Theorem stoweidlem18 46047

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 11235	. . . 4 ⊢ 1 ∈ ℝ
3		stoweidlem18.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴)
4	3	stoweidlem4 46033	. . . 4 ⊢ ((𝜑 ∧ 1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
5	2, 4	mpan2 691	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
6	1, 5	eqeltrid 2838	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
7		stoweidlem18.2	. . 3 ⊢ Ⅎ𝑡𝜑
8		0le1 11760	. . . . . 6 ⊢ 0 ≤ 1
9		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
10	1	fvmpt2 6997	. . . . . . 7 ⊢ ((𝑡 ∈ 𝑇 ∧ 1 ∈ ℝ) → (𝐹‘𝑡) = 1)
11	9, 2, 10	sylancl 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) = 1)
12	8, 11	breqtrrid 5157	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐹‘𝑡))
13		1le1 11865	. . . . . 6 ⊢ 1 ≤ 1
14	11, 13	eqbrtrdi 5158	. . . . 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 3240	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1))
18		stoweidlem18.8	. . 3 ⊢ (𝜑 → 𝐷 = ∅)
19		stoweidlem18.1	. . . . 5 ⊢ Ⅎ𝑡𝐷
20		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑡∅
21	19, 20	nfeq 2912	. . . 4 ⊢ Ⅎ𝑡 𝐷 = ∅
22	21	rzalf 45041	. . 3 ⊢ (𝐷 = ∅ → ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸)
23	18, 22	syl 17	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸)
24		1red 11236	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
25		stoweidlem18.7	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
26	24, 25	ltsubrpd 13083	. . . . . 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 22967	. . . . . . . 8 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ 𝑇)
31	28, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝑇)
32	31	sselda 3958	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑡 ∈ 𝑇)
33	32, 2, 10	sylancl 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝐹‘𝑡) = 1)
34	27, 33	breqtrrd 5147	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (𝐹‘𝑡))
35	34	ex 412	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝐵 → (1 − 𝐸) < (𝐹‘𝑡)))
36	7, 35	ralrimi 3240	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝐹‘𝑡))
37		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑡𝑥
38		nfmpt1 5220	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ 1)
39	1, 38	nfcxfr 2896	. . . . . 6 ⊢ Ⅎ𝑡𝐹
40	37, 39	nfeq 2912	. . . . 5 ⊢ Ⅎ𝑡 𝑥 = 𝐹
41		fveq1 6875	. . . . . . 7 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑡) = (𝐹‘𝑡))
42	41	breq2d 5131	. . . . . 6 ⊢ (𝑥 = 𝐹 → (0 ≤ (𝑥‘𝑡) ↔ 0 ≤ (𝐹‘𝑡)))
43	41	breq1d 5129	. . . . . 6 ⊢ (𝑥 = 𝐹 → ((𝑥‘𝑡) ≤ 1 ↔ (𝐹‘𝑡) ≤ 1))
44	42, 43	anbi12d 632	. . . . 5 ⊢ (𝑥 = 𝐹 → ((0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1)))
45	40, 44	ralbid 3255	. . . 4 ⊢ (𝑥 = 𝐹 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1)))
46	41	breq1d 5129	. . . . 5 ⊢ (𝑥 = 𝐹 → ((𝑥‘𝑡) < 𝐸 ↔ (𝐹‘𝑡) < 𝐸))
47	40, 46	ralbid 3255	. . . 4 ⊢ (𝑥 = 𝐹 → (∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ↔ ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸))
48	41	breq2d 5131	. . . . 5 ⊢ (𝑥 = 𝐹 → ((1 − 𝐸) < (𝑥‘𝑡) ↔ (1 − 𝐸) < (𝐹‘𝑡)))
49	40, 48	ralbid 3255	. . . 4 ⊢ (𝑥 = 𝐹 → (∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡) ↔ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝐹‘𝑡)))
50	45, 47, 49	3anbi123d 1438	. . 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 3601	. 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 1374	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 1086   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108  Ⅎwnfc 2883  ∀wral 3051  ∃wrex 3060   ⊆ wss 3926  ∅c0 4308  ∪ cuni 4883   class class class wbr 5119   ↦ cmpt 5201  ‘cfv 6531  (class class class)co 7405  ℝcr 11128  0cc0 11129  1c1 11130   < clt 11269   ≤ cle 11270   − cmin 11466  ℝ⁺crp 13008  Clsdccld 22954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  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 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-po 5561  df-so 5562  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-rp 13009  df-top 22832  df-cld 22957

This theorem is referenced by:  stoweidlem58  46087