stoweidlem58 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem58 46042

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 1899	. . 3 ⊢ Ⅎ𝑡(𝜑 ∧ 𝐷 = ∅)
5		eqid 2737	. . 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 46002	. 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 2905	. . . . 5 ⊢ Ⅎ𝑡∅
17	1, 16	nfne 3043	. . . 4 ⊢ Ⅎ𝑡 𝐷 ≠ ∅
18	2, 17	nfan 1899	. . 3 ⊢ Ⅎ𝑡(𝜑 ∧ 𝐷 ≠ ∅)
19		eqid 2737	. . 3 ⊢ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
20		eqid 2737	. . 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 1178	. . 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 1178	. . 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 46041	. 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 3029	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 1087 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2108 Ⅎwnfc 2890 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3436 ∖ cdif 3963 ∩ cin 3965 ⊆ wss 3966 ∅c0 4342 ∪ cuni 4915 class class class wbr 5151 ↦ cmpt 5234 ran crn 5694 ‘cfv 6569 (class class class)co 7438 ℝcr 11161 0cc0 11162 1c1 11163 + caddc 11165 · cmul 11167 < clt 11302 ≤ cle 11303 − cmin 11499 / cdiv 11927 3c3 12329 ℝ⁺crp 13041 (,)cioo 13393 topGenctg 17493 Clsdccld 23049 Cn ccn 23257 Compccmp 23419

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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-inf2 9688 ax-cnex 11218 ax-resscn 11219 ax-1cn 11220 ax-icn 11221 ax-addcl 11222 ax-addrcl 11223 ax-mulcl 11224 ax-mulrcl 11225 ax-mulcom 11226 ax-addass 11227 ax-mulass 11228 ax-distr 11229 ax-i2m1 11230 ax-1ne0 11231 ax-1rid 11232 ax-rnegex 11233 ax-rrecex 11234 ax-cnre 11235 ax-pre-lttri 11236 ax-pre-lttrn 11237 ax-pre-ltadd 11238 ax-pre-mulgt0 11239 ax-pre-sup 11240

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-pss 3986 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-tp 4639 df-op 4641 df-uni 4916 df-int 4955 df-iun 5001 df-iin 5002 df-br 5152 df-opab 5214 df-mpt 5235 df-tr 5269 df-id 5587 df-eprel 5593 df-po 5601 df-so 5602 df-fr 5645 df-se 5646 df-we 5647 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-pred 6329 df-ord 6395 df-on 6396 df-lim 6397 df-suc 6398 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-isom 6578 df-riota 7395 df-ov 7441 df-oprab 7442 df-mpo 7443 df-of 7704 df-om 7895 df-1st 8022 df-2nd 8023 df-supp 8194 df-frecs 8314 df-wrecs 8345 df-recs 8419 df-rdg 8458 df-1o 8514 df-2o 8515 df-er 8753 df-map 8876 df-pm 8877 df-ixp 8946 df-en 8994 df-dom 8995 df-sdom 8996 df-fin 8997 df-fsupp 9409 df-fi 9458 df-sup 9489 df-inf 9490 df-oi 9557 df-card 9986 df-pnf 11304 df-mnf 11305 df-xr 11306 df-ltxr 11307 df-le 11308 df-sub 11501 df-neg 11502 df-div 11928 df-nn 12274 df-2 12336 df-3 12337 df-4 12338 df-5 12339 df-6 12340 df-7 12341 df-8 12342 df-9 12343 df-n0 12534 df-z 12621 df-dec 12741 df-uz 12886 df-q 12998 df-rp 13042 df-xneg 13161 df-xadd 13162 df-xmul 13163 df-ioo 13397 df-ico 13399 df-icc 13400 df-fz 13554 df-fzo 13701 df-fl 13838 df-seq 14049 df-exp 14109 df-hash 14376 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-clim 15530 df-rlim 15531 df-sum 15729 df-struct 17190 df-sets 17207 df-slot 17225 df-ndx 17237 df-base 17255 df-ress 17284 df-plusg 17320 df-mulr 17321 df-starv 17322 df-sca 17323 df-vsca 17324 df-ip 17325 df-tset 17326 df-ple 17327 df-ds 17329 df-unif 17330 df-hom 17331 df-cco 17332 df-rest 17478 df-topn 17479 df-0g 17497 df-gsum 17498 df-topgen 17499 df-pt 17500 df-prds 17503 df-xrs 17558 df-qtop 17563 df-imas 17564 df-xps 17566 df-mre 17640 df-mrc 17641 df-acs 17643 df-mgm 18675 df-sgrp 18754 df-mnd 18770 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21383 df-xmet 21384 df-met 21385 df-bl 21386 df-mopn 21387 df-cnfld 21392 df-top 22925 df-topon 22942 df-topsp 22964 df-bases 22978 df-cld 23052 df-cn 23260 df-cnp 23261 df-cmp 23420 df-tx 23595 df-hmeo 23788 df-xms 24355 df-ms 24356 df-tms 24357

This theorem is referenced by: stoweidlem59 46043

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