Theorem stoweidlem15 45999

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_(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem15.1	⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
stoweidlem15.3	⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝑄)
stoweidlem15.4	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)

Assertion

Ref	Expression
stoweidlem15	⊢ (((𝜑 ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) → (((𝐺‘𝐼)‘𝑆) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝐼)‘𝑆) ∧ ((𝐺‘𝐼)‘𝑆) ≤ 1))

Distinct variable groups:   𝐴,𝑓   𝑓,𝐺   𝑓,𝐼   𝑇,𝑓   𝜑,𝑓   𝑡,ℎ,𝐺   𝐴,ℎ   ℎ,𝐼,𝑡   𝑇,ℎ,𝑡   ℎ,𝑍

Allowed substitution hints:   𝜑(𝑡,ℎ)   𝐴(𝑡)   𝑄(𝑡,𝑓,ℎ)   𝑆(𝑡,𝑓,ℎ)   𝑀(𝑡,𝑓,ℎ)   𝑍(𝑡,𝑓)

Proof of Theorem stoweidlem15

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ (1...𝑀)) → 𝜑)
2		stoweidlem15.3	. . . . . 6 ⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝑄)
3	2	ffvelcdmda 7111	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ (1...𝑀)) → (𝐺‘𝐼) ∈ 𝑄)
4		elrabi 3693	. . . . . 6 ⊢ ((𝐺‘𝐼) ∈ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} → (𝐺‘𝐼) ∈ 𝐴)
5		stoweidlem15.1	. . . . . 6 ⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
6	4, 5	eleq2s 2859	. . . . 5 ⊢ ((𝐺‘𝐼) ∈ 𝑄 → (𝐺‘𝐼) ∈ 𝐴)
7	3, 6	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ (1...𝑀)) → (𝐺‘𝐼) ∈ 𝐴)
8		eleq1 2829	. . . . . . . 8 ⊢ (𝑓 = (𝐺‘𝐼) → (𝑓 ∈ 𝐴 ↔ (𝐺‘𝐼) ∈ 𝐴))
9	8	anbi2d 630	. . . . . . 7 ⊢ (𝑓 = (𝐺‘𝐼) → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ (𝐺‘𝐼) ∈ 𝐴)))
10		feq1 6724	. . . . . . 7 ⊢ (𝑓 = (𝐺‘𝐼) → (𝑓:𝑇⟶ℝ ↔ (𝐺‘𝐼):𝑇⟶ℝ))
11	9, 10	imbi12d 344	. . . . . 6 ⊢ (𝑓 = (𝐺‘𝐼) → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺‘𝐼) ∈ 𝐴) → (𝐺‘𝐼):𝑇⟶ℝ)))
12		stoweidlem15.4	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
13	11, 12	vtoclg 3557	. . . . 5 ⊢ ((𝐺‘𝐼) ∈ 𝐴 → ((𝜑 ∧ (𝐺‘𝐼) ∈ 𝐴) → (𝐺‘𝐼):𝑇⟶ℝ))
14	7, 13	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ (1...𝑀)) → ((𝜑 ∧ (𝐺‘𝐼) ∈ 𝐴) → (𝐺‘𝐼):𝑇⟶ℝ))
15	1, 7, 14	mp2and 699	. . 3 ⊢ ((𝜑 ∧ 𝐼 ∈ (1...𝑀)) → (𝐺‘𝐼):𝑇⟶ℝ)
16	15	ffvelcdmda 7111	. 2 ⊢ (((𝜑 ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) → ((𝐺‘𝐼)‘𝑆) ∈ ℝ)
17	3, 5	eleqtrdi 2851	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐼 ∈ (1...𝑀)) → (𝐺‘𝐼) ∈ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))})
18		fveq1 6913	. . . . . . . . . 10 ⊢ (ℎ = (𝐺‘𝐼) → (ℎ‘𝑍) = ((𝐺‘𝐼)‘𝑍))
19	18	eqeq1d 2739	. . . . . . . . 9 ⊢ (ℎ = (𝐺‘𝐼) → ((ℎ‘𝑍) = 0 ↔ ((𝐺‘𝐼)‘𝑍) = 0))
20		fveq1 6913	. . . . . . . . . . . 12 ⊢ (ℎ = (𝐺‘𝐼) → (ℎ‘𝑡) = ((𝐺‘𝐼)‘𝑡))
21	20	breq2d 5163	. . . . . . . . . . 11 ⊢ (ℎ = (𝐺‘𝐼) → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ ((𝐺‘𝐼)‘𝑡)))
22	20	breq1d 5161	. . . . . . . . . . 11 ⊢ (ℎ = (𝐺‘𝐼) → ((ℎ‘𝑡) ≤ 1 ↔ ((𝐺‘𝐼)‘𝑡) ≤ 1))
23	21, 22	anbi12d 632	. . . . . . . . . 10 ⊢ (ℎ = (𝐺‘𝐼) → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ ((𝐺‘𝐼)‘𝑡) ∧ ((𝐺‘𝐼)‘𝑡) ≤ 1)))
24	23	ralbidv 3178	. . . . . . . . 9 ⊢ (ℎ = (𝐺‘𝐼) → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝐺‘𝐼)‘𝑡) ∧ ((𝐺‘𝐼)‘𝑡) ≤ 1)))
25	19, 24	anbi12d 632	. . . . . . . 8 ⊢ (ℎ = (𝐺‘𝐼) → (((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)) ↔ (((𝐺‘𝐼)‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝐺‘𝐼)‘𝑡) ∧ ((𝐺‘𝐼)‘𝑡) ≤ 1))))
26	25	elrab 3698	. . . . . . 7 ⊢ ((𝐺‘𝐼) ∈ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} ↔ ((𝐺‘𝐼) ∈ 𝐴 ∧ (((𝐺‘𝐼)‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝐺‘𝐼)‘𝑡) ∧ ((𝐺‘𝐼)‘𝑡) ≤ 1))))
27	17, 26	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ 𝐼 ∈ (1...𝑀)) → ((𝐺‘𝐼) ∈ 𝐴 ∧ (((𝐺‘𝐼)‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝐺‘𝐼)‘𝑡) ∧ ((𝐺‘𝐼)‘𝑡) ≤ 1))))
28	27	simprd 495	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ (1...𝑀)) → (((𝐺‘𝐼)‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝐺‘𝐼)‘𝑡) ∧ ((𝐺‘𝐼)‘𝑡) ≤ 1)))
29	28	simprd 495	. . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ (1...𝑀)) → ∀𝑡 ∈ 𝑇 (0 ≤ ((𝐺‘𝐼)‘𝑡) ∧ ((𝐺‘𝐼)‘𝑡) ≤ 1))
30		fveq2 6914	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → ((𝐺‘𝐼)‘𝑠) = ((𝐺‘𝐼)‘𝑡))
31	30	breq2d 5163	. . . . . . 7 ⊢ (𝑠 = 𝑡 → (0 ≤ ((𝐺‘𝐼)‘𝑠) ↔ 0 ≤ ((𝐺‘𝐼)‘𝑡)))
32	30	breq1d 5161	. . . . . . 7 ⊢ (𝑠 = 𝑡 → (((𝐺‘𝐼)‘𝑠) ≤ 1 ↔ ((𝐺‘𝐼)‘𝑡) ≤ 1))
33	31, 32	anbi12d 632	. . . . . 6 ⊢ (𝑠 = 𝑡 → ((0 ≤ ((𝐺‘𝐼)‘𝑠) ∧ ((𝐺‘𝐼)‘𝑠) ≤ 1) ↔ (0 ≤ ((𝐺‘𝐼)‘𝑡) ∧ ((𝐺‘𝐼)‘𝑡) ≤ 1)))
34	33	cbvralvw 3237	. . . . 5 ⊢ (∀𝑠 ∈ 𝑇 (0 ≤ ((𝐺‘𝐼)‘𝑠) ∧ ((𝐺‘𝐼)‘𝑠) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝐺‘𝐼)‘𝑡) ∧ ((𝐺‘𝐼)‘𝑡) ≤ 1))
35		fveq2 6914	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → ((𝐺‘𝐼)‘𝑠) = ((𝐺‘𝐼)‘𝑆))
36	35	breq2d 5163	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (0 ≤ ((𝐺‘𝐼)‘𝑠) ↔ 0 ≤ ((𝐺‘𝐼)‘𝑆)))
37	35	breq1d 5161	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (((𝐺‘𝐼)‘𝑠) ≤ 1 ↔ ((𝐺‘𝐼)‘𝑆) ≤ 1))
38	36, 37	anbi12d 632	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((0 ≤ ((𝐺‘𝐼)‘𝑠) ∧ ((𝐺‘𝐼)‘𝑠) ≤ 1) ↔ (0 ≤ ((𝐺‘𝐼)‘𝑆) ∧ ((𝐺‘𝐼)‘𝑆) ≤ 1)))
39	38	rspccva 3624	. . . . 5 ⊢ ((∀𝑠 ∈ 𝑇 (0 ≤ ((𝐺‘𝐼)‘𝑠) ∧ ((𝐺‘𝐼)‘𝑠) ≤ 1) ∧ 𝑆 ∈ 𝑇) → (0 ≤ ((𝐺‘𝐼)‘𝑆) ∧ ((𝐺‘𝐼)‘𝑆) ≤ 1))
40	34, 39	sylanbr 582	. . . 4 ⊢ ((∀𝑡 ∈ 𝑇 (0 ≤ ((𝐺‘𝐼)‘𝑡) ∧ ((𝐺‘𝐼)‘𝑡) ≤ 1) ∧ 𝑆 ∈ 𝑇) → (0 ≤ ((𝐺‘𝐼)‘𝑆) ∧ ((𝐺‘𝐼)‘𝑆) ≤ 1))
41	29, 40	sylan 580	. . 3 ⊢ (((𝜑 ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) → (0 ≤ ((𝐺‘𝐼)‘𝑆) ∧ ((𝐺‘𝐼)‘𝑆) ≤ 1))
42	41	simpld 494	. 2 ⊢ (((𝜑 ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) → 0 ≤ ((𝐺‘𝐼)‘𝑆))
43	41	simprd 495	. 2 ⊢ (((𝜑 ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) → ((𝐺‘𝐼)‘𝑆) ≤ 1)
44	16, 42, 43	3jca 1129	1 ⊢ (((𝜑 ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) → (((𝐺‘𝐼)‘𝑆) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝐼)‘𝑆) ∧ ((𝐺‘𝐼)‘𝑆) ≤ 1))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1539   ∈ wcel 2108  ∀wral 3061  {crab 3436   class class class wbr 5151  ⟶wf 6565  ‘cfv 6569  (class class class)co 7438  ℝcr 11161  0cc0 11162  1c1 11163   ≤ cle 11303  ...cfz 13553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-fv 6577

This theorem is referenced by:  stoweidlem30  46014  stoweidlem38  46022  stoweidlem44  46028