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Theorem stoweidlem18 45244
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 11212 . . . 4 1 ∈ ℝ
3 stoweidlem18.5 . . . . 5 ((𝜑𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)
43stoweidlem4 45230 . . . 4 ((𝜑 ∧ 1 ∈ ℝ) → (𝑡𝑇 ↦ 1) ∈ 𝐴)
52, 4mpan2 688 . . 3 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
61, 5eqeltrid 2829 . 2 (𝜑𝐹𝐴)
7 stoweidlem18.2 . . 3 𝑡𝜑
8 0le1 11735 . . . . . 6 0 ≤ 1
9 simpr 484 . . . . . . 7 ((𝜑𝑡𝑇) → 𝑡𝑇)
101fvmpt2 7000 . . . . . . 7 ((𝑡𝑇 ∧ 1 ∈ ℝ) → (𝐹𝑡) = 1)
119, 2, 10sylancl 585 . . . . . 6 ((𝜑𝑡𝑇) → (𝐹𝑡) = 1)
128, 11breqtrrid 5177 . . . . 5 ((𝜑𝑡𝑇) → 0 ≤ (𝐹𝑡))
13 1le1 11840 . . . . . 6 1 ≤ 1
1411, 13eqbrtrdi 5178 . . . . 5 ((𝜑𝑡𝑇) → (𝐹𝑡) ≤ 1)
1512, 14jca 511 . . . 4 ((𝜑𝑡𝑇) → (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1))
1615ex 412 . . 3 (𝜑 → (𝑡𝑇 → (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1)))
177, 16ralrimi 3246 . 2 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1))
18 stoweidlem18.8 . . 3 (𝜑𝐷 = ∅)
19 stoweidlem18.1 . . . . 5 𝑡𝐷
20 nfcv 2895 . . . . 5 𝑡
2119, 20nfeq 2908 . . . 4 𝑡 𝐷 = ∅
2221rzalf 44215 . . 3 (𝐷 = ∅ → ∀𝑡𝐷 (𝐹𝑡) < 𝐸)
2318, 22syl 17 . 2 (𝜑 → ∀𝑡𝐷 (𝐹𝑡) < 𝐸)
24 1red 11213 . . . . . . 7 (𝜑 → 1 ∈ ℝ)
25 stoweidlem18.7 . . . . . . 7 (𝜑𝐸 ∈ ℝ+)
2624, 25ltsubrpd 13046 . . . . . 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 22857 . . . . . . . 8 (𝐵 ∈ (Clsd‘𝐽) → 𝐵𝑇)
3128, 30syl 17 . . . . . . 7 (𝜑𝐵𝑇)
3231sselda 3975 . . . . . 6 ((𝜑𝑡𝐵) → 𝑡𝑇)
3332, 2, 10sylancl 585 . . . . 5 ((𝜑𝑡𝐵) → (𝐹𝑡) = 1)
3427, 33breqtrrd 5167 . . . 4 ((𝜑𝑡𝐵) → (1 − 𝐸) < (𝐹𝑡))
3534ex 412 . . 3 (𝜑 → (𝑡𝐵 → (1 − 𝐸) < (𝐹𝑡)))
367, 35ralrimi 3246 . 2 (𝜑 → ∀𝑡𝐵 (1 − 𝐸) < (𝐹𝑡))
37 nfcv 2895 . . . . . 6 𝑡𝑥
38 nfmpt1 5247 . . . . . . 7 𝑡(𝑡𝑇 ↦ 1)
391, 38nfcxfr 2893 . . . . . 6 𝑡𝐹
4037, 39nfeq 2908 . . . . 5 𝑡 𝑥 = 𝐹
41 fveq1 6881 . . . . . . 7 (𝑥 = 𝐹 → (𝑥𝑡) = (𝐹𝑡))
4241breq2d 5151 . . . . . 6 (𝑥 = 𝐹 → (0 ≤ (𝑥𝑡) ↔ 0 ≤ (𝐹𝑡)))
4341breq1d 5149 . . . . . 6 (𝑥 = 𝐹 → ((𝑥𝑡) ≤ 1 ↔ (𝐹𝑡) ≤ 1))
4442, 43anbi12d 630 . . . . 5 (𝑥 = 𝐹 → ((0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1)))
4540, 44ralbid 3262 . . . 4 (𝑥 = 𝐹 → (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝐹𝑡) ∧ (𝐹𝑡) ≤ 1)))
4641breq1d 5149 . . . . 5 (𝑥 = 𝐹 → ((𝑥𝑡) < 𝐸 ↔ (𝐹𝑡) < 𝐸))
4740, 46ralbid 3262 . . . 4 (𝑥 = 𝐹 → (∀𝑡𝐷 (𝑥𝑡) < 𝐸 ↔ ∀𝑡𝐷 (𝐹𝑡) < 𝐸))
4841breq2d 5151 . . . . 5 (𝑥 = 𝐹 → ((1 − 𝐸) < (𝑥𝑡) ↔ (1 − 𝐸) < (𝐹𝑡)))
4940, 48ralbid 3262 . . . 4 (𝑥 = 𝐹 → (∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡) ↔ ∀𝑡𝐵 (1 − 𝐸) < (𝐹𝑡)))
5045, 47, 493anbi123d 1432 . . 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 3604 . 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 1369 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 1084   = wceq 1533  wnf 1777  wcel 2098  wnfc 2875  wral 3053  wrex 3062  wss 3941  c0 4315   cuni 4900   class class class wbr 5139  cmpt 5222  cfv 6534  (class class class)co 7402  cr 11106  0cc0 11107  1c1 11108   < clt 11246  cle 11247  cmin 11442  +crp 12972  Clsdccld 22844
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-po 5579  df-so 5580  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11444  df-neg 11445  df-rp 12973  df-top 22720  df-cld 22847
This theorem is referenced by:  stoweidlem58  45284
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