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Theorem stoweidlem18 45974
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 11259 . . . 4 1 ∈ ℝ
3 stoweidlem18.5 . . . . 5 ((𝜑𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)
43stoweidlem4 45960 . . . 4 ((𝜑 ∧ 1 ∈ ℝ) → (𝑡𝑇 ↦ 1) ∈ 𝐴)
52, 4mpan2 691 . . 3 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
61, 5eqeltrid 2843 . 2 (𝜑𝐹𝐴)
7 stoweidlem18.2 . . 3 𝑡𝜑
8 0le1 11784 . . . . . 6 0 ≤ 1
9 simpr 484 . . . . . . 7 ((𝜑𝑡𝑇) → 𝑡𝑇)
101fvmpt2 7027 . . . . . . 7 ((𝑡𝑇 ∧ 1 ∈ ℝ) → (𝐹𝑡) = 1)
119, 2, 10sylancl 586 . . . . . 6 ((𝜑𝑡𝑇) → (𝐹𝑡) = 1)
128, 11breqtrrid 5186 . . . . 5 ((𝜑𝑡𝑇) → 0 ≤ (𝐹𝑡))
13 1le1 11889 . . . . . 6 1 ≤ 1
1411, 13eqbrtrdi 5187 . . . . 5 ((𝜑𝑡𝑇) → (𝐹𝑡) ≤ 1)
1512, 14jca 511 . . . 4 ((𝜑𝑡𝑇) → (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1))
1615ex 412 . . 3 (𝜑 → (𝑡𝑇 → (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1)))
177, 16ralrimi 3255 . 2 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1))
18 stoweidlem18.8 . . 3 (𝜑𝐷 = ∅)
19 stoweidlem18.1 . . . . 5 𝑡𝐷
20 nfcv 2903 . . . . 5 𝑡
2119, 20nfeq 2917 . . . 4 𝑡 𝐷 = ∅
2221rzalf 44955 . . 3 (𝐷 = ∅ → ∀𝑡𝐷 (𝐹𝑡) < 𝐸)
2318, 22syl 17 . 2 (𝜑 → ∀𝑡𝐷 (𝐹𝑡) < 𝐸)
24 1red 11260 . . . . . . 7 (𝜑 → 1 ∈ ℝ)
25 stoweidlem18.7 . . . . . . 7 (𝜑𝐸 ∈ ℝ+)
2624, 25ltsubrpd 13107 . . . . . 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 23053 . . . . . . . 8 (𝐵 ∈ (Clsd‘𝐽) → 𝐵𝑇)
3128, 30syl 17 . . . . . . 7 (𝜑𝐵𝑇)
3231sselda 3995 . . . . . 6 ((𝜑𝑡𝐵) → 𝑡𝑇)
3332, 2, 10sylancl 586 . . . . 5 ((𝜑𝑡𝐵) → (𝐹𝑡) = 1)
3427, 33breqtrrd 5176 . . . 4 ((𝜑𝑡𝐵) → (1 − 𝐸) < (𝐹𝑡))
3534ex 412 . . 3 (𝜑 → (𝑡𝐵 → (1 − 𝐸) < (𝐹𝑡)))
367, 35ralrimi 3255 . 2 (𝜑 → ∀𝑡𝐵 (1 − 𝐸) < (𝐹𝑡))
37 nfcv 2903 . . . . . 6 𝑡𝑥
38 nfmpt1 5256 . . . . . . 7 𝑡(𝑡𝑇 ↦ 1)
391, 38nfcxfr 2901 . . . . . 6 𝑡𝐹
4037, 39nfeq 2917 . . . . 5 𝑡 𝑥 = 𝐹
41 fveq1 6906 . . . . . . 7 (𝑥 = 𝐹 → (𝑥𝑡) = (𝐹𝑡))
4241breq2d 5160 . . . . . 6 (𝑥 = 𝐹 → (0 ≤ (𝑥𝑡) ↔ 0 ≤ (𝐹𝑡)))
4341breq1d 5158 . . . . . 6 (𝑥 = 𝐹 → ((𝑥𝑡) ≤ 1 ↔ (𝐹𝑡) ≤ 1))
4442, 43anbi12d 632 . . . . 5 (𝑥 = 𝐹 → ((0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1)))
4540, 44ralbid 3271 . . . 4 (𝑥 = 𝐹 → (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1)))
4641breq1d 5158 . . . . 5 (𝑥 = 𝐹 → ((𝑥𝑡) < 𝐸 ↔ (𝐹𝑡) < 𝐸))
4740, 46ralbid 3271 . . . 4 (𝑥 = 𝐹 → (∀𝑡𝐷 (𝑥𝑡) < 𝐸 ↔ ∀𝑡𝐷 (𝐹𝑡) < 𝐸))
4841breq2d 5160 . . . . 5 (𝑥 = 𝐹 → ((1 − 𝐸) < (𝑥𝑡) ↔ (1 − 𝐸) < (𝐹𝑡)))
4940, 48ralbid 3271 . . . 4 (𝑥 = 𝐹 → (∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡) ↔ ∀𝑡𝐵 (1 − 𝐸) < (𝐹𝑡)))
5045, 47, 493anbi123d 1435 . . 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 3622 . 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 1371 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 395  w3a 1086   = wceq 1537  wnf 1780  wcel 2106  wnfc 2888  wral 3059  wrex 3068  wss 3963  c0 4339   cuni 4912   class class class wbr 5148  cmpt 5231  cfv 6563  (class class class)co 7431  cr 11152  0cc0 11153  1c1 11154   < clt 11293  cle 11294  cmin 11490  +crp 13032  Clsdccld 23040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-rp 13033  df-top 22916  df-cld 23043
This theorem is referenced by:  stoweidlem58  46014
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