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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 | ⊢ Ⅎ𝑡𝐷 |
stoweidlem58.2 | ⊢ Ⅎ𝑡𝑈 |
stoweidlem58.3 | ⊢ Ⅎ𝑡𝜑 |
stoweidlem58.4 | ⊢ 𝐾 = (topGen‘ran (,)) |
stoweidlem58.5 | ⊢ 𝑇 = ∪ 𝐽 |
stoweidlem58.6 | ⊢ 𝐶 = (𝐽 Cn 𝐾) |
stoweidlem58.7 | ⊢ (𝜑 → 𝐽 ∈ Comp) |
stoweidlem58.8 | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
stoweidlem58.9 | ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
stoweidlem58.10 | ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
stoweidlem58.11 | ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴) |
stoweidlem58.12 | ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) |
stoweidlem58.13 | ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽)) |
stoweidlem58.14 | ⊢ (𝜑 → 𝐷 ∈ (Clsd‘𝐽)) |
stoweidlem58.15 | ⊢ (𝜑 → (𝐵 ∩ 𝐷) = ∅) |
stoweidlem58.16 | ⊢ 𝑈 = (𝑇 ∖ 𝐵) |
stoweidlem58.17 | ⊢ (𝜑 → 𝐸 ∈ ℝ+) |
stoweidlem58.18 | ⊢ (𝜑 → 𝐸 < (1 / 3)) |
Ref | Expression |
---|---|
stoweidlem58 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stoweidlem58.1 | . . 3 ⊢ Ⅎ𝑡𝐷 | |
2 | stoweidlem58.3 | . . . 4 ⊢ Ⅎ𝑡𝜑 | |
3 | 1 | nfeq1 2922 | . . . 4 ⊢ Ⅎ𝑡 𝐷 = ∅ |
4 | 2, 3 | nfan 1902 | . . 3 ⊢ Ⅎ𝑡(𝜑 ∧ 𝐷 = ∅) |
5 | eqid 2738 | . . 3 ⊢ (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ 𝑇 ↦ 1) | |
6 | stoweidlem58.5 | . . 3 ⊢ 𝑇 = ∪ 𝐽 | |
7 | stoweidlem58.11 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴) | |
8 | 7 | adantlr 712 | . . 3 ⊢ (((𝜑 ∧ 𝐷 = ∅) ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴) |
9 | stoweidlem58.13 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽)) | |
10 | 9 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐷 = ∅) → 𝐵 ∈ (Clsd‘𝐽)) |
11 | stoweidlem58.17 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
12 | 11 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐷 = ∅) → 𝐸 ∈ ℝ+) |
13 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝐷 = ∅) → 𝐷 = ∅) | |
14 | 1, 4, 5, 6, 8, 10, 12, 13 | stoweidlem18 43529 | . 2 ⊢ ((𝜑 ∧ 𝐷 = ∅) → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))) |
15 | stoweidlem58.2 | . . 3 ⊢ Ⅎ𝑡𝑈 | |
16 | nfcv 2907 | . . . . 5 ⊢ Ⅎ𝑡∅ | |
17 | 1, 16 | nfne 3045 | . . . 4 ⊢ Ⅎ𝑡 𝐷 ≠ ∅ |
18 | 2, 17 | nfan 1902 | . . 3 ⊢ Ⅎ𝑡(𝜑 ∧ 𝐷 ≠ ∅) |
19 | eqid 2738 | . . 3 ⊢ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} | |
20 | eqid 2738 | . . 3 ⊢ {𝑤 ∈ 𝐽 ∣ ∀𝑒 ∈ ℝ+ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))} = {𝑤 ∈ 𝐽 ∣ ∀𝑒 ∈ ℝ+ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))} | |
21 | stoweidlem58.4 | . . 3 ⊢ 𝐾 = (topGen‘ran (,)) | |
22 | stoweidlem58.6 | . . 3 ⊢ 𝐶 = (𝐽 Cn 𝐾) | |
23 | stoweidlem58.16 | . . 3 ⊢ 𝑈 = (𝑇 ∖ 𝐵) | |
24 | stoweidlem58.7 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Comp) | |
25 | 24 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐽 ∈ Comp) |
26 | stoweidlem58.8 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
27 | 26 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐴 ⊆ 𝐶) |
28 | stoweidlem58.9 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) | |
29 | 28 | 3adant1r 1176 | . . 3 ⊢ (((𝜑 ∧ 𝐷 ≠ ∅) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
30 | stoweidlem58.10 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) | |
31 | 30 | 3adant1r 1176 | . . 3 ⊢ (((𝜑 ∧ 𝐷 ≠ ∅) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
32 | 7 | adantlr 712 | . . 3 ⊢ (((𝜑 ∧ 𝐷 ≠ ∅) ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴) |
33 | stoweidlem58.12 | . . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) | |
34 | 33 | adantlr 712 | . . 3 ⊢ (((𝜑 ∧ 𝐷 ≠ ∅) ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) |
35 | 9 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐵 ∈ (Clsd‘𝐽)) |
36 | stoweidlem58.14 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (Clsd‘𝐽)) | |
37 | 36 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐷 ∈ (Clsd‘𝐽)) |
38 | stoweidlem58.15 | . . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐷) = ∅) | |
39 | 38 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → (𝐵 ∩ 𝐷) = ∅) |
40 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐷 ≠ ∅) | |
41 | 11 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐸 ∈ ℝ+) |
42 | stoweidlem58.18 | . . . 4 ⊢ (𝜑 → 𝐸 < (1 / 3)) | |
43 | 42 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐸 < (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 43568 | . 2 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))) |
45 | 14, 44 | pm2.61dane 3032 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 Ⅎwnf 1786 ∈ wcel 2106 Ⅎwnfc 2887 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 {crab 3068 ∖ cdif 3885 ∩ cin 3887 ⊆ wss 3888 ∅c0 4258 ∪ cuni 4841 class class class wbr 5076 ↦ cmpt 5159 ran crn 5592 ‘cfv 6435 (class class class)co 7277 ℝcr 10868 0cc0 10869 1c1 10870 + caddc 10872 · cmul 10874 < clt 11007 ≤ cle 11008 − cmin 11203 / cdiv 11630 3c3 12027 ℝ+crp 12728 (,)cioo 13077 topGenctg 17146 Clsdccld 22165 Cn ccn 22373 Compccmp 22535 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-inf2 9397 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 ax-pre-sup 10947 ax-mulf 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-isom 6444 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7976 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-2o 8296 df-er 8496 df-map 8615 df-pm 8616 df-ixp 8684 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-fsupp 9127 df-fi 9168 df-sup 9199 df-inf 9200 df-oi 9267 df-card 9695 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12436 df-uz 12581 df-q 12687 df-rp 12729 df-xneg 12846 df-xadd 12847 df-xmul 12848 df-ioo 13081 df-ico 13083 df-icc 13084 df-fz 13238 df-fzo 13381 df-fl 13510 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-clim 15195 df-rlim 15196 df-sum 15396 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-0g 17150 df-gsum 17151 df-topgen 17152 df-pt 17153 df-prds 17156 df-xrs 17211 df-qtop 17216 df-imas 17217 df-xps 17219 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-mulg 18699 df-cntz 18921 df-cmn 19386 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cld 22168 df-cn 22376 df-cnp 22377 df-cmp 22536 df-tx 22711 df-hmeo 22904 df-xms 23471 df-ms 23472 df-tms 23473 |
This theorem is referenced by: stoweidlem59 43570 |
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