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Theorem stoweidlem18 42188
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 𝑡𝐷
stoweidlem18.2 𝑡𝜑
stoweidlem18.3 𝐹 = (𝑡𝑇 ↦ 1)
stoweidlem18.4 𝑇 = 𝐽
stoweidlem18.5 ((𝜑𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)
stoweidlem18.6 (𝜑𝐵 ∈ (Clsd‘𝐽))
stoweidlem18.7 (𝜑𝐸 ∈ ℝ+)
stoweidlem18.8 (𝜑𝐷 = ∅)
Assertion
Ref Expression
stoweidlem18 (𝜑 → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡)))
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
StepHypRef Expression
1 stoweidlem18.3 . . 3 𝐹 = (𝑡𝑇 ↦ 1)
2 1re 10635 . . . 4 1 ∈ ℝ
3 stoweidlem18.5 . . . . 5 ((𝜑𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)
43stoweidlem4 42174 . . . 4 ((𝜑 ∧ 1 ∈ ℝ) → (𝑡𝑇 ↦ 1) ∈ 𝐴)
52, 4mpan2 687 . . 3 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
61, 5eqeltrid 2922 . 2 (𝜑𝐹𝐴)
7 stoweidlem18.2 . . 3 𝑡𝜑
8 0le1 11157 . . . . . 6 0 ≤ 1
9 simpr 485 . . . . . . 7 ((𝜑𝑡𝑇) → 𝑡𝑇)
101fvmpt2 6777 . . . . . . 7 ((𝑡𝑇 ∧ 1 ∈ ℝ) → (𝐹𝑡) = 1)
119, 2, 10sylancl 586 . . . . . 6 ((𝜑𝑡𝑇) → (𝐹𝑡) = 1)
128, 11breqtrrid 5101 . . . . 5 ((𝜑𝑡𝑇) → 0 ≤ (𝐹𝑡))
13 1le1 11262 . . . . . 6 1 ≤ 1
1411, 13eqbrtrdi 5102 . . . . 5 ((𝜑𝑡𝑇) → (𝐹𝑡) ≤ 1)
1512, 14jca 512 . . . 4 ((𝜑𝑡𝑇) → (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1))
1615ex 413 . . 3 (𝜑 → (𝑡𝑇 → (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1)))
177, 16ralrimi 3221 . 2 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1))
18 stoweidlem18.8 . . 3 (𝜑𝐷 = ∅)
19 stoweidlem18.1 . . . . 5 𝑡𝐷
20 nfcv 2982 . . . . 5 𝑡
2119, 20nfeq 2996 . . . 4 𝑡 𝐷 = ∅
2221rzalf 41158 . . 3 (𝐷 = ∅ → ∀𝑡𝐷 (𝐹𝑡) < 𝐸)
2318, 22syl 17 . 2 (𝜑 → ∀𝑡𝐷 (𝐹𝑡) < 𝐸)
24 1red 10636 . . . . . . 7 (𝜑 → 1 ∈ ℝ)
25 stoweidlem18.7 . . . . . . 7 (𝜑𝐸 ∈ ℝ+)
2624, 25ltsubrpd 12458 . . . . . 6 (𝜑 → (1 − 𝐸) < 1)
2726adantr 481 . . . . 5 ((𝜑𝑡𝐵) → (1 − 𝐸) < 1)
28 stoweidlem18.6 . . . . . . . 8 (𝜑𝐵 ∈ (Clsd‘𝐽))
29 stoweidlem18.4 . . . . . . . . 9 𝑇 = 𝐽
3029cldss 21572 . . . . . . . 8 (𝐵 ∈ (Clsd‘𝐽) → 𝐵𝑇)
3128, 30syl 17 . . . . . . 7 (𝜑𝐵𝑇)
3231sselda 3971 . . . . . 6 ((𝜑𝑡𝐵) → 𝑡𝑇)
3332, 2, 10sylancl 586 . . . . 5 ((𝜑𝑡𝐵) → (𝐹𝑡) = 1)
3427, 33breqtrrd 5091 . . . 4 ((𝜑𝑡𝐵) → (1 − 𝐸) < (𝐹𝑡))
3534ex 413 . . 3 (𝜑 → (𝑡𝐵 → (1 − 𝐸) < (𝐹𝑡)))
367, 35ralrimi 3221 . 2 (𝜑 → ∀𝑡𝐵 (1 − 𝐸) < (𝐹𝑡))
37 nfcv 2982 . . . . . 6 𝑡𝑥
38 nfmpt1 5161 . . . . . . 7 𝑡(𝑡𝑇 ↦ 1)
391, 38nfcxfr 2980 . . . . . 6 𝑡𝐹
4037, 39nfeq 2996 . . . . 5 𝑡 𝑥 = 𝐹
41 fveq1 6668 . . . . . . 7 (𝑥 = 𝐹 → (𝑥𝑡) = (𝐹𝑡))
4241breq2d 5075 . . . . . 6 (𝑥 = 𝐹 → (0 ≤ (𝑥𝑡) ↔ 0 ≤ (𝐹𝑡)))
4341breq1d 5073 . . . . . 6 (𝑥 = 𝐹 → ((𝑥𝑡) ≤ 1 ↔ (𝐹𝑡) ≤ 1))
4442, 43anbi12d 630 . . . . 5 (𝑥 = 𝐹 → ((0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1)))
4540, 44ralbid 3236 . . . 4 (𝑥 = 𝐹 → (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1)))
4641breq1d 5073 . . . . 5 (𝑥 = 𝐹 → ((𝑥𝑡) < 𝐸 ↔ (𝐹𝑡) < 𝐸))
4740, 46ralbid 3236 . . . 4 (𝑥 = 𝐹 → (∀𝑡𝐷 (𝑥𝑡) < 𝐸 ↔ ∀𝑡𝐷 (𝐹𝑡) < 𝐸))
4841breq2d 5075 . . . . 5 (𝑥 = 𝐹 → ((1 − 𝐸) < (𝑥𝑡) ↔ (1 − 𝐸) < (𝐹𝑡)))
4940, 48ralbid 3236 . . . 4 (𝑥 = 𝐹 → (∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡) ↔ ∀𝑡𝐵 (1 − 𝐸) < (𝐹𝑡)))
5045, 47, 493anbi123d 1429 . . 3 (𝑥 = 𝐹 → ((∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡)) ↔ (∀𝑡𝑇 (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝐹𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝐹𝑡))))
5150rspcev 3627 . 2 ((𝐹𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝐹𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝐹𝑡))) → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡)))
526, 17, 23, 36, 51syl13anc 1366 1 (𝜑 → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wnf 1777  wcel 2107  wnfc 2966  wral 3143  wrex 3144  wss 3940  c0 4295   cuni 4837   class class class wbr 5063  cmpt 5143  cfv 6354  (class class class)co 7150  cr 10530  0cc0 10531  1c1 10532   < clt 10669  cle 10670  cmin 10864  +crp 12384  Clsdccld 21559
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-po 5473  df-so 5474  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-rp 12385  df-top 21437  df-cld 21562
This theorem is referenced by:  stoweidlem58  42228
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