Theorem stoweidlem18 38911

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 9918	. . . 4 ⊢ 1 ∈ ℝ
3		stoweidlem18.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴)
4	3	stoweidlem4 38897	. . . 4 ⊢ ((𝜑 ∧ 1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
5	2, 4	mpan2 703	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
6	1, 5	syl5eqel 2692	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
7		stoweidlem18.2	. . 3 ⊢ Ⅎ𝑡𝜑
8		0le1 10430	. . . . . 6 ⊢ 0 ≤ 1
9		simpr 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
10	1	fvmpt2 6200	. . . . . . 7 ⊢ ((𝑡 ∈ 𝑇 ∧ 1 ∈ ℝ) → (𝐹‘𝑡) = 1)
11	9, 2, 10	sylancl 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) = 1)
12	8, 11	syl5breqr 4621	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐹‘𝑡))
13		1le1 10534	. . . . . 6 ⊢ 1 ≤ 1
14	11, 13	syl6eqbr 4622	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ≤ 1)
15	12, 14	jca 553	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1))
16	15	ex 449	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1)))
17	7, 16	ralrimi 2940	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1))
18		stoweidlem18.8	. . 3 ⊢ (𝜑 → 𝐷 = ∅)
19		stoweidlem18.1	. . . . 5 ⊢ Ⅎ𝑡𝐷
20		nfcv 2751	. . . . 5 ⊢ Ⅎ𝑡∅
21	19, 20	nfeq 2762	. . . 4 ⊢ Ⅎ𝑡 𝐷 = ∅
22	21	rzalf 38199	. . 3 ⊢ (𝐷 = ∅ → ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸)
23	18, 22	syl 17	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸)
24		1red 9934	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
25		stoweidlem18.7	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
26	24, 25	ltsubrpd 11780	. . . . . 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 20643	. . . . . . . 8 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ 𝑇)
31	28, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝑇)
32	31	sselda 3568	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑡 ∈ 𝑇)
33	32, 2, 10	sylancl 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝐹‘𝑡) = 1)
34	27, 33	breqtrrd 4611	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (𝐹‘𝑡))
35	34	ex 449	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝐵 → (1 − 𝐸) < (𝐹‘𝑡)))
36	7, 35	ralrimi 2940	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝐹‘𝑡))
37		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑡𝑥
38		nfmpt1 4675	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ 1)
39	1, 38	nfcxfr 2749	. . . . . 6 ⊢ Ⅎ𝑡𝐹
40	37, 39	nfeq 2762	. . . . 5 ⊢ Ⅎ𝑡 𝑥 = 𝐹
41		fveq1 6102	. . . . . . 7 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑡) = (𝐹‘𝑡))
42	41	breq2d 4595	. . . . . 6 ⊢ (𝑥 = 𝐹 → (0 ≤ (𝑥‘𝑡) ↔ 0 ≤ (𝐹‘𝑡)))
43	41	breq1d 4593	. . . . . 6 ⊢ (𝑥 = 𝐹 → ((𝑥‘𝑡) ≤ 1 ↔ (𝐹‘𝑡) ≤ 1))
44	42, 43	anbi12d 743	. . . . 5 ⊢ (𝑥 = 𝐹 → ((0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1)))
45	40, 44	ralbid 2966	. . . 4 ⊢ (𝑥 = 𝐹 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1)))
46	41	breq1d 4593	. . . . 5 ⊢ (𝑥 = 𝐹 → ((𝑥‘𝑡) < 𝐸 ↔ (𝐹‘𝑡) < 𝐸))
47	40, 46	ralbid 2966	. . . 4 ⊢ (𝑥 = 𝐹 → (∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ↔ ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸))
48	41	breq2d 4595	. . . . 5 ⊢ (𝑥 = 𝐹 → ((1 − 𝐸) < (𝑥‘𝑡) ↔ (1 − 𝐸) < (𝐹‘𝑡)))
49	40, 48	ralbid 2966	. . . 4 ⊢ (𝑥 = 𝐹 → (∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡) ↔ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝐹‘𝑡)))
50	45, 47, 49	3anbi123d 1391	. . 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 3282	. 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 1320	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 383   ∧ w3a 1031   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   < clt 9953   ≤ cle 9954   − cmin 10145  ℝ⁺crp 11708  Clsdccld 20630

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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  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

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  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-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-rp 11709  df-top 20521  df-cld 20633

This theorem is referenced by:  stoweidlem58  38951