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Theorem stoweidlem15 38732
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 471 . . . 4 ((𝜑𝐼 ∈ (1...𝑀)) → 𝜑)
2 stoweidlem15.3 . . . . . 6 (𝜑𝐺:(1...𝑀)⟶𝑄)
32fnvinran 38020 . . . . 5 ((𝜑𝐼 ∈ (1...𝑀)) → (𝐺𝐼) ∈ 𝑄)
4 elrabi 3327 . . . . . 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 2705 . . . . 5 ((𝐺𝐼) ∈ 𝑄 → (𝐺𝐼) ∈ 𝐴)
73, 6syl 17 . . . 4 ((𝜑𝐼 ∈ (1...𝑀)) → (𝐺𝐼) ∈ 𝐴)
8 eleq1 2675 . . . . . . . 8 (𝑓 = (𝐺𝐼) → (𝑓𝐴 ↔ (𝐺𝐼) ∈ 𝐴))
98anbi2d 735 . . . . . . 7 (𝑓 = (𝐺𝐼) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝐺𝐼) ∈ 𝐴)))
10 feq1 5925 . . . . . . 7 (𝑓 = (𝐺𝐼) → (𝑓:𝑇⟶ℝ ↔ (𝐺𝐼):𝑇⟶ℝ))
119, 10imbi12d 332 . . . . . 6 (𝑓 = (𝐺𝐼) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺𝐼) ∈ 𝐴) → (𝐺𝐼):𝑇⟶ℝ)))
12 stoweidlem15.4 . . . . . 6 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
1311, 12vtoclg 3238 . . . . 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 710 . . 3 ((𝜑𝐼 ∈ (1...𝑀)) → (𝐺𝐼):𝑇⟶ℝ)
1615fnvinran 38020 . 2 (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → ((𝐺𝐼)‘𝑆) ∈ ℝ)
173, 5syl6eleq 2697 . . . . . . 7 ((𝜑𝐼 ∈ (1...𝑀)) → (𝐺𝐼) ∈ {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))})
18 fveq1 6087 . . . . . . . . . 10 ( = (𝐺𝐼) → (𝑍) = ((𝐺𝐼)‘𝑍))
1918eqeq1d 2611 . . . . . . . . 9 ( = (𝐺𝐼) → ((𝑍) = 0 ↔ ((𝐺𝐼)‘𝑍) = 0))
20 fveq1 6087 . . . . . . . . . . . 12 ( = (𝐺𝐼) → (𝑡) = ((𝐺𝐼)‘𝑡))
2120breq2d 4589 . . . . . . . . . . 11 ( = (𝐺𝐼) → (0 ≤ (𝑡) ↔ 0 ≤ ((𝐺𝐼)‘𝑡)))
2220breq1d 4587 . . . . . . . . . . 11 ( = (𝐺𝐼) → ((𝑡) ≤ 1 ↔ ((𝐺𝐼)‘𝑡) ≤ 1))
2321, 22anbi12d 742 . . . . . . . . . 10 ( = (𝐺𝐼) → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1)))
2423ralbidv 2968 . . . . . . . . 9 ( = (𝐺𝐼) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1)))
2519, 24anbi12d 742 . . . . . . . 8 ( = (𝐺𝐼) → (((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)) ↔ (((𝐺𝐼)‘𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))))
2625elrab 3330 . . . . . . 7 ((𝐺𝐼) ∈ {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))} ↔ ((𝐺𝐼) ∈ 𝐴 ∧ (((𝐺𝐼)‘𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))))
2717, 26sylib 206 . . . . . 6 ((𝜑𝐼 ∈ (1...𝑀)) → ((𝐺𝐼) ∈ 𝐴 ∧ (((𝐺𝐼)‘𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))))
2827simprd 477 . . . . 5 ((𝜑𝐼 ∈ (1...𝑀)) → (((𝐺𝐼)‘𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1)))
2928simprd 477 . . . 4 ((𝜑𝐼 ∈ (1...𝑀)) → ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))
30 fveq2 6088 . . . . . . . 8 (𝑠 = 𝑡 → ((𝐺𝐼)‘𝑠) = ((𝐺𝐼)‘𝑡))
3130breq2d 4589 . . . . . . 7 (𝑠 = 𝑡 → (0 ≤ ((𝐺𝐼)‘𝑠) ↔ 0 ≤ ((𝐺𝐼)‘𝑡)))
3230breq1d 4587 . . . . . . 7 (𝑠 = 𝑡 → (((𝐺𝐼)‘𝑠) ≤ 1 ↔ ((𝐺𝐼)‘𝑡) ≤ 1))
3331, 32anbi12d 742 . . . . . 6 (𝑠 = 𝑡 → ((0 ≤ ((𝐺𝐼)‘𝑠) ∧ ((𝐺𝐼)‘𝑠) ≤ 1) ↔ (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1)))
3433cbvralv 3146 . . . . 5 (∀𝑠𝑇 (0 ≤ ((𝐺𝐼)‘𝑠) ∧ ((𝐺𝐼)‘𝑠) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1))
35 fveq2 6088 . . . . . . . 8 (𝑠 = 𝑆 → ((𝐺𝐼)‘𝑠) = ((𝐺𝐼)‘𝑆))
3635breq2d 4589 . . . . . . 7 (𝑠 = 𝑆 → (0 ≤ ((𝐺𝐼)‘𝑠) ↔ 0 ≤ ((𝐺𝐼)‘𝑆)))
3735breq1d 4587 . . . . . . 7 (𝑠 = 𝑆 → (((𝐺𝐼)‘𝑠) ≤ 1 ↔ ((𝐺𝐼)‘𝑆) ≤ 1))
3836, 37anbi12d 742 . . . . . 6 (𝑠 = 𝑆 → ((0 ≤ ((𝐺𝐼)‘𝑠) ∧ ((𝐺𝐼)‘𝑠) ≤ 1) ↔ (0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1)))
3938rspccva 3280 . . . . 5 ((∀𝑠𝑇 (0 ≤ ((𝐺𝐼)‘𝑠) ∧ ((𝐺𝐼)‘𝑠) ≤ 1) ∧ 𝑆𝑇) → (0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1))
4034, 39sylanbr 488 . . . 4 ((∀𝑡𝑇 (0 ≤ ((𝐺𝐼)‘𝑡) ∧ ((𝐺𝐼)‘𝑡) ≤ 1) ∧ 𝑆𝑇) → (0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1))
4129, 40sylan 486 . . 3 (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → (0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1))
4241simpld 473 . 2 (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → 0 ≤ ((𝐺𝐼)‘𝑆))
4341simprd 477 . 2 (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → ((𝐺𝐼)‘𝑆) ≤ 1)
4416, 42, 433jca 1234 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 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  {crab 2899   class class class wbr 4577  wf 5786  cfv 5790  (class class class)co 6527  cr 9792  0cc0 9793  1c1 9794  cle 9932  ...cfz 12155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798
This theorem is referenced by:  stoweidlem30  38747  stoweidlem38  38755  stoweidlem44  38761
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