (((π β§ π· β β ) β§ π β π΄ β§ π β π΄) β (π‘ β π β¦ ((πβπ‘) + (πβπ‘))) β π΄) |
30 | stoweidlem58.10 | . . . 4 β’ ((π β§ π β π΄ β§ π β π΄) β (π‘ β π β¦ ((πβπ‘) Β· (πβπ‘))) β π΄) | |
31 | 30 | 3adant1r 1178 | . . 3 β’ (((π β§ π· β β ) β§ π β π΄ β§ π β π΄) β (π‘ β π β¦ ((πβπ‘) Β· (πβπ‘))) β π΄) |
32 | 7 | adantlr 714 | . . 3 β’ (((π β§ π· β β ) β§ π β β) β (π‘ β π β¦ π) β π΄) |
33 | stoweidlem58.12 | . . . 4 β’ ((π β§ (π β π β§ π‘ β π β§ π β π‘)) β βπ β π΄ (πβπ) β (πβπ‘)) | |
34 | 33 | adantlr 714 | . . 3 β’ (((π β§ π· β β ) β§ (π β π β§ π‘ β π β§ π β π‘)) β βπ β π΄ (πβπ) β (πβπ‘)) |
35 | 9 | adantr 482 | . . 3 β’ ((π β§ π· β β ) β π΅ β (Clsdβπ½)) |
36 | stoweidlem58.14 | . . . 4 β’ (π β π· β (Clsdβπ½)) | |
37 | 36 | adantr 482 | . . 3 β’ ((π β§ π· β β ) β π· β (Clsdβπ½)) |
38 | stoweidlem58.15 | . . . 4 β’ (π β (π΅ β© π·) = β ) | |
39 | 38 | adantr 482 | . . 3 β’ ((π β§ π· β β ) β (π΅ β© π·) = β ) |
40 | simpr 486 | . . 3 β’ ((π β§ π· β β ) β π· β β ) | |
41 | 11 | adantr 482 | . . 3 β’ ((π β§ π· β β ) β πΈ β β+) |
42 | stoweidlem58.18 | . . . 4 β’ (π β πΈ < (1 / 3)) | |
43 | 42 | adantr 482 | . . 3 β’ ((π β§ π· β β ) β πΈ < (1 / 3)) |
44 | 1, 15, 18, 19, 20, 21, 6, 22, 23, 25, 27, 29, 31, 32, 34, 35, 37, 39, 40, 41, 43 | stoweidlem57 44384 | . 2 β’ ((π β§ π· β β ) β βπ₯ β π΄ (βπ‘ β π (0 β€ (π₯βπ‘) β§ (π₯βπ‘) β€ 1) β§ βπ‘ β π· (π₯βπ‘) < πΈ β§ βπ‘ β π΅ (1 β πΈ) < (π₯βπ‘))) |
45 | 14, 44 | pm2.61dane 3029 | 1 β’ (π β βπ₯ β π΄ (βπ‘ β π (0 β€ (π₯βπ‘) β§ (π₯βπ‘) β€ 1) β§ βπ‘ β π· (π₯βπ‘) < πΈ β§ βπ‘ β π΅ (1 β πΈ) < (π₯βπ‘))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β²wnf 1786 β wcel 2107 β²wnfc 2884 β wne 2940 βwral 3061 βwrex 3070 {crab 3406 β cdif 3908 β© cin 3910 β wss 3911 β c0 4283 βͺ cuni 4866 class class class wbr 5106 β¦ cmpt 5189 ran crn 5635 βcfv 6497 (class class class)co 7358 βcr 11055 0cc0 11056 1c1 11057 + caddc 11059 Β· cmul 11061 < clt 11194 β€ cle 11195 β cmin 11390 / cdiv 11817 3c3 12214 β+crp 12920 (,)cioo 13270 topGenctg 17324 Clsdccld 22383 Cn ccn 22591 Compccmp 22753 |
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 ax-mulf 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 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-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 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-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8651 df-map 8770 df-pm 8771 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-fi 9352 df-sup 9383 df-inf 9384 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-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-q 12879 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13274 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-fl 13703 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-rlim 15377 df-sum 15577 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-hom 17162 df-cco 17163 df-rest 17309 df-topn 17310 df-0g 17328 df-gsum 17329 df-topgen 17330 df-pt 17331 df-prds 17334 df-xrs 17389 df-qtop 17394 df-imas 17395 df-xps 17397 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-submnd 18607 df-mulg 18878 df-cntz 19102 df-cmn 19569 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-cn 22594 df-cnp 22595 df-cmp 22754 df-tx 22929 df-hmeo 23122 df-xms 23689 df-ms 23690 df-tms 23691 |
This theorem is referenced by: stoweidlem59 44386 |
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