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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  wnf 1748  wcel 2030  wnfc 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  cr 9973  0cc0 9974  1c1 9975   < clt 10112  cle 10113  cmin 10304  +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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