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

Theorem stoweidlem15 38908
 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(t_0) = 0, and p > 0 on T - U. Here (𝐺‘𝐼) is used to represent p(t_i) 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 472 . . . 4 ((𝜑𝐼 ∈ (1...𝑀)) → 𝜑)
2 stoweidlem15.3 . . . . . 6 (𝜑𝐺:(1...𝑀)⟶𝑄)
32fnvinran 38196 . . . . 5 ((𝜑𝐼 ∈ (1...𝑀)) → (𝐺𝐼) ∈ 𝑄)
4 elrabi 3328 . . . . . 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 2706 . . . . 5 ((𝐺𝐼) ∈ 𝑄 → (𝐺𝐼) ∈ 𝐴)
73, 6syl 17 . . . 4 ((𝜑𝐼 ∈ (1...𝑀)) → (𝐺𝐼) ∈ 𝐴)
8 eleq1 2676 . . . . . . . 8 (𝑓 = (𝐺𝐼) → (𝑓𝐴 ↔ (𝐺𝐼) ∈ 𝐴))
98anbi2d 736 . . . . . . 7 (𝑓 = (𝐺𝐼) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝐺𝐼) ∈ 𝐴)))
10 feq1 5939 . . . . . . 7 (𝑓 = (𝐺𝐼) → (𝑓:𝑇⟶ℝ ↔ (𝐺𝐼):𝑇⟶ℝ))
119, 10imbi12d 333 . . . . . 6 (𝑓 = (𝐺𝐼) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺𝐼) ∈ 𝐴) → (𝐺𝐼):𝑇⟶ℝ)))
12 stoweidlem15.4 . . . . . 6 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
1311, 12vtoclg 3239 . . . . 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 711 . . 3 ((𝜑𝐼 ∈ (1...𝑀)) → (𝐺𝐼):𝑇⟶ℝ)
1615fnvinran 38196 . 2 (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → ((𝐺𝐼)‘𝑆) ∈ ℝ)
173, 5syl6eleq 2698 . . . . . . 7 ((𝜑𝐼 ∈ (1...𝑀)) → (𝐺𝐼) ∈ {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))})
18 fveq1 6102 . . . . . . . . . 10 ( = (𝐺𝐼) → (𝑍) = ((𝐺𝐼)‘𝑍))
1918eqeq1d 2612 . . . . . . . . 9 ( = (𝐺𝐼) → ((𝑍) = 0 ↔ ((𝐺𝐼)‘𝑍) = 0))
20 fveq1 6102 . . . . . . . . . . . 12 ( = (𝐺𝐼) → (𝑡) = ((𝐺𝐼)‘𝑡))
2120breq2d 4595 . . . . . . . . . . 11 ( = (𝐺𝐼) → (0 ≤ (𝑡) ↔ 0 ≤ ((𝐺𝐼)‘𝑡)))
2220breq1d 4593 . . . . . . . . . . 11 ( = (𝐺𝐼) → ((𝑡) ≤ 1 ↔ ((𝐺𝐼)‘𝑡) ≤ 1))
2321, 22anbi12d 743 . . . . . . . . . 10 ( = (𝐺𝐼) → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1)))
2423ralbidv 2969 . . . . . . . . 9 ( = (𝐺𝐼) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1)))
2519, 24anbi12d 743 . . . . . . . 8 ( = (𝐺𝐼) → (((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)) ↔ (((𝐺𝐼)‘𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))))
2625elrab 3331 . . . . . . 7 ((𝐺𝐼) ∈ {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))} ↔ ((𝐺𝐼) ∈ 𝐴 ∧ (((𝐺𝐼)‘𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))))
2717, 26sylib 207 . . . . . 6 ((𝜑𝐼 ∈ (1...𝑀)) → ((𝐺𝐼) ∈ 𝐴 ∧ (((𝐺𝐼)‘𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))))
2827simprd 478 . . . . 5 ((𝜑𝐼 ∈ (1...𝑀)) → (((𝐺𝐼)‘𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1)))
2928simprd 478 . . . 4 ((𝜑𝐼 ∈ (1...𝑀)) → ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))
30 fveq2 6103 . . . . . . . 8 (𝑠 = 𝑡 → ((𝐺𝐼)‘𝑠) = ((𝐺𝐼)‘𝑡))
3130breq2d 4595 . . . . . . 7 (𝑠 = 𝑡 → (0 ≤ ((𝐺𝐼)‘𝑠) ↔ 0 ≤ ((𝐺𝐼)‘𝑡)))
3230breq1d 4593 . . . . . . 7 (𝑠 = 𝑡 → (((𝐺𝐼)‘𝑠) ≤ 1 ↔ ((𝐺𝐼)‘𝑡) ≤ 1))
3331, 32anbi12d 743 . . . . . 6 (𝑠 = 𝑡 → ((0 ≤ ((𝐺𝐼)‘𝑠) ∧ ((𝐺𝐼)‘𝑠) ≤ 1) ↔ (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1)))
3433cbvralv 3147 . . . . 5 (∀𝑠𝑇 (0 ≤ ((𝐺𝐼)‘𝑠) ∧ ((𝐺𝐼)‘𝑠) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))
35 fveq2 6103 . . . . . . . 8 (𝑠 = 𝑆 → ((𝐺𝐼)‘𝑠) = ((𝐺𝐼)‘𝑆))
3635breq2d 4595 . . . . . . 7 (𝑠 = 𝑆 → (0 ≤ ((𝐺𝐼)‘𝑠) ↔ 0 ≤ ((𝐺𝐼)‘𝑆)))
3735breq1d 4593 . . . . . . 7 (𝑠 = 𝑆 → (((𝐺𝐼)‘𝑠) ≤ 1 ↔ ((𝐺𝐼)‘𝑆) ≤ 1))
3836, 37anbi12d 743 . . . . . 6 (𝑠 = 𝑆 → ((0 ≤ ((𝐺𝐼)‘𝑠) ∧ ((𝐺𝐼)‘𝑠) ≤ 1) ↔ (0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1)))
3938rspccva 3281 . . . . 5 ((∀𝑠𝑇 (0 ≤ ((𝐺𝐼)‘𝑠) ∧ ((𝐺𝐼)‘𝑠) ≤ 1) ∧ 𝑆𝑇) → (0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1))
4034, 39sylanbr 489 . . . 4 ((∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1) ∧ 𝑆𝑇) → (0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1))
4129, 40sylan 487 . . 3 (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → (0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1))
4241simpld 474 . 2 (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → 0 ≤ ((𝐺𝐼)‘𝑆))
4341simprd 478 . 2 (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → ((𝐺𝐼)‘𝑆) ≤ 1)
4416, 42, 433jca 1235 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   class class class wbr 4583  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   ≤ cle 9954  ...cfz 12197 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812 This theorem is referenced by:  stoweidlem30  38923  stoweidlem38  38931  stoweidlem44  38937
 Copyright terms: Public domain W3C validator