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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 2764 | . . . 4 ⊢ Ⅎ𝑡 𝐷 = ∅ |
4 | 2, 3 | nfan 1816 | . . 3 ⊢ Ⅎ𝑡(𝜑 ∧ 𝐷 = ∅) |
5 | eqid 2610 | . . 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 747 | . . 3 ⊢ (((𝜑 ∧ 𝐷 = ∅) ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴) |
9 | stoweidlem58.13 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽)) | |
10 | 9 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐷 = ∅) → 𝐵 ∈ (Clsd‘𝐽)) |
11 | stoweidlem58.17 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
12 | 11 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐷 = ∅) → 𝐸 ∈ ℝ+) |
13 | simpr 476 | . . 3 ⊢ ((𝜑 ∧ 𝐷 = ∅) → 𝐷 = ∅) | |
14 | 1, 4, 5, 6, 8, 10, 12, 13 | stoweidlem18 38911 | . 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 2751 | . . . . 5 ⊢ Ⅎ𝑡∅ | |
17 | 1, 16 | nfne 2882 | . . . 4 ⊢ Ⅎ𝑡 𝐷 ≠ ∅ |
18 | 2, 17 | nfan 1816 | . . 3 ⊢ Ⅎ𝑡(𝜑 ∧ 𝐷 ≠ ∅) |
19 | eqid 2610 | . . 3 ⊢ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} | |
20 | eqid 2610 | . . 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 480 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐽 ∈ Comp) |
26 | stoweidlem58.8 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
27 | 26 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐴 ⊆ 𝐶) |
28 | stoweidlem58.9 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) | |
29 | 28 | 3adant1r 1311 | . . 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 1311 | . . 3 ⊢ (((𝜑 ∧ 𝐷 ≠ ∅) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
32 | 7 | adantlr 747 | . . 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 747 | . . 3 ⊢ (((𝜑 ∧ 𝐷 ≠ ∅) ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) |
35 | 9 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐵 ∈ (Clsd‘𝐽)) |
36 | stoweidlem58.14 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (Clsd‘𝐽)) | |
37 | 36 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐷 ∈ (Clsd‘𝐽)) |
38 | stoweidlem58.15 | . . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐷) = ∅) | |
39 | 38 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → (𝐵 ∩ 𝐷) = ∅) |
40 | simpr 476 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐷 ≠ ∅) | |
41 | 11 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ ∅) → 𝐸 ∈ ℝ+) |
42 | stoweidlem58.18 | . . . 4 ⊢ (𝜑 → 𝐸 < (1 / 3)) | |
43 | 42 | adantr 480 | . . 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 38950 | . 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 2869 | 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 383 ∧ w3a 1031 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 3c3 10948 ℝ+crp 11708 (,)cioo 12046 topGenctg 15921 Clsdccld 20630 Cn ccn 20838 Compccmp 20999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-cn 20841 df-cnp 20842 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 |
This theorem is referenced by: stoweidlem59 38952 |
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