stoweidlem58 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem58 46030

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 2913	. . . 4 ⊢ Ⅎ𝑡 𝐷 = ∅
4	2, 3	nfan 1898	. . 3 ⊢ Ⅎ𝑡(𝜑 ∧ 𝐷 = ∅)
5		eqid 2734	. . 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 715	. . 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 45990	. 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 2897	. . . . 5 ⊢ Ⅎ𝑡∅
17	1, 16	nfne 3032	. . . 4 ⊢ Ⅎ𝑡 𝐷 ≠ ∅
18	2, 17	nfan 1898	. . 3 ⊢ Ⅎ𝑡(𝜑 ∧ 𝐷 ≠ ∅)
19		eqid 2734	. . 3 ⊢ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
20		eqid 2734	. . 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 1177	. . 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 1177	. . 3 ⊢ (((𝜑 ∧ 𝐷 ≠ ∅) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
32	7	adantlr 715	. . 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 715	. . 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 46029	. 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 3018	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 1086 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2107 Ⅎwnfc 2882 ≠ wne 2931 ∀wral 3050 ∃wrex 3059 {crab 3419 ∖ cdif 3928 ∩ cin 3930 ⊆ wss 3931 ∅c0 4313 ∪ cuni 4887 class class class wbr 5123 ↦ cmpt 5205 ran crn 5666 ‘cfv 6541 (class class class)co 7413 ℝcr 11136 0cc0 11137 1c1 11138 + caddc 11140 · cmul 11142 < clt 11277 ≤ cle 11278 − cmin 11474 / cdiv 11902 3c3 12304 ℝ⁺crp 13016 (,)cioo 13369 topGenctg 17453 Clsdccld 22970 Cn ccn 23178 Compccmp 23340

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-inf2 9663 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214 ax-pre-sup 11215

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4888 df-int 4927 df-iun 4973 df-iin 4974 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-se 5618 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7870 df-1st 7996 df-2nd 7997 df-supp 8168 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8727 df-map 8850 df-pm 8851 df-ixp 8920 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-fsupp 9384 df-fi 9433 df-sup 9464 df-inf 9465 df-oi 9532 df-card 9961 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-div 11903 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12510 df-z 12597 df-dec 12717 df-uz 12861 df-q 12973 df-rp 13017 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-ioo 13373 df-ico 13375 df-icc 13376 df-fz 13530 df-fzo 13677 df-fl 13814 df-seq 14025 df-exp 14085 df-hash 14352 df-cj 15120 df-re 15121 df-im 15122 df-sqrt 15256 df-abs 15257 df-clim 15506 df-rlim 15507 df-sum 15705 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17230 df-ress 17253 df-plusg 17286 df-mulr 17287 df-starv 17288 df-sca 17289 df-vsca 17290 df-ip 17291 df-tset 17292 df-ple 17293 df-ds 17295 df-unif 17296 df-hom 17297 df-cco 17298 df-rest 17438 df-topn 17439 df-0g 17457 df-gsum 17458 df-topgen 17459 df-pt 17460 df-prds 17463 df-xrs 17518 df-qtop 17523 df-imas 17524 df-xps 17526 df-mre 17600 df-mrc 17601 df-acs 17603 df-mgm 18622 df-sgrp 18701 df-mnd 18717 df-submnd 18766 df-mulg 19055 df-cntz 19304 df-cmn 19768 df-psmet 21318 df-xmet 21319 df-met 21320 df-bl 21321 df-mopn 21322 df-cnfld 21327 df-top 22848 df-topon 22865 df-topsp 22887 df-bases 22900 df-cld 22973 df-cn 23181 df-cnp 23182 df-cmp 23341 df-tx 23516 df-hmeo 23709 df-xms 24275 df-ms 24276 df-tms 24277

This theorem is referenced by: stoweidlem59 46031

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