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Theorem stoweidlem15 40550
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 (𝐺𝐼) 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)
32ffvelrnda 6399 . . . . 5 ((𝜑𝐼 ∈ (1...𝑀)) → (𝐺𝐼) ∈ 𝑄)
4 elrabi 3391 . . . . . 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 2748 . . . . 5 ((𝐺𝐼) ∈ 𝑄 → (𝐺𝐼) ∈ 𝐴)
73, 6syl 17 . . . 4 ((𝜑𝐼 ∈ (1...𝑀)) → (𝐺𝐼) ∈ 𝐴)
8 eleq1 2718 . . . . . . . 8 (𝑓 = (𝐺𝐼) → (𝑓𝐴 ↔ (𝐺𝐼) ∈ 𝐴))
98anbi2d 740 . . . . . . 7 (𝑓 = (𝐺𝐼) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝐺𝐼) ∈ 𝐴)))
10 feq1 6064 . . . . . . 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 3297 . . . . 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 715 . . 3 ((𝜑𝐼 ∈ (1...𝑀)) → (𝐺𝐼):𝑇⟶ℝ)
1615ffvelrnda 6399 . 2 (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → ((𝐺𝐼)‘𝑆) ∈ ℝ)
173, 5syl6eleq 2740 . . . . . . 7 ((𝜑𝐼 ∈ (1...𝑀)) → (𝐺𝐼) ∈ {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))})
18 fveq1 6228 . . . . . . . . . 10 ( = (𝐺𝐼) → (𝑍) = ((𝐺𝐼)‘𝑍))
1918eqeq1d 2653 . . . . . . . . 9 ( = (𝐺𝐼) → ((𝑍) = 0 ↔ ((𝐺𝐼)‘𝑍) = 0))
20 fveq1 6228 . . . . . . . . . . . 12 ( = (𝐺𝐼) → (𝑡) = ((𝐺𝐼)‘𝑡))
2120breq2d 4697 . . . . . . . . . . 11 ( = (𝐺𝐼) → (0 ≤ (𝑡) ↔ 0 ≤ ((𝐺𝐼)‘𝑡)))
2220breq1d 4695 . . . . . . . . . . 11 ( = (𝐺𝐼) → ((𝑡) ≤ 1 ↔ ((𝐺𝐼)‘𝑡) ≤ 1))
2321, 22anbi12d 747 . . . . . . . . . 10 ( = (𝐺𝐼) → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1)))
2423ralbidv 3015 . . . . . . . . 9 ( = (𝐺𝐼) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1)))
2519, 24anbi12d 747 . . . . . . . 8 ( = (𝐺𝐼) → (((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)) ↔ (((𝐺𝐼)‘𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))))
2625elrab 3396 . . . . . . 7 ((𝐺𝐼) ∈ {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))} ↔ ((𝐺𝐼) ∈ 𝐴 ∧ (((𝐺𝐼)‘𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))))
2717, 26sylib 208 . . . . . 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 6229 . . . . . . . 8 (𝑠 = 𝑡 → ((𝐺𝐼)‘𝑠) = ((𝐺𝐼)‘𝑡))
3130breq2d 4697 . . . . . . 7 (𝑠 = 𝑡 → (0 ≤ ((𝐺𝐼)‘𝑠) ↔ 0 ≤ ((𝐺𝐼)‘𝑡)))
3230breq1d 4695 . . . . . . 7 (𝑠 = 𝑡 → (((𝐺𝐼)‘𝑠) ≤ 1 ↔ ((𝐺𝐼)‘𝑡) ≤ 1))
3331, 32anbi12d 747 . . . . . 6 (𝑠 = 𝑡 → ((0 ≤ ((𝐺𝐼)‘𝑠) ∧ ((𝐺𝐼)‘𝑠) ≤ 1) ↔ (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1)))
3433cbvralv 3201 . . . . 5 (∀𝑠𝑇 (0 ≤ ((𝐺𝐼)‘𝑠) ∧ ((𝐺𝐼)‘𝑠) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))
35 fveq2 6229 . . . . . . . 8 (𝑠 = 𝑆 → ((𝐺𝐼)‘𝑠) = ((𝐺𝐼)‘𝑆))
3635breq2d 4697 . . . . . . 7 (𝑠 = 𝑆 → (0 ≤ ((𝐺𝐼)‘𝑠) ↔ 0 ≤ ((𝐺𝐼)‘𝑆)))
3735breq1d 4695 . . . . . . 7 (𝑠 = 𝑆 → (((𝐺𝐼)‘𝑠) ≤ 1 ↔ ((𝐺𝐼)‘𝑆) ≤ 1))
3836, 37anbi12d 747 . . . . . 6 (𝑠 = 𝑆 → ((0 ≤ ((𝐺𝐼)‘𝑠) ∧ ((𝐺𝐼)‘𝑠) ≤ 1) ↔ (0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1)))
3938rspccva 3339 . . . . 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 1261 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 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945   class class class wbr 4685  wf 5922  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974  1c1 9975  cle 10113  ...cfz 12364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934
This theorem is referenced by:  stoweidlem30  40565  stoweidlem38  40573  stoweidlem44  40579
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