Theorem stoweidlem18 46329

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 11136	. . . 4 ⊢ 1 ∈ ℝ
3		stoweidlem18.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴)
4	3	stoweidlem4 46315	. . . 4 ⊢ ((𝜑 ∧ 1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
5	2, 4	mpan2 692	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
6	1, 5	eqeltrid 2841	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
7		stoweidlem18.2	. . 3 ⊢ Ⅎ𝑡𝜑
8		0le1 11664	. . . . . 6 ⊢ 0 ≤ 1
9		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
10	1	fvmpt2 6954	. . . . . . 7 ⊢ ((𝑡 ∈ 𝑇 ∧ 1 ∈ ℝ) → (𝐹‘𝑡) = 1)
11	9, 2, 10	sylancl 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) = 1)
12	8, 11	breqtrrid 5137	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐹‘𝑡))
13		1le1 11769	. . . . . 6 ⊢ 1 ≤ 1
14	11, 13	eqbrtrdi 5138	. . . . 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 3235	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1))
18		stoweidlem18.8	. . 3 ⊢ (𝜑 → 𝐷 = ∅)
19		stoweidlem18.1	. . . . 5 ⊢ Ⅎ𝑡𝐷
20		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑡∅
21	19, 20	nfeq 2913	. . . 4 ⊢ Ⅎ𝑡 𝐷 = ∅
22	21	rzalf 45329	. . 3 ⊢ (𝐷 = ∅ → ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸)
23	18, 22	syl 17	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸)
24		1red 11137	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
25		stoweidlem18.7	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
26	24, 25	ltsubrpd 12985	. . . . . 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 22977	. . . . . . . 8 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ 𝑇)
31	28, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝑇)
32	31	sselda 3934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑡 ∈ 𝑇)
33	32, 2, 10	sylancl 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝐹‘𝑡) = 1)
34	27, 33	breqtrrd 5127	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (𝐹‘𝑡))
35	34	ex 412	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝐵 → (1 − 𝐸) < (𝐹‘𝑡)))
36	7, 35	ralrimi 3235	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝐹‘𝑡))
37		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑡𝑥
38		nfmpt1 5198	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ 1)
39	1, 38	nfcxfr 2897	. . . . . 6 ⊢ Ⅎ𝑡𝐹
40	37, 39	nfeq 2913	. . . . 5 ⊢ Ⅎ𝑡 𝑥 = 𝐹
41		fveq1 6834	. . . . . . 7 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑡) = (𝐹‘𝑡))
42	41	breq2d 5111	. . . . . 6 ⊢ (𝑥 = 𝐹 → (0 ≤ (𝑥‘𝑡) ↔ 0 ≤ (𝐹‘𝑡)))
43	41	breq1d 5109	. . . . . 6 ⊢ (𝑥 = 𝐹 → ((𝑥‘𝑡) ≤ 1 ↔ (𝐹‘𝑡) ≤ 1))
44	42, 43	anbi12d 633	. . . . 5 ⊢ (𝑥 = 𝐹 → ((0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1)))
45	40, 44	ralbid 3250	. . . 4 ⊢ (𝑥 = 𝐹 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ 1)))
46	41	breq1d 5109	. . . . 5 ⊢ (𝑥 = 𝐹 → ((𝑥‘𝑡) < 𝐸 ↔ (𝐹‘𝑡) < 𝐸))
47	40, 46	ralbid 3250	. . . 4 ⊢ (𝑥 = 𝐹 → (∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ↔ ∀𝑡 ∈ 𝐷 (𝐹‘𝑡) < 𝐸))
48	41	breq2d 5111	. . . . 5 ⊢ (𝑥 = 𝐹 → ((1 − 𝐸) < (𝑥‘𝑡) ↔ (1 − 𝐸) < (𝐹‘𝑡)))
49	40, 48	ralbid 3250	. . . 4 ⊢ (𝑥 = 𝐹 → (∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡) ↔ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝐹‘𝑡)))
50	45, 47, 49	3anbi123d 1439	. . 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 3577	. 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 1375	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 1087   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2884  ∀wral 3052  ∃wrex 3061   ⊆ wss 3902  ∅c0 4286  ∪ cuni 4864   class class class wbr 5099   ↦ cmpt 5180  ‘cfv 6493  (class class class)co 7360  ℝcr 11029  0cc0 11030  1c1 11031   < clt 11170   ≤ cle 11171   − cmin 11368  ℝ⁺crp 12909  Clsdccld 22964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-po 5533  df-so 5534  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-rp 12910  df-top 22842  df-cld 22967

This theorem is referenced by:  stoweidlem58  46369