stoweidlem58 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem58 43489

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.)

**Hypotheses**
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))

**Assertion**
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)))

Distinct variable groups: 𝑓,𝑎,𝑟,𝑡,𝐴,𝑞 𝐷,𝑎,𝑓,𝑟 𝑇,𝑎,𝑓,𝑟,𝑡 𝑈,𝑎,𝑓,𝑟 𝜑,𝑎,𝑓,𝑟 𝑓,𝑔,𝑟,𝑡,𝐴 𝑓,𝐸,𝑔,𝑟,𝑡 𝑥,𝑓,𝑔,𝑡,𝐴 𝐵,𝑓,𝑔,𝑟 𝑓,𝐽,𝑔,𝑟,𝑡 𝑔,𝑞,𝐷 𝑇,𝑔 𝑈,𝑔 𝜑,𝑔 𝐷,𝑞 𝑇,𝑞 𝑈,𝑞 𝜑,𝑞 𝑡,𝐾 𝑥,𝐵 𝑥,𝐷 𝑥,𝐸 𝑥,𝑇 Allowed substitution hints: 𝜑(𝑥,𝑡) 𝐵(𝑡,𝑞,𝑎) 𝐶(𝑥,𝑡,𝑓,𝑔,𝑟,𝑞,𝑎) 𝐷(𝑡) 𝑈(𝑥,𝑡) 𝐸(𝑞,𝑎) 𝐽(𝑥,𝑞,𝑎) 𝐾(𝑥,𝑓,𝑔,𝑟,𝑞,𝑎)

Proof of Theorem stoweidlem58 Dummy variables 𝑒 ℎ 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem58.1	. . 3 ⊢ Ⅎ𝑡𝐷
2		stoweidlem58.3	. . . 4 ⊢ Ⅎ𝑡𝜑
3	1	nfeq1 2921	. . . 4 ⊢ Ⅎ𝑡 𝐷 = ∅
4	2, 3	nfan 1903	. . 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 711	. . 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 484	. . 3 ⊢ ((𝜑 ∧ 𝐷 = ∅) → 𝐷 = ∅)
14	1, 4, 5, 6, 8, 10, 12, 13	stoweidlem18 43449	. 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 2906	. . . . 5 ⊢ Ⅎ𝑡∅
17	1, 16	nfne 3044	. . . 4 ⊢ Ⅎ𝑡 𝐷 ≠ ∅
18	2, 17	nfan 1903	. . 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 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 1175	. . 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 1175	. . 3 ⊢ (((𝜑 ∧ 𝐷 ≠ ∅) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
32	7	adantlr 711	. . 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 711	. . 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 484	. . 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 43488	. 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 3031	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 395 ∧ w3a 1085 = wceq 1539 Ⅎwnf 1787 ∈ wcel 2108 Ⅎwnfc 2886 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 {crab 3067 ∖ cdif 3880 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 ∪ cuni 4836 class class class wbr 5070 ↦ cmpt 5153 ran crn 5581 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 / cdiv 11562 3c3 11959 ℝ⁺crp 12659 (,)cioo 13008 topGenctg 17065 Clsdccld 22075 Cn ccn 22283 Compccmp 22445

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-rlim 15126 df-sum 15326 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-cn 22286 df-cnp 22287 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-xms 23381 df-ms 23382 df-tms 23383

This theorem is referenced by: stoweidlem59 43490

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