Theorem stoweidlem18 43449

Description: This theorem proves Lemma 2 in [BrosowskiDeutsh] p. 92 when A is empty, the trivial case. Here D is used to denote the set A of Lemma 2, because the variable A is used for the subalgebra. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem18.1	⊢ Ⅎ𝑡𝐷
stoweidlem18.2	⊢ Ⅎ𝑡𝜑
stoweidlem18.3	⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ 1)
stoweidlem18.4	⊢ 𝑇 = ∪ 𝐽
stoweidlem18.5	⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴)
stoweidlem18.6	⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽))
stoweidlem18.7	⊢ (𝜑 → 𝐸 ∈ ℝ+)
stoweidlem18.8	⊢ (𝜑 → 𝐷 = ∅)

Assertion

Ref	Expression
stoweidlem18	⊢ (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:   𝑡,𝑎,𝑇   𝐴,𝑎   𝜑,𝑎   𝑥,𝑡   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝐸   𝑥,𝐹   𝑥,𝑇

Allowed substitution hints:   𝜑(𝑥,𝑡)   𝐴(𝑡)   𝐵(𝑡,𝑎)   𝐷(𝑡,𝑎)   𝐸(𝑡,𝑎)   𝐹(𝑡,𝑎)   𝐽(𝑥,𝑡,𝑎)

Proof of Theorem stoweidlem18

Step	Hyp	Ref	Expression
1		stoweidlem18.3	. . 3 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ 1)
2		1re 10906	. . . 4 ⊢ 1 ∈ ℝ
3		stoweidlem18.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴)
4	3	stoweidlem4 43435	. . . 4 ⊢ ((𝜑 ∧ 1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
5	2, 4	mpan2 687	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
6	1, 5	eqeltrid 2843	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
7		stoweidlem18.2	. . 3 ⊢ Ⅎ𝑡𝜑
8		0le1 11428	. . . . . 6 ⊢ 0 ≤ 1
9		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
10	1	fvmpt2 6868	. . . . . . 7 ⊢ ((𝑡 ∈ 𝑇 ∧ 1 ∈ ℝ) → (𝐹‘𝑡) = 1)
11	9, 2, 10	sylancl 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) = 1)
12	8, 11	breqtrrid 5108	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐹‘𝑡))
13		1le1 11533	. . . . . 6 ⊢ 1 ≤ 1
14	11, 13	eqbrtrdi 5109	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ≤ 1)
15	12, 14	jca 511	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1))
16	15	ex 412	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1)))
17	7, 16	ralrimi 3139	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1))
18		stoweidlem18.8	. . 3 ⊢ (𝜑 → 𝐷 = ∅)
19		stoweidlem18.1	. . . . 5 ⊢ Ⅎ𝑡𝐷
20		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑡∅
21	19, 20	nfeq 2919	. . . 4 ⊢ Ⅎ𝑡 𝐷 = ∅
22	21	rzalf 42449	. . 3 ⊢ (𝐷 = ∅ → ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸)
23	18, 22	syl 17	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸)
24		1red 10907	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
25		stoweidlem18.7	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
26	24, 25	ltsubrpd 12733	. . . . . 6 ⊢ (𝜑 → (1 − 𝐸) < 1)
27	26	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < 1)
28		stoweidlem18.6	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽))
29		stoweidlem18.4	. . . . . . . . 9 ⊢ 𝑇 = ∪ 𝐽
30	29	cldss 22088	. . . . . . . 8 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ 𝑇)
31	28, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝑇)
32	31	sselda 3917	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑡 ∈ 𝑇)
33	32, 2, 10	sylancl 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝐹‘𝑡) = 1)
34	27, 33	breqtrrd 5098	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (𝐹‘𝑡))
35	34	ex 412	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝐵 → (1 − 𝐸) < (𝐹‘𝑡)))
36	7, 35	ralrimi 3139	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝐹‘𝑡))
37		nfcv 2906	. . . . . 6 ⊢ Ⅎ𝑡𝑥
38		nfmpt1 5178	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ 1)
39	1, 38	nfcxfr 2904	. . . . . 6 ⊢ Ⅎ𝑡𝐹
40	37, 39	nfeq 2919	. . . . 5 ⊢ Ⅎ𝑡 𝑥 = 𝐹
41		fveq1 6755	. . . . . . 7 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑡) = (𝐹‘𝑡))
42	41	breq2d 5082	. . . . . 6 ⊢ (𝑥 = 𝐹 → (0 ≤ (𝑥‘𝑡) ↔ 0 ≤ (𝐹‘𝑡)))
43	41	breq1d 5080	. . . . . 6 ⊢ (𝑥 = 𝐹 → ((𝑥‘𝑡) ≤ 1 ↔ (𝐹‘𝑡) ≤ 1))
44	42, 43	anbi12d 630	. . . . 5 ⊢ (𝑥 = 𝐹 → ((0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1)))
45	40, 44	ralbid 3158	. . . 4 ⊢ (𝑥 = 𝐹 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1)))
46	41	breq1d 5080	. . . . 5 ⊢ (𝑥 = 𝐹 → ((𝑥‘𝑡) < 𝐸 ↔ (𝐹‘𝑡) < 𝐸))
47	40, 46	ralbid 3158	. . . 4 ⊢ (𝑥 = 𝐹 → (∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ↔ ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸))
48	41	breq2d 5082	. . . . 5 ⊢ (𝑥 = 𝐹 → ((1 − 𝐸) < (𝑥‘𝑡) ↔ (1 − 𝐸) < (𝐹‘𝑡)))
49	40, 48	ralbid 3158	. . . 4 ⊢ (𝑥 = 𝐹 → (∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡) ↔ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝐹‘𝑡)))
50	45, 47, 49	3anbi123d 1434	. . 3 ⊢ (𝑥 = 𝐹 → ((∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡)) ↔ (∀𝑡 ∈ 𝑇 (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝐹‘𝑡))))
51	50	rspcev 3552	. 2 ⊢ ((𝐹 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝐹‘𝑡))) → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡)))
52	6, 17, 23, 36, 51	syl13anc 1370	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539  Ⅎwnf 1787   ∈ wcel 2108  Ⅎwnfc 2886  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836   class class class wbr 5070   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255  ℝcr 10801  0cc0 10802  1c1 10803   < clt 10940   ≤ cle 10941   − cmin 11135  ℝ⁺crp 12659  Clsdccld 22075

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-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  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

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-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  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-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-rp 12660  df-top 21951  df-cld 22078

This theorem is referenced by:  stoweidlem58  43489