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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 2993 | . . . 4 ⊢ Ⅎ𝑡 𝐷 = ∅ |
4 | 2, 3 | nfan 1900 | . . 3 ⊢ Ⅎ𝑡(𝜑 ∧ 𝐷 = ∅) |
5 | eqid 2821 | . . 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 713 | . . 3 ⊢ (((𝜑 ∧ 𝐷 = ∅) ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴) |
9 | stoweidlem58.13 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽)) | |
10 | 9 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐷 = ∅) → 𝐵 ∈ (Clsd‘𝐽)) |
11 | stoweidlem58.17 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
12 | 11 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐷 = ∅) → 𝐸 ∈ ℝ+) |
13 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝐷 = ∅) → 𝐷 = ∅) | |
14 | 1, 4, 5, 6, 8, 10, 12, 13 | stoweidlem18 42323 | . 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 2977 | . . . . 5 ⊢ Ⅎ𝑡∅ | |
17 | 1, 16 | nfne 3119 | . . . 4 ⊢ Ⅎ𝑡 𝐷 ≠ ∅ |
18 | 2, 17 | nfan 1900 | . . 3 ⊢ Ⅎ𝑡(𝜑 ∧ 𝐷 ≠ ∅) |
19 | eqid 2821 | . . 3 ⊢ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} | |
20 | eqid 2821 | . . 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 483 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐽 ∈ Comp) |
26 | stoweidlem58.8 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
27 | 26 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐴 ⊆ 𝐶) |
28 | stoweidlem58.9 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) | |
29 | 28 | 3adant1r 1173 | . . 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 1173 | . . 3 ⊢ (((𝜑 ∧ 𝐷 ≠ ∅) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
32 | 7 | adantlr 713 | . . 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 713 | . . 3 ⊢ (((𝜑 ∧ 𝐷 ≠ ∅) ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) |
35 | 9 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐵 ∈ (Clsd‘𝐽)) |
36 | stoweidlem58.14 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (Clsd‘𝐽)) | |
37 | 36 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐷 ∈ (Clsd‘𝐽)) |
38 | stoweidlem58.15 | . . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐷) = ∅) | |
39 | 38 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → (𝐵 ∩ 𝐷) = ∅) |
40 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐷 ≠ ∅) | |
41 | 11 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐸 ∈ ℝ+) |
42 | stoweidlem58.18 | . . . 4 ⊢ (𝜑 → 𝐸 < (1 / 3)) | |
43 | 42 | adantr 483 | . . 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 42362 | . 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 3104 | 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 398 ∧ w3a 1083 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2961 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 ∪ cuni 4838 class class class wbr 5066 ↦ cmpt 5146 ran crn 5556 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 < clt 10675 ≤ cle 10676 − cmin 10870 / cdiv 11297 3c3 11694 ℝ+crp 12390 (,)cioo 12739 topGenctg 16711 Clsdccld 21624 Cn ccn 21832 Compccmp 21994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-rlim 14846 df-sum 15043 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-cn 21835 df-cnp 21836 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-xms 22930 df-ms 22931 df-tms 22932 |
This theorem is referenced by: stoweidlem59 42364 |
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