Theorem stoweidlem18 45465

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 11239	. . . 4 ⊢ 1 ∈ ℝ
3		stoweidlem18.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴)
4	3	stoweidlem4 45451	. . . 4 ⊢ ((𝜑 ∧ 1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
5	2, 4	mpan2 689	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
6	1, 5	eqeltrid 2829	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
7		stoweidlem18.2	. . 3 ⊢ Ⅎ𝑡𝜑
8		0le1 11762	. . . . . 6 ⊢ 0 ≤ 1
9		simpr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
10	1	fvmpt2 7009	. . . . . . 7 ⊢ ((𝑡 ∈ 𝑇 ∧ 1 ∈ ℝ) → (𝐹‘𝑡) = 1)
11	9, 2, 10	sylancl 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) = 1)
12	8, 11	breqtrrid 5182	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐹‘𝑡))
13		1le1 11867	. . . . . 6 ⊢ 1 ≤ 1
14	11, 13	eqbrtrdi 5183	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ≤ 1)
15	12, 14	jca 510	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1))
16	15	ex 411	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1)))
17	7, 16	ralrimi 3245	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1))
18		stoweidlem18.8	. . 3 ⊢ (𝜑 → 𝐷 = ∅)
19		stoweidlem18.1	. . . . 5 ⊢ Ⅎ𝑡𝐷
20		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑡∅
21	19, 20	nfeq 2906	. . . 4 ⊢ Ⅎ𝑡 𝐷 = ∅
22	21	rzalf 44440	. . 3 ⊢ (𝐷 = ∅ → ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸)
23	18, 22	syl 17	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸)
24		1red 11240	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
25		stoweidlem18.7	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
26	24, 25	ltsubrpd 13075	. . . . . 6 ⊢ (𝜑 → (1 − 𝐸) < 1)
27	26	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < 1)
28		stoweidlem18.6	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽))
29		stoweidlem18.4	. . . . . . . . 9 ⊢ 𝑇 = ∪ 𝐽
30	29	cldss 22946	. . . . . . . 8 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ 𝑇)
31	28, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝑇)
32	31	sselda 3973	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑡 ∈ 𝑇)
33	32, 2, 10	sylancl 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝐹‘𝑡) = 1)
34	27, 33	breqtrrd 5172	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (𝐹‘𝑡))
35	34	ex 411	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝐵 → (1 − 𝐸) < (𝐹‘𝑡)))
36	7, 35	ralrimi 3245	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝐹‘𝑡))
37		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑡𝑥
38		nfmpt1 5252	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ 1)
39	1, 38	nfcxfr 2890	. . . . . 6 ⊢ Ⅎ𝑡𝐹
40	37, 39	nfeq 2906	. . . . 5 ⊢ Ⅎ𝑡 𝑥 = 𝐹
41		fveq1 6889	. . . . . . 7 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑡) = (𝐹‘𝑡))
42	41	breq2d 5156	. . . . . 6 ⊢ (𝑥 = 𝐹 → (0 ≤ (𝑥‘𝑡) ↔ 0 ≤ (𝐹‘𝑡)))
43	41	breq1d 5154	. . . . . 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 3261	. . . 4 ⊢ (𝑥 = 𝐹 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1)))
46	41	breq1d 5154	. . . . 5 ⊢ (𝑥 = 𝐹 → ((𝑥‘𝑡) < 𝐸 ↔ (𝐹‘𝑡) < 𝐸))
47	40, 46	ralbid 3261	. . . 4 ⊢ (𝑥 = 𝐹 → (∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ↔ ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸))
48	41	breq2d 5156	. . . . 5 ⊢ (𝑥 = 𝐹 → ((1 − 𝐸) < (𝑥‘𝑡) ↔ (1 − 𝐸) < (𝐹‘𝑡)))
49	40, 48	ralbid 3261	. . . 4 ⊢ (𝑥 = 𝐹 → (∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡) ↔ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝐹‘𝑡)))
50	45, 47, 49	3anbi123d 1432	. . 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 3603	. 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 1369	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 394   ∧ w3a 1084   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098  Ⅎwnfc 2875  ∀wral 3051  ∃wrex 3060   ⊆ wss 3941  ∅c0 4319  ∪ cuni 4904   class class class wbr 5144   ↦ cmpt 5227  ‘cfv 6543  (class class class)co 7413  ℝcr 11132  0cc0 11133  1c1 11134   < clt 11273   ≤ cle 11274   − cmin 11469  ℝ⁺crp 13001  Clsdccld 22933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-po 5585  df-so 5586  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-rp 13002  df-top 22809  df-cld 22936

This theorem is referenced by:  stoweidlem58  45505