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Theorem stoweidlem18 40553
 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 10077 . . . 4 1 ∈ ℝ
3 stoweidlem18.5 . . . . 5 ((𝜑𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)
43stoweidlem4 40539 . . . 4 ((𝜑 ∧ 1 ∈ ℝ) → (𝑡𝑇 ↦ 1) ∈ 𝐴)
52, 4mpan2 707 . . 3 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
61, 5syl5eqel 2734 . 2 (𝜑𝐹𝐴)
7 stoweidlem18.2 . . 3 𝑡𝜑
8 0le1 10589 . . . . . 6 0 ≤ 1
9 simpr 476 . . . . . . 7 ((𝜑𝑡𝑇) → 𝑡𝑇)
101fvmpt2 6330 . . . . . . 7 ((𝑡𝑇 ∧ 1 ∈ ℝ) → (𝐹𝑡) = 1)
119, 2, 10sylancl 695 . . . . . 6 ((𝜑𝑡𝑇) → (𝐹𝑡) = 1)
128, 11syl5breqr 4723 . . . . 5 ((𝜑𝑡𝑇) → 0 ≤ (𝐹𝑡))
13 1le1 10693 . . . . . 6 1 ≤ 1
1411, 13syl6eqbr 4724 . . . . 5 ((𝜑𝑡𝑇) → (𝐹𝑡) ≤ 1)
1512, 14jca 553 . . . 4 ((𝜑𝑡𝑇) → (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1))
1615ex 449 . . 3 (𝜑 → (𝑡𝑇 → (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1)))
177, 16ralrimi 2986 . 2 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1))
18 stoweidlem18.8 . . 3 (𝜑𝐷 = ∅)
19 stoweidlem18.1 . . . . 5 𝑡𝐷
20 nfcv 2793 . . . . 5 𝑡
2119, 20nfeq 2805 . . . 4 𝑡 𝐷 = ∅
2221rzalf 39490 . . 3 (𝐷 = ∅ → ∀𝑡𝐷 (𝐹𝑡) < 𝐸)
2318, 22syl 17 . 2 (𝜑 → ∀𝑡𝐷 (𝐹𝑡) < 𝐸)
24 1red 10093 . . . . . . 7 (𝜑 → 1 ∈ ℝ)
25 stoweidlem18.7 . . . . . . 7 (𝜑𝐸 ∈ ℝ+)
2624, 25ltsubrpd 11942 . . . . . 6 (𝜑 → (1 − 𝐸) < 1)
2726adantr 480 . . . . 5 ((𝜑𝑡𝐵) → (1 − 𝐸) < 1)
28 stoweidlem18.6 . . . . . . . 8 (𝜑𝐵 ∈ (Clsd‘𝐽))
29 stoweidlem18.4 . . . . . . . . 9 𝑇 = 𝐽
3029cldss 20881 . . . . . . . 8 (𝐵 ∈ (Clsd‘𝐽) → 𝐵𝑇)
3128, 30syl 17 . . . . . . 7 (𝜑𝐵𝑇)
3231sselda 3636 . . . . . 6 ((𝜑𝑡𝐵) → 𝑡𝑇)
3332, 2, 10sylancl 695 . . . . 5 ((𝜑𝑡𝐵) → (𝐹𝑡) = 1)
3427, 33breqtrrd 4713 . . . 4 ((𝜑𝑡𝐵) → (1 − 𝐸) < (𝐹𝑡))
3534ex 449 . . 3 (𝜑 → (𝑡𝐵 → (1 − 𝐸) < (𝐹𝑡)))
367, 35ralrimi 2986 . 2 (𝜑 → ∀𝑡𝐵 (1 − 𝐸) < (𝐹𝑡))
37 nfcv 2793 . . . . . 6 𝑡𝑥
38 nfmpt1 4780 . . . . . . 7 𝑡(𝑡𝑇 ↦ 1)
391, 38nfcxfr 2791 . . . . . 6 𝑡𝐹
4037, 39nfeq 2805 . . . . 5 𝑡 𝑥 = 𝐹
41 fveq1 6228 . . . . . . 7 (𝑥 = 𝐹 → (𝑥𝑡) = (𝐹𝑡))
4241breq2d 4697 . . . . . 6 (𝑥 = 𝐹 → (0 ≤ (𝑥𝑡) ↔ 0 ≤ (𝐹𝑡)))
4341breq1d 4695 . . . . . 6 (𝑥 = 𝐹 → ((𝑥𝑡) ≤ 1 ↔ (𝐹𝑡) ≤ 1))
4442, 43anbi12d 747 . . . . 5 (𝑥 = 𝐹 → ((0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1)))
4540, 44ralbid 3012 . . . 4 (𝑥 = 𝐹 → (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1)))
4641breq1d 4695 . . . . 5 (𝑥 = 𝐹 → ((𝑥𝑡) < 𝐸 ↔ (𝐹𝑡) < 𝐸))
4740, 46ralbid 3012 . . . 4 (𝑥 = 𝐹 → (∀𝑡𝐷 (𝑥𝑡) < 𝐸 ↔ ∀𝑡𝐷 (𝐹𝑡) < 𝐸))
4841breq2d 4697 . . . . 5 (𝑥 = 𝐹 → ((1 − 𝐸) < (𝑥𝑡) ↔ (1 − 𝐸) < (𝐹𝑡)))
4940, 48ralbid 3012 . . . 4 (𝑥 = 𝐹 → (∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡) ↔ ∀𝑡𝐵 (1 − 𝐸) < (𝐹𝑡)))
5045, 47, 493anbi123d 1439 . . 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 3340 . 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 1368 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 383   ∧ w3a 1054   = wceq 1523  Ⅎwnf 1748   ∈ wcel 2030  Ⅎwnfc 2780  ∀wral 2941  ∃wrex 2942   ⊆ wss 3607  ∅c0 3948  ∪ cuni 4468   class class class wbr 4685   ↦ cmpt 4762  ‘cfv 5926  (class class class)co 6690  ℝcr 9973  0cc0 9974  1c1 9975   < clt 10112   ≤ cle 10113   − cmin 10304  ℝ+crp 11870  Clsdccld 20868 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-rp 11871  df-top 20747  df-cld 20871 This theorem is referenced by:  stoweidlem58  40593
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