Theorem stoweidlem15 46280

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 7029	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ (1...𝑀)) → (𝐺‘𝐼) ∈ 𝑄)
4		elrabi 3642	. . . . . 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 2854	. . . . 5 ⊢ ((𝐺‘𝐼) ∈ 𝑄 → (𝐺‘𝐼) ∈ 𝐴)
7	3, 6	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ (1...𝑀)) → (𝐺‘𝐼) ∈ 𝐴)
8		eleq1 2824	. . . . . . . 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 6640	. . . . . . 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 3511	. . . . 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 7029	. 2 ⊢ (((𝜑 ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) → ((𝐺‘𝐼)‘𝑆) ∈ ℝ)
17	3, 5	eleqtrdi 2846	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐼 ∈ (1...𝑀)) → (𝐺‘𝐼) ∈ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))})
18		fveq1 6833	. . . . . . . . . 10 ⊢ (ℎ = (𝐺‘𝐼) → (ℎ‘𝑍) = ((𝐺‘𝐼)‘𝑍))
19	18	eqeq1d 2738	. . . . . . . . 9 ⊢ (ℎ = (𝐺‘𝐼) → ((ℎ‘𝑍) = 0 ↔ ((𝐺‘𝐼)‘𝑍) = 0))
20		fveq1 6833	. . . . . . . . . . . 12 ⊢ (ℎ = (𝐺‘𝐼) → (ℎ‘𝑡) = ((𝐺‘𝐼)‘𝑡))
21	20	breq2d 5110	. . . . . . . . . . 11 ⊢ (ℎ = (𝐺‘𝐼) → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ ((𝐺‘𝐼)‘𝑡)))
22	20	breq1d 5108	. . . . . . . . . . 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 3159	. . . . . . . . 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 3646	. . . . . . 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 6834	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → ((𝐺‘𝐼)‘𝑠) = ((𝐺‘𝐼)‘𝑡))
31	30	breq2d 5110	. . . . . . 7 ⊢ (𝑠 = 𝑡 → (0 ≤ ((𝐺‘𝐼)‘𝑠) ↔ 0 ≤ ((𝐺‘𝐼)‘𝑡)))
32	30	breq1d 5108	. . . . . . 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 3214	. . . . 5 ⊢ (∀𝑠 ∈ 𝑇 (0 ≤ ((𝐺‘𝐼)‘𝑠) ∧ ((𝐺‘𝐼)‘𝑠) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝐺‘𝐼)‘𝑡) ∧ ((𝐺‘𝐼)‘𝑡) ≤ 1))
35		fveq2 6834	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → ((𝐺‘𝐼)‘𝑠) = ((𝐺‘𝐼)‘𝑆))
36	35	breq2d 5110	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (0 ≤ ((𝐺‘𝐼)‘𝑠) ↔ 0 ≤ ((𝐺‘𝐼)‘𝑆)))
37	35	breq1d 5108	. . . . . . 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 3575	. . . . 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 1128	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 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  {crab 3399   class class class wbr 5098  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  ℝcr 11027  0cc0 11028  1c1 11029   ≤ cle 11169  ...cfz 13425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by:  stoweidlem30  46295  stoweidlem38  46303  stoweidlem44  46309