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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 (𝐺𝐼) 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  cr 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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