Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem15 Structured version   Visualization version   GIF version

Theorem stoweidlem15 42583
 Description: This lemma is used to prove the existence of a function 𝑝 as in Lemma 1 from [BrosowskiDeutsh] p. 90: 𝑝 is in the subalgebra, such that 0 ≤ p ≤ 1, p_(t0) = 0, and p > 0 on T - U. Here (𝐺‘𝐼) is used to represent p_(ti) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem15.1 𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}
stoweidlem15.3 (𝜑𝐺:(1...𝑀)⟶𝑄)
stoweidlem15.4 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
Assertion
Ref Expression
stoweidlem15 (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → (((𝐺𝐼)‘𝑆) ∈ ℝ ∧ 0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1))
Distinct variable groups:   𝐴,𝑓   𝑓,𝐺   𝑓,𝐼   𝑇,𝑓   𝜑,𝑓   𝑡,,𝐺   𝐴,   ,𝐼,𝑡   𝑇,,𝑡   ,𝑍
Allowed substitution hints:   𝜑(𝑡,)   𝐴(𝑡)   𝑄(𝑡,𝑓,)   𝑆(𝑡,𝑓,)   𝑀(𝑡,𝑓,)   𝑍(𝑡,𝑓)

Proof of Theorem stoweidlem15
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 simpl 486 . . . 4 ((𝜑𝐼 ∈ (1...𝑀)) → 𝜑)
2 stoweidlem15.3 . . . . . 6 (𝜑𝐺:(1...𝑀)⟶𝑄)
32ffvelrnda 6842 . . . . 5 ((𝜑𝐼 ∈ (1...𝑀)) → (𝐺𝐼) ∈ 𝑄)
4 elrabi 3661 . . . . . 6 ((𝐺𝐼) ∈ {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))} → (𝐺𝐼) ∈ 𝐴)
5 stoweidlem15.1 . . . . . 6 𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}
64, 5eleq2s 2934 . . . . 5 ((𝐺𝐼) ∈ 𝑄 → (𝐺𝐼) ∈ 𝐴)
73, 6syl 17 . . . 4 ((𝜑𝐼 ∈ (1...𝑀)) → (𝐺𝐼) ∈ 𝐴)
8 eleq1 2903 . . . . . . . 8 (𝑓 = (𝐺𝐼) → (𝑓𝐴 ↔ (𝐺𝐼) ∈ 𝐴))
98anbi2d 631 . . . . . . 7 (𝑓 = (𝐺𝐼) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝐺𝐼) ∈ 𝐴)))
10 feq1 6484 . . . . . . 7 (𝑓 = (𝐺𝐼) → (𝑓:𝑇⟶ℝ ↔ (𝐺𝐼):𝑇⟶ℝ))
119, 10imbi12d 348 . . . . . 6 (𝑓 = (𝐺𝐼) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺𝐼) ∈ 𝐴) → (𝐺𝐼):𝑇⟶ℝ)))
12 stoweidlem15.4 . . . . . 6 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
1311, 12vtoclg 3553 . . . . 5 ((𝐺𝐼) ∈ 𝐴 → ((𝜑 ∧ (𝐺𝐼) ∈ 𝐴) → (𝐺𝐼):𝑇⟶ℝ))
147, 13syl 17 . . . 4 ((𝜑𝐼 ∈ (1...𝑀)) → ((𝜑 ∧ (𝐺𝐼) ∈ 𝐴) → (𝐺𝐼):𝑇⟶ℝ))
151, 7, 14mp2and 698 . . 3 ((𝜑𝐼 ∈ (1...𝑀)) → (𝐺𝐼):𝑇⟶ℝ)
1615ffvelrnda 6842 . 2 (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → ((𝐺𝐼)‘𝑆) ∈ ℝ)
173, 5eleqtrdi 2926 . . . . . . 7 ((𝜑𝐼 ∈ (1...𝑀)) → (𝐺𝐼) ∈ {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))})
18 fveq1 6660 . . . . . . . . . 10 ( = (𝐺𝐼) → (𝑍) = ((𝐺𝐼)‘𝑍))
1918eqeq1d 2826 . . . . . . . . 9 ( = (𝐺𝐼) → ((𝑍) = 0 ↔ ((𝐺𝐼)‘𝑍) = 0))
20 fveq1 6660 . . . . . . . . . . . 12 ( = (𝐺𝐼) → (𝑡) = ((𝐺𝐼)‘𝑡))
2120breq2d 5064 . . . . . . . . . . 11 ( = (𝐺𝐼) → (0 ≤ (𝑡) ↔ 0 ≤ ((𝐺𝐼)‘𝑡)))
2220breq1d 5062 . . . . . . . . . . 11 ( = (𝐺𝐼) → ((𝑡) ≤ 1 ↔ ((𝐺𝐼)‘𝑡) ≤ 1))
2321, 22anbi12d 633 . . . . . . . . . 10 ( = (𝐺𝐼) → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1)))
2423ralbidv 3192 . . . . . . . . 9 ( = (𝐺𝐼) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1)))
2519, 24anbi12d 633 . . . . . . . 8 ( = (𝐺𝐼) → (((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)) ↔ (((𝐺𝐼)‘𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))))
2625elrab 3666 . . . . . . 7 ((𝐺𝐼) ∈ {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))} ↔ ((𝐺𝐼) ∈ 𝐴 ∧ (((𝐺𝐼)‘𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))))
2717, 26sylib 221 . . . . . 6 ((𝜑𝐼 ∈ (1...𝑀)) → ((𝐺𝐼) ∈ 𝐴 ∧ (((𝐺𝐼)‘𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))))
2827simprd 499 . . . . 5 ((𝜑𝐼 ∈ (1...𝑀)) → (((𝐺𝐼)‘𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1)))
2928simprd 499 . . . 4 ((𝜑𝐼 ∈ (1...𝑀)) → ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))
30 fveq2 6661 . . . . . . . 8 (𝑠 = 𝑡 → ((𝐺𝐼)‘𝑠) = ((𝐺𝐼)‘𝑡))
3130breq2d 5064 . . . . . . 7 (𝑠 = 𝑡 → (0 ≤ ((𝐺𝐼)‘𝑠) ↔ 0 ≤ ((𝐺𝐼)‘𝑡)))
3230breq1d 5062 . . . . . . 7 (𝑠 = 𝑡 → (((𝐺𝐼)‘𝑠) ≤ 1 ↔ ((𝐺𝐼)‘𝑡) ≤ 1))
3331, 32anbi12d 633 . . . . . 6 (𝑠 = 𝑡 → ((0 ≤ ((𝐺𝐼)‘𝑠) ∧ ((𝐺𝐼)‘𝑠) ≤ 1) ↔ (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1)))
3433cbvralvw 3434 . . . . 5 (∀𝑠𝑇 (0 ≤ ((𝐺𝐼)‘𝑠) ∧ ((𝐺𝐼)‘𝑠) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))
35 fveq2 6661 . . . . . . . 8 (𝑠 = 𝑆 → ((𝐺𝐼)‘𝑠) = ((𝐺𝐼)‘𝑆))
3635breq2d 5064 . . . . . . 7 (𝑠 = 𝑆 → (0 ≤ ((𝐺𝐼)‘𝑠) ↔ 0 ≤ ((𝐺𝐼)‘𝑆)))
3735breq1d 5062 . . . . . . 7 (𝑠 = 𝑆 → (((𝐺𝐼)‘𝑠) ≤ 1 ↔ ((𝐺𝐼)‘𝑆) ≤ 1))
3836, 37anbi12d 633 . . . . . 6 (𝑠 = 𝑆 → ((0 ≤ ((𝐺𝐼)‘𝑠) ∧ ((𝐺𝐼)‘𝑠) ≤ 1) ↔ (0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1)))
3938rspccva 3608 . . . . 5 ((∀𝑠𝑇 (0 ≤ ((𝐺𝐼)‘𝑠) ∧ ((𝐺𝐼)‘𝑠) ≤ 1) ∧ 𝑆𝑇) → (0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1))
4034, 39sylanbr 585 . . . 4 ((∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1) ∧ 𝑆𝑇) → (0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1))
4129, 40sylan 583 . . 3 (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → (0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1))
4241simpld 498 . 2 (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → 0 ≤ ((𝐺𝐼)‘𝑆))
4341simprd 499 . 2 (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → ((𝐺𝐼)‘𝑆) ≤ 1)
4416, 42, 433jca 1125 1 (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → (((𝐺𝐼)‘𝑆) ∈ ℝ ∧ 0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∀wral 3133  {crab 3137   class class class wbr 5052  ⟶wf 6339  ‘cfv 6343  (class class class)co 7149  ℝcr 10534  0cc0 10535  1c1 10536   ≤ cle 10674  ...cfz 12894 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351 This theorem is referenced by:  stoweidlem30  42598  stoweidlem38  42606  stoweidlem44  42612
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