stoweidlem31 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem31 38924

Description: This lemma is used to prove that there exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91: assuming that 𝑅 is a finite subset of 𝑉, 𝑥 indexes a finite set of functions in the subalgebra (of the Stone Weierstrass theorem), such that for all 𝑖 ranging in the finite indexing set, 0 ≤ x_i ≤ 1, x_i < ε / m on V(t_i), and x_i > 1 - ε / m on 𝐵. Here M is used to represent m in the paper, 𝐸 is used to represent ε in the paper, v_i is used to represent V(t_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

**Hypotheses**
Ref	Expression
stoweidlem31.1	⊢ Ⅎℎ𝜑
stoweidlem31.2	⊢ Ⅎ𝑡𝜑
stoweidlem31.3	⊢ Ⅎ𝑤𝜑
stoweidlem31.4	⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
stoweidlem31.5	⊢ 𝑉 = {𝑤 ∈ 𝐽 ∣ ∀𝑒 ∈ ℝ⁺ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))}
stoweidlem31.6	⊢ 𝐺 = (𝑤 ∈ 𝑅 ↦ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
stoweidlem31.7	⊢ (𝜑 → 𝑅 ⊆ 𝑉)
stoweidlem31.8	⊢ (𝜑 → 𝑀 ∈ ℕ)
stoweidlem31.9	⊢ (𝜑 → 𝑣:(1...𝑀)–1-1-onto→𝑅)
stoweidlem31.10	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
stoweidlem31.11	⊢ (𝜑 → 𝐵 ⊆ (𝑇 ∖ 𝑈))
stoweidlem31.12	⊢ (𝜑 → 𝑉 ∈ V)
stoweidlem31.13	⊢ (𝜑 → 𝐴 ∈ V)
stoweidlem31.14	⊢ (𝜑 → ran 𝐺 ∈ Fin)

**Assertion**
Ref	Expression
stoweidlem31	⊢ (𝜑 → ∃𝑥(𝑥:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥‘𝑖)‘𝑡))))

Distinct variable groups: ℎ,𝑖,𝑡,𝑣,𝑤 𝑖,𝐺 𝑤,𝑌 𝜑,𝑖 𝑒,ℎ,𝑡,𝑤,𝐴 𝑒,𝐸,ℎ,𝑡,𝑤 𝑒,𝑀,ℎ,𝑡,𝑤 𝑇,𝑒,ℎ,𝑤 𝑈,𝑒,ℎ,𝑤 𝑅,ℎ,𝑡,𝑤 𝑥,𝑖,𝑡,𝑣 𝑖,𝑀 𝑥,𝐵 𝑥,𝐸 𝑥,𝐺 𝑥,𝑀 𝑥,𝑌 Allowed substitution hints: 𝜑(𝑥,𝑤,𝑣,𝑡,𝑒,ℎ) 𝐴(𝑥,𝑣,𝑖) 𝐵(𝑤,𝑣,𝑡,𝑒,ℎ,𝑖) 𝑅(𝑥,𝑣,𝑒,𝑖) 𝑇(𝑥,𝑣,𝑡,𝑖) 𝑈(𝑥,𝑣,𝑡,𝑖) 𝐸(𝑣,𝑖) 𝐺(𝑤,𝑣,𝑡,𝑒,ℎ) 𝐽(𝑥,𝑤,𝑣,𝑡,𝑒,ℎ,𝑖) 𝑀(𝑣) 𝑉(𝑥,𝑤,𝑣,𝑡,𝑒,ℎ,𝑖) 𝑌(𝑣,𝑡,𝑒,ℎ,𝑖)

Proof of Theorem stoweidlem31 Dummy variables 𝑏 𝑙 𝑢 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem31.14	. . 3 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
2		fnchoice 38211	. . 3 ⊢ (ran 𝐺 ∈ Fin → ∃𝑙(𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏)))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → ∃𝑙(𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏)))
4		vex 3176	. . . . 5 ⊢ 𝑙 ∈ V
5		stoweidlem31.6	. . . . . . 7 ⊢ 𝐺 = (𝑤 ∈ 𝑅 ↦ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
6		stoweidlem31.12	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ V)
7		stoweidlem31.7	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ⊆ 𝑉)
8	6, 7	ssexd 4733	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ V)
9		mptexg 6389	. . . . . . . 8 ⊢ (𝑅 ∈ V → (𝑤 ∈ 𝑅 ↦ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))}) ∈ V)
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ 𝑅 ↦ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))}) ∈ V)
11	5, 10	syl5eqel 2692	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V)
12		vex 3176	. . . . . 6 ⊢ 𝑣 ∈ V
13		coexg 7010	. . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝑣 ∈ V) → (𝐺 ∘ 𝑣) ∈ V)
14	11, 12, 13	sylancl 693	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ 𝑣) ∈ V)
15		coexg 7010	. . . . 5 ⊢ ((𝑙 ∈ V ∧ (𝐺 ∘ 𝑣) ∈ V) → (𝑙 ∘ (𝐺 ∘ 𝑣)) ∈ V)
16	4, 14, 15	sylancr 694	. . . 4 ⊢ (𝜑 → (𝑙 ∘ (𝐺 ∘ 𝑣)) ∈ V)
17	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) → (𝑙 ∘ (𝐺 ∘ 𝑣)) ∈ V)
18		simprl 790	. . . . . 6 ⊢ ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) → 𝑙 Fn ran 𝐺)
19		stoweidlem31.1	. . . . . . . . 9 ⊢ Ⅎℎ𝜑
20		nfcv 2751	. . . . . . . . . . 11 ⊢ Ⅎℎ𝑙
21		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎℎ𝑅
22		nfrab1 3099	. . . . . . . . . . . . . 14 ⊢ Ⅎℎ{ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))}
23	21, 22	nfmpt 4674	. . . . . . . . . . . . 13 ⊢ Ⅎℎ(𝑤 ∈ 𝑅 ↦ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
24	5, 23	nfcxfr 2749	. . . . . . . . . . . 12 ⊢ Ⅎℎ𝐺
25	24	nfrn 5289	. . . . . . . . . . 11 ⊢ Ⅎℎran 𝐺
26	20, 25	nffn 5901	. . . . . . . . . 10 ⊢ Ⅎℎ 𝑙 Fn ran 𝐺
27		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎℎ(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏)
28	25, 27	nfral 2929	. . . . . . . . . 10 ⊢ Ⅎℎ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏)
29	26, 28	nfan 1816	. . . . . . . . 9 ⊢ Ⅎℎ(𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))
30	19, 29	nfan 1816	. . . . . . . 8 ⊢ Ⅎℎ(𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏)))
31		fvelrnb 6153	. . . . . . . . . . . . 13 ⊢ (𝑙 Fn ran 𝐺 → (ℎ ∈ ran 𝑙 ↔ ∃𝑏 ∈ ran 𝐺(𝑙‘𝑏) = ℎ))
32	18, 31	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) → (ℎ ∈ ran 𝑙 ↔ ∃𝑏 ∈ ran 𝐺(𝑙‘𝑏) = ℎ))
33	32	biimpa 500	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ ℎ ∈ ran 𝑙) → ∃𝑏 ∈ ran 𝐺(𝑙‘𝑏) = ℎ)
34		nfv 1830	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑏𝜑
35		nfv 1830	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑏 𝑙 Fn ran 𝐺
36		nfra1 2925	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑏∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏)
37	35, 36	nfan 1816	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑏(𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))
38	34, 37	nfan 1816	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏(𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏)))
39		nfv 1830	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏 ℎ ∈ ran 𝑙
40	38, 39	nfan 1816	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ ℎ ∈ ran 𝑙)
41		simp3 1056	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ ℎ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙‘𝑏) = ℎ) → (𝑙‘𝑏) = ℎ)
42		simp1ll 1117	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ ℎ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙‘𝑏) = ℎ) → 𝜑)
43		simplrr 797	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ ℎ ∈ ran 𝑙) → ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))
44	43	3ad2ant1 1075	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ ℎ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙‘𝑏) = ℎ) → ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))
45		simp2 1055	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ ℎ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙‘𝑏) = ℎ) → 𝑏 ∈ ran 𝐺)
46		simp3 1056	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → 𝑏 ∈ ran 𝐺)
47		3simpc 1053	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → (∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺))
48		simpr 476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑏 ∈ ran 𝐺) → 𝑏 ∈ ran 𝐺)
49		stoweidlem31.3	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑤𝜑
50		stoweidlem31.13	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐴 ∈ V)
51		rabexg 4739	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐴 ∈ V → {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ∈ V)
52	50, 51	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ∈ V)
53	52	a1d 25	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑤 ∈ 𝑅 → {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ∈ V))
54	49, 53	ralrimi 2940	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ∈ V)
55	5	fnmpt 5933	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑤 ∈ 𝑅 {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ∈ V → 𝐺 Fn 𝑅)
56	54, 55	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐺 Fn 𝑅)
57	56	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑏 ∈ ran 𝐺) → 𝐺 Fn 𝑅)
58		fvelrnb 6153	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐺 Fn 𝑅 → (𝑏 ∈ ran 𝐺 ↔ ∃𝑢 ∈ 𝑅 (𝐺‘𝑢) = 𝑏))
59		nfmpt1 4675	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ Ⅎ𝑤(𝑤 ∈ 𝑅 ↦ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
60	5, 59	nfcxfr 2749	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑤𝐺
61		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑤𝑢
62	60, 61	nffv 6110	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑤(𝐺‘𝑢)
63		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑤𝑏
64	62, 63	nfeq 2762	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑤(𝐺‘𝑢) = 𝑏
65		nfv 1830	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑢(𝐺‘𝑤) = 𝑏
66		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑢 = 𝑤 → (𝐺‘𝑢) = (𝐺‘𝑤))
67	66	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = 𝑤 → ((𝐺‘𝑢) = 𝑏 ↔ (𝐺‘𝑤) = 𝑏))
68	64, 65, 67	cbvrex 3144	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃𝑢 ∈ 𝑅 (𝐺‘𝑢) = 𝑏 ↔ ∃𝑤 ∈ 𝑅 (𝐺‘𝑤) = 𝑏)
69	58, 68	syl6bb 275	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐺 Fn 𝑅 → (𝑏 ∈ ran 𝐺 ↔ ∃𝑤 ∈ 𝑅 (𝐺‘𝑤) = 𝑏))
70	57, 69	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑏 ∈ ran 𝐺) → (𝑏 ∈ ran 𝐺 ↔ ∃𝑤 ∈ 𝑅 (𝐺‘𝑤) = 𝑏))
71	48, 70	mpbid 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑏 ∈ ran 𝐺) → ∃𝑤 ∈ 𝑅 (𝐺‘𝑤) = 𝑏)
72	60	nfrn 5289	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑤ran 𝐺
73	72	nfcri 2745	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑤 𝑏 ∈ ran 𝐺
74	49, 73	nfan 1816	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑏 ∈ ran 𝐺)
75		nfv 1830	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑤 𝑏 ≠ ∅
76		simp3 1056	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑅 ∧ (𝐺‘𝑤) = 𝑏) → (𝐺‘𝑤) = 𝑏)
77		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑅) → 𝑤 ∈ 𝑅)
78	50	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑅) → 𝐴 ∈ V)
79	78, 51	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑅) → {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ∈ V)
80	5	fvmpt2 6200	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑤 ∈ 𝑅 ∧ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ∈ V) → (𝐺‘𝑤) = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
81	77, 79, 80	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑅) → (𝐺‘𝑤) = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
82	7	sselda 3568	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑅) → 𝑤 ∈ 𝑉)
83		stoweidlem31.5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 𝑉 = {𝑤 ∈ 𝐽 ∣ ∀𝑒 ∈ ℝ⁺ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))}
84	83	rabeq2i 3170	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 ∈ 𝑉 ↔ (𝑤 ∈ 𝐽 ∧ ∀𝑒 ∈ ℝ⁺ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))))
85	82, 84	sylib 207	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑅) → (𝑤 ∈ 𝐽 ∧ ∀𝑒 ∈ ℝ⁺ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))))
86	85	simprd 478	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑅) → ∀𝑒 ∈ ℝ⁺ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡)))
87		stoweidlem31.10	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
88		stoweidlem31.8	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → 𝑀 ∈ ℕ)
89	88	nnrpd 11746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
90	87, 89	rpdivcld 11765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (𝐸 / 𝑀) ∈ ℝ⁺)
91	90	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑅) → (𝐸 / 𝑀) ∈ ℝ⁺)
92		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 = (𝐸 / 𝑀) → ((ℎ‘𝑡) < 𝑒 ↔ (ℎ‘𝑡) < (𝐸 / 𝑀)))
93	92	ralbidv 2969	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑒 = (𝐸 / 𝑀) → (∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ↔ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀)))
94		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑒 = (𝐸 / 𝑀) → (1 − 𝑒) = (1 − (𝐸 / 𝑀)))
95	94	breq1d 4593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 = (𝐸 / 𝑀) → ((1 − 𝑒) < (ℎ‘𝑡) ↔ (1 − (𝐸 / 𝑀)) < (ℎ‘𝑡)))
96	95	ralbidv 2969	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑒 = (𝐸 / 𝑀) → (∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡)))
97	93, 96	3anbi23d 1394	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑒 = (𝐸 / 𝑀) → ((∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡)) ↔ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))))
98	97	rexbidv 3034	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑒 = (𝐸 / 𝑀) → (∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡)) ↔ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))))
99	98	rspccva 3281	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((∀𝑒 ∈ ℝ⁺ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡)) ∧ (𝐸 / 𝑀) ∈ ℝ⁺) → ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡)))
100	86, 91, 99	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑅) → ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡)))
101		nfv 1830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ Ⅎℎ 𝑤 ∈ 𝑅
102	19, 101	nfan 1816	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎℎ(𝜑 ∧ 𝑤 ∈ 𝑅)
103		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ Ⅎℎ∅
104	22, 103	nfne 2882	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎℎ{ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ≠ ∅
105		3simpc 1053	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑅) ∧ ℎ ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))) → (ℎ ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))))
106		rabid 3095	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (ℎ ∈ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ↔ (ℎ ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))))
107	105, 106	sylibr 223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑅) ∧ ℎ ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))) → ℎ ∈ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
108		ne0i 3880	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (ℎ ∈ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} → {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ≠ ∅)
109	107, 108	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑅) ∧ ℎ ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))) → {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ≠ ∅)
110	109	3exp 1256	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑅) → (ℎ ∈ 𝐴 → ((∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡)) → {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ≠ ∅)))
111	102, 104, 110	rexlimd 3008	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑅) → (∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡)) → {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ≠ ∅))
112	100, 111	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑅) → {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ≠ ∅)
113	81, 112	eqnetrd 2849	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑅) → (𝐺‘𝑤) ≠ ∅)
114	113	3adant3 1074	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑅 ∧ (𝐺‘𝑤) = 𝑏) → (𝐺‘𝑤) ≠ ∅)
115	76, 114	eqnetrrd 2850	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑅 ∧ (𝐺‘𝑤) = 𝑏) → 𝑏 ≠ ∅)
116	115	3adant1r 1311	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑏 ∈ ran 𝐺) ∧ 𝑤 ∈ 𝑅 ∧ (𝐺‘𝑤) = 𝑏) → 𝑏 ≠ ∅)
117	116	3exp 1256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑏 ∈ ran 𝐺) → (𝑤 ∈ 𝑅 → ((𝐺‘𝑤) = 𝑏 → 𝑏 ≠ ∅)))
118	74, 75, 117	rexlimd 3008	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑏 ∈ ran 𝐺) → (∃𝑤 ∈ 𝑅 (𝐺‘𝑤) = 𝑏 → 𝑏 ≠ ∅))
119	71, 118	mpd 15	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑏 ∈ ran 𝐺) → 𝑏 ≠ ∅)
120	119	3adant2 1073	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → 𝑏 ≠ ∅)
121		rspa 2914	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → (𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))
122	47, 120, 121	sylc 63	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → (𝑙‘𝑏) ∈ 𝑏)
123	46, 122	jca 553	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → (𝑏 ∈ ran 𝐺 ∧ (𝑙‘𝑏) ∈ 𝑏))
124		vex 3176	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑏 ∈ V
125	5	elrnmpt 5293	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ V → (𝑏 ∈ ran 𝐺 ↔ ∃𝑤 ∈ 𝑅 𝑏 = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))}))
126	124, 125	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ ran 𝐺 ↔ ∃𝑤 ∈ 𝑅 𝑏 = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
127	46, 126	sylib 207	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → ∃𝑤 ∈ 𝑅 𝑏 = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
128		nfv 1830	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑤(𝑙‘𝑏) ∈ 𝑏
129	73, 128	nfan 1816	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑤(𝑏 ∈ ran 𝐺 ∧ (𝑙‘𝑏) ∈ 𝑏)
130		nfv 1830	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑤(𝑙‘𝑏) ∈ 𝑌
131		simp1r 1079	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑏 ∈ ran 𝐺 ∧ (𝑙‘𝑏) ∈ 𝑏) ∧ 𝑤 ∈ 𝑅 ∧ 𝑏 = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))}) → (𝑙‘𝑏) ∈ 𝑏)
132		simp3 1056	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑏 ∈ ran 𝐺 ∧ (𝑙‘𝑏) ∈ 𝑏) ∧ 𝑤 ∈ 𝑅 ∧ 𝑏 = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))}) → 𝑏 = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
133		simpl 472	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑙‘𝑏) ∈ 𝑏 ∧ 𝑏 = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))}) → (𝑙‘𝑏) ∈ 𝑏)
134		simpr 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑙‘𝑏) ∈ 𝑏 ∧ 𝑏 = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))}) → 𝑏 = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
135	133, 134	eleqtrd 2690	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑙‘𝑏) ∈ 𝑏 ∧ 𝑏 = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))}) → (𝑙‘𝑏) ∈ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
136		elrabi 3328	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑙‘𝑏) ∈ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} → (𝑙‘𝑏) ∈ 𝐴)
137		fveq1 6102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (ℎ = (𝑙‘𝑏) → (ℎ‘𝑡) = ((𝑙‘𝑏)‘𝑡))
138	137	breq2d 4595	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (ℎ = (𝑙‘𝑏) → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ ((𝑙‘𝑏)‘𝑡)))
139	137	breq1d 4593	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (ℎ = (𝑙‘𝑏) → ((ℎ‘𝑡) ≤ 1 ↔ ((𝑙‘𝑏)‘𝑡) ≤ 1))
140	138, 139	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (ℎ = (𝑙‘𝑏) → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ ((𝑙‘𝑏)‘𝑡) ∧ ((𝑙‘𝑏)‘𝑡) ≤ 1)))
141	140	ralbidv 2969	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℎ = (𝑙‘𝑏) → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝑙‘𝑏)‘𝑡) ∧ ((𝑙‘𝑏)‘𝑡) ≤ 1)))
142	137	breq1d 4593	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (ℎ = (𝑙‘𝑏) → ((ℎ‘𝑡) < (𝐸 / 𝑀) ↔ ((𝑙‘𝑏)‘𝑡) < (𝐸 / 𝑀)))
143	142	ralbidv 2969	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℎ = (𝑙‘𝑏) → (∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ↔ ∀𝑡 ∈ 𝑤 ((𝑙‘𝑏)‘𝑡) < (𝐸 / 𝑀)))
144	137	breq2d 4595	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (ℎ = (𝑙‘𝑏) → ((1 − (𝐸 / 𝑀)) < (ℎ‘𝑡) ↔ (1 − (𝐸 / 𝑀)) < ((𝑙‘𝑏)‘𝑡)))
145	144	ralbidv 2969	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℎ = (𝑙‘𝑏) → (∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < ((𝑙‘𝑏)‘𝑡)))
146	141, 143, 145	3anbi123d 1391	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ = (𝑙‘𝑏) → ((∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡)) ↔ (∀𝑡 ∈ 𝑇 (0 ≤ ((𝑙‘𝑏)‘𝑡) ∧ ((𝑙‘𝑏)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 ((𝑙‘𝑏)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < ((𝑙‘𝑏)‘𝑡))))
147	146	elrab 3331	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑙‘𝑏) ∈ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ↔ ((𝑙‘𝑏) ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ ((𝑙‘𝑏)‘𝑡) ∧ ((𝑙‘𝑏)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 ((𝑙‘𝑏)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < ((𝑙‘𝑏)‘𝑡))))
148	147	simprbi 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑙‘𝑏) ∈ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} → (∀𝑡 ∈ 𝑇 (0 ≤ ((𝑙‘𝑏)‘𝑡) ∧ ((𝑙‘𝑏)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 ((𝑙‘𝑏)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < ((𝑙‘𝑏)‘𝑡)))
149	148	simp1d 1066	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑙‘𝑏) ∈ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} → ∀𝑡 ∈ 𝑇 (0 ≤ ((𝑙‘𝑏)‘𝑡) ∧ ((𝑙‘𝑏)‘𝑡) ≤ 1))
150	141	elrab 3331	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑙‘𝑏) ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} ↔ ((𝑙‘𝑏) ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝑙‘𝑏)‘𝑡) ∧ ((𝑙‘𝑏)‘𝑡) ≤ 1)))
151	136, 149, 150	sylanbrc 695	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑙‘𝑏) ∈ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} → (𝑙‘𝑏) ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)})
152	135, 151	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑙‘𝑏) ∈ 𝑏 ∧ 𝑏 = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))}) → (𝑙‘𝑏) ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)})
153		stoweidlem31.4	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
154	152, 153	syl6eleqr 2699	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑙‘𝑏) ∈ 𝑏 ∧ 𝑏 = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))}) → (𝑙‘𝑏) ∈ 𝑌)
155	131, 132, 154	syl2anc 691	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑏 ∈ ran 𝐺 ∧ (𝑙‘𝑏) ∈ 𝑏) ∧ 𝑤 ∈ 𝑅 ∧ 𝑏 = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))}) → (𝑙‘𝑏) ∈ 𝑌)
156	155	3exp 1256	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ ran 𝐺 ∧ (𝑙‘𝑏) ∈ 𝑏) → (𝑤 ∈ 𝑅 → (𝑏 = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} → (𝑙‘𝑏) ∈ 𝑌)))
157	129, 130, 156	rexlimd 3008	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ∈ ran 𝐺 ∧ (𝑙‘𝑏) ∈ 𝑏) → (∃𝑤 ∈ 𝑅 𝑏 = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} → (𝑙‘𝑏) ∈ 𝑌))
158	123, 127, 157	sylc 63	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → (𝑙‘𝑏) ∈ 𝑌)
159	42, 44, 45, 158	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ ℎ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙‘𝑏) = ℎ) → (𝑙‘𝑏) ∈ 𝑌)
160	41, 159	eqeltrrd 2689	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ ℎ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙‘𝑏) = ℎ) → ℎ ∈ 𝑌)
161	160	3exp 1256	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ ℎ ∈ ran 𝑙) → (𝑏 ∈ ran 𝐺 → ((𝑙‘𝑏) = ℎ → ℎ ∈ 𝑌)))
162	40, 161	reximdai 2995	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ ℎ ∈ ran 𝑙) → (∃𝑏 ∈ ran 𝐺(𝑙‘𝑏) = ℎ → ∃𝑏 ∈ ran 𝐺 ℎ ∈ 𝑌))
163	33, 162	mpd 15	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ ℎ ∈ ran 𝑙) → ∃𝑏 ∈ ran 𝐺 ℎ ∈ 𝑌)
164		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑏 ℎ ∈ 𝑌
165		idd 24	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ran 𝐺 → (ℎ ∈ 𝑌 → ℎ ∈ 𝑌))
166	165	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ ℎ ∈ ran 𝑙) → (𝑏 ∈ ran 𝐺 → (ℎ ∈ 𝑌 → ℎ ∈ 𝑌)))
167	40, 164, 166	rexlimd 3008	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ ℎ ∈ ran 𝑙) → (∃𝑏 ∈ ran 𝐺 ℎ ∈ 𝑌 → ℎ ∈ 𝑌))
168	163, 167	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ ℎ ∈ ran 𝑙) → ℎ ∈ 𝑌)
169	168	ex 449	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) → (ℎ ∈ ran 𝑙 → ℎ ∈ 𝑌))
170	30, 169	ralrimi 2940	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) → ∀ℎ ∈ ran 𝑙 ℎ ∈ 𝑌)
171		dfss3 3558	. . . . . . . 8 ⊢ (ran 𝑙 ⊆ 𝑌 ↔ ∀𝑧 ∈ ran 𝑙 𝑧 ∈ 𝑌)
172		nfrab1 3099	. . . . . . . . . . 11 ⊢ Ⅎℎ{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
173	153, 172	nfcxfr 2749	. . . . . . . . . 10 ⊢ Ⅎℎ𝑌
174	173	nfcri 2745	. . . . . . . . 9 ⊢ Ⅎℎ 𝑧 ∈ 𝑌
175		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑧 ℎ ∈ 𝑌
176		eleq1 2676	. . . . . . . . 9 ⊢ (𝑧 = ℎ → (𝑧 ∈ 𝑌 ↔ ℎ ∈ 𝑌))
177	174, 175, 176	cbvral 3143	. . . . . . . 8 ⊢ (∀𝑧 ∈ ran 𝑙 𝑧 ∈ 𝑌 ↔ ∀ℎ ∈ ran 𝑙 ℎ ∈ 𝑌)
178	171, 177	bitri 263	. . . . . . 7 ⊢ (ran 𝑙 ⊆ 𝑌 ↔ ∀ℎ ∈ ran 𝑙 ℎ ∈ 𝑌)
179	170, 178	sylibr 223	. . . . . 6 ⊢ ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) → ran 𝑙 ⊆ 𝑌)
180		df-f 5808	. . . . . 6 ⊢ (𝑙:ran 𝐺⟶𝑌 ↔ (𝑙 Fn ran 𝐺 ∧ ran 𝑙 ⊆ 𝑌))
181	18, 179, 180	sylanbrc 695	. . . . 5 ⊢ ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) → 𝑙:ran 𝐺⟶𝑌)
182		dffn3 5967	. . . . . . . 8 ⊢ (𝐺 Fn 𝑅 ↔ 𝐺:𝑅⟶ran 𝐺)
183	56, 182	sylib 207	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝑅⟶ran 𝐺)
184	183	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) → 𝐺:𝑅⟶ran 𝐺)
185		stoweidlem31.9	. . . . . . . 8 ⊢ (𝜑 → 𝑣:(1...𝑀)–1-1-onto→𝑅)
186		f1of 6050	. . . . . . . 8 ⊢ (𝑣:(1...𝑀)–1-1-onto→𝑅 → 𝑣:(1...𝑀)⟶𝑅)
187	185, 186	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑣:(1...𝑀)⟶𝑅)
188	187	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) → 𝑣:(1...𝑀)⟶𝑅)
189		fco 5971	. . . . . 6 ⊢ ((𝐺:𝑅⟶ran 𝐺 ∧ 𝑣:(1...𝑀)⟶𝑅) → (𝐺 ∘ 𝑣):(1...𝑀)⟶ran 𝐺)
190	184, 188, 189	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) → (𝐺 ∘ 𝑣):(1...𝑀)⟶ran 𝐺)
191		fco 5971	. . . . 5 ⊢ ((𝑙:ran 𝐺⟶𝑌 ∧ (𝐺 ∘ 𝑣):(1...𝑀)⟶ran 𝐺) → (𝑙 ∘ (𝐺 ∘ 𝑣)):(1...𝑀)⟶𝑌)
192	181, 190, 191	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) → (𝑙 ∘ (𝐺 ∘ 𝑣)):(1...𝑀)⟶𝑌)
193		fvco3 6185	. . . . . . . . 9 ⊢ (((𝐺 ∘ 𝑣):(1...𝑀)⟶ran 𝐺 ∧ 𝑖 ∈ (1...𝑀)) → ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) = (𝑙‘((𝐺 ∘ 𝑣)‘𝑖)))
194	190, 193	sylan 487	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) = (𝑙‘((𝐺 ∘ 𝑣)‘𝑖)))
195		simpll 786	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
196		simplrr 797	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))
197	190	fnvinran 38196	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺 ∘ 𝑣)‘𝑖) ∈ ran 𝐺)
198		simp3 1056	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ ((𝐺 ∘ 𝑣)‘𝑖) ∈ ran 𝐺) → ((𝐺 ∘ 𝑣)‘𝑖) ∈ ran 𝐺)
199		nfv 1830	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏((𝐺 ∘ 𝑣)‘𝑖) ∈ ran 𝐺
200	34, 36, 199	nf3an 1819	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏(𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ ((𝐺 ∘ 𝑣)‘𝑖) ∈ ran 𝐺)
201		nfv 1830	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏(𝑙‘((𝐺 ∘ 𝑣)‘𝑖)) ∈ ((𝐺 ∘ 𝑣)‘𝑖)
202	200, 201	nfim 1813	. . . . . . . . . . 11 ⊢ Ⅎ𝑏((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ ((𝐺 ∘ 𝑣)‘𝑖) ∈ ran 𝐺) → (𝑙‘((𝐺 ∘ 𝑣)‘𝑖)) ∈ ((𝐺 ∘ 𝑣)‘𝑖))
203		eleq1 2676	. . . . . . . . . . . . 13 ⊢ (𝑏 = ((𝐺 ∘ 𝑣)‘𝑖) → (𝑏 ∈ ran 𝐺 ↔ ((𝐺 ∘ 𝑣)‘𝑖) ∈ ran 𝐺))
204	203	3anbi3d 1397	. . . . . . . . . . . 12 ⊢ (𝑏 = ((𝐺 ∘ 𝑣)‘𝑖) → ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) ↔ (𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ ((𝐺 ∘ 𝑣)‘𝑖) ∈ ran 𝐺)))
205		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑏 = ((𝐺 ∘ 𝑣)‘𝑖) → (𝑙‘𝑏) = (𝑙‘((𝐺 ∘ 𝑣)‘𝑖)))
206		id 22	. . . . . . . . . . . . 13 ⊢ (𝑏 = ((𝐺 ∘ 𝑣)‘𝑖) → 𝑏 = ((𝐺 ∘ 𝑣)‘𝑖))
207	205, 206	eleq12d 2682	. . . . . . . . . . . 12 ⊢ (𝑏 = ((𝐺 ∘ 𝑣)‘𝑖) → ((𝑙‘𝑏) ∈ 𝑏 ↔ (𝑙‘((𝐺 ∘ 𝑣)‘𝑖)) ∈ ((𝐺 ∘ 𝑣)‘𝑖)))
208	204, 207	imbi12d 333	. . . . . . . . . . 11 ⊢ (𝑏 = ((𝐺 ∘ 𝑣)‘𝑖) → (((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → (𝑙‘𝑏) ∈ 𝑏) ↔ ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ ((𝐺 ∘ 𝑣)‘𝑖) ∈ ran 𝐺) → (𝑙‘((𝐺 ∘ 𝑣)‘𝑖)) ∈ ((𝐺 ∘ 𝑣)‘𝑖))))
209	202, 208, 122	vtoclg1f 3238	. . . . . . . . . 10 ⊢ (((𝐺 ∘ 𝑣)‘𝑖) ∈ ran 𝐺 → ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ ((𝐺 ∘ 𝑣)‘𝑖) ∈ ran 𝐺) → (𝑙‘((𝐺 ∘ 𝑣)‘𝑖)) ∈ ((𝐺 ∘ 𝑣)‘𝑖)))
210	198, 209	mpcom 37	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏) ∧ ((𝐺 ∘ 𝑣)‘𝑖) ∈ ran 𝐺) → (𝑙‘((𝐺 ∘ 𝑣)‘𝑖)) ∈ ((𝐺 ∘ 𝑣)‘𝑖))
211	195, 196, 197, 210	syl3anc 1318	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → (𝑙‘((𝐺 ∘ 𝑣)‘𝑖)) ∈ ((𝐺 ∘ 𝑣)‘𝑖))
212	194, 211	eqeltrd 2688	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) ∈ ((𝐺 ∘ 𝑣)‘𝑖))
213		fvco3 6185	. . . . . . . . . . . 12 ⊢ ((𝑣:(1...𝑀)⟶𝑅 ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺 ∘ 𝑣)‘𝑖) = (𝐺‘(𝑣‘𝑖)))
214	187, 213	sylan 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺 ∘ 𝑣)‘𝑖) = (𝐺‘(𝑣‘𝑖)))
215	187	fnvinran 38196	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑣‘𝑖) ∈ 𝑅)
216	50	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐴 ∈ V)
217		rabexg 4739	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ V → {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ∈ V)
218	216, 217	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ∈ V)
219		raleq 3115	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑣‘𝑖) → (∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ↔ ∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀)))
220	219	3anbi2d 1396	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑣‘𝑖) → ((∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡)) ↔ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))))
221	220	rabbidv 3164	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑣‘𝑖) → {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
222	221, 5	fvmptg 6189	. . . . . . . . . . . 12 ⊢ (((𝑣‘𝑖) ∈ 𝑅 ∧ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ∈ V) → (𝐺‘(𝑣‘𝑖)) = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
223	215, 218, 222	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘(𝑣‘𝑖)) = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
224	214, 223	eqtrd 2644	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺 ∘ 𝑣)‘𝑖) = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
225	224	adantlr 747	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺 ∘ 𝑣)‘𝑖) = {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
226	225	eleq2d 2673	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) ∈ ((𝐺 ∘ 𝑣)‘𝑖) ↔ ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) ∈ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))}))
227		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎℎ𝑣
228	24, 227	nfco 5209	. . . . . . . . . . . . 13 ⊢ Ⅎℎ(𝐺 ∘ 𝑣)
229	20, 228	nfco 5209	. . . . . . . . . . . 12 ⊢ Ⅎℎ(𝑙 ∘ (𝐺 ∘ 𝑣))
230		nfcv 2751	. . . . . . . . . . . 12 ⊢ Ⅎℎ𝑖
231	229, 230	nffv 6110	. . . . . . . . . . 11 ⊢ Ⅎℎ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)
232		nfcv 2751	. . . . . . . . . . 11 ⊢ Ⅎℎ𝐴
233		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎℎ𝑇
234		nfcv 2751	. . . . . . . . . . . . . . 15 ⊢ Ⅎℎ0
235		nfcv 2751	. . . . . . . . . . . . . . 15 ⊢ Ⅎℎ ≤
236		nfcv 2751	. . . . . . . . . . . . . . . 16 ⊢ Ⅎℎ𝑡
237	231, 236	nffv 6110	. . . . . . . . . . . . . . 15 ⊢ Ⅎℎ(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡)
238	234, 235, 237	nfbr 4629	. . . . . . . . . . . . . 14 ⊢ Ⅎℎ0 ≤ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡)
239		nfcv 2751	. . . . . . . . . . . . . . 15 ⊢ Ⅎℎ1
240	237, 235, 239	nfbr 4629	. . . . . . . . . . . . . 14 ⊢ Ⅎℎ(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ≤ 1
241	238, 240	nfan 1816	. . . . . . . . . . . . 13 ⊢ Ⅎℎ(0 ≤ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ≤ 1)
242	233, 241	nfral 2929	. . . . . . . . . . . 12 ⊢ Ⅎℎ∀𝑡 ∈ 𝑇 (0 ≤ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ≤ 1)
243		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎℎ(𝑣‘𝑖)
244		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎℎ <
245		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎℎ(𝐸 / 𝑀)
246	237, 244, 245	nfbr 4629	. . . . . . . . . . . . 13 ⊢ Ⅎℎ(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)
247	243, 246	nfral 2929	. . . . . . . . . . . 12 ⊢ Ⅎℎ∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)
248		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎℎ(𝑇 ∖ 𝑈)
249		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎℎ(1 − (𝐸 / 𝑀))
250	249, 244, 237	nfbr 4629	. . . . . . . . . . . . 13 ⊢ Ⅎℎ(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡)
251	248, 250	nfral 2929	. . . . . . . . . . . 12 ⊢ Ⅎℎ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡)
252	242, 247, 251	nf3an 1819	. . . . . . . . . . 11 ⊢ Ⅎℎ(∀𝑡 ∈ 𝑇 (0 ≤ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡))
253		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡ℎ
254		nfcv 2751	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡𝑙
255		nfcv 2751	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑡𝑅
256		nfra1 2925	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
257		nfra1 2925	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀)
258		nfra1 2925	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡)
259	256, 257, 258	nf3an 1819	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑡(∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))
260		nfcv 2751	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑡𝐴
261	259, 260	nfrab 3100	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))}
262	255, 261	nfmpt 4674	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡(𝑤 ∈ 𝑅 ↦ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))})
263	5, 262	nfcxfr 2749	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡𝐺
264		nfcv 2751	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡𝑣
265	263, 264	nfco 5209	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡(𝐺 ∘ 𝑣)
266	254, 265	nfco 5209	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡(𝑙 ∘ (𝐺 ∘ 𝑣))
267		nfcv 2751	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡𝑖
268	266, 267	nffv 6110	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)
269	253, 268	nfeq 2762	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡 ℎ = ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)
270		fveq1 6102	. . . . . . . . . . . . . . 15 ⊢ (ℎ = ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) → (ℎ‘𝑡) = (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡))
271	270	breq2d 4595	. . . . . . . . . . . . . 14 ⊢ (ℎ = ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡)))
272	270	breq1d 4593	. . . . . . . . . . . . . 14 ⊢ (ℎ = ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) → ((ℎ‘𝑡) ≤ 1 ↔ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ≤ 1))
273	271, 272	anbi12d 743	. . . . . . . . . . . . 13 ⊢ (ℎ = ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ≤ 1)))
274	269, 273	ralbid 2966	. . . . . . . . . . . 12 ⊢ (ℎ = ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ≤ 1)))
275	270	breq1d 4593	. . . . . . . . . . . . 13 ⊢ (ℎ = ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) → ((ℎ‘𝑡) < (𝐸 / 𝑀) ↔ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
276	269, 275	ralbid 2966	. . . . . . . . . . . 12 ⊢ (ℎ = ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) → (∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀) ↔ ∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
277	270	breq2d 4595	. . . . . . . . . . . . 13 ⊢ (ℎ = ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) → ((1 − (𝐸 / 𝑀)) < (ℎ‘𝑡) ↔ (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡)))
278	269, 277	ralbid 2966	. . . . . . . . . . . 12 ⊢ (ℎ = ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) → (∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡)))
279	274, 276, 278	3anbi123d 1391	. . . . . . . . . . 11 ⊢ (ℎ = ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) → ((∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡)) ↔ (∀𝑡 ∈ 𝑇 (0 ≤ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡))))
280	231, 232, 252, 279	elrabf 3329	. . . . . . . . . 10 ⊢ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) ∈ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} ↔ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡))))
281	280	simprbi 479	. . . . . . . . 9 ⊢ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) ∈ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} → (∀𝑡 ∈ 𝑇 (0 ≤ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡)))
282	281	simp2d 1067	. . . . . . . 8 ⊢ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) ∈ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} → ∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀))
283	226, 282	syl6bi 242	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) ∈ ((𝐺 ∘ 𝑣)‘𝑖) → ∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
284	212, 283	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀))
285		stoweidlem31.2	. . . . . . . . 9 ⊢ Ⅎ𝑡𝜑
286	263	nfrn 5289	. . . . . . . . . . 11 ⊢ Ⅎ𝑡ran 𝐺
287	254, 286	nffn 5901	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑙 Fn ran 𝐺
288		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏)
289	286, 288	nfral 2929	. . . . . . . . . 10 ⊢ Ⅎ𝑡∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏)
290	287, 289	nfan 1816	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))
291	285, 290	nfan 1816	. . . . . . . 8 ⊢ Ⅎ𝑡(𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏)))
292		nfv 1830	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑖 ∈ (1...𝑀)
293	291, 292	nfan 1816	. . . . . . 7 ⊢ Ⅎ𝑡((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀))
294		stoweidlem31.11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ⊆ (𝑇 ∖ 𝑈))
295	294	ad3antrrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝐵) → 𝐵 ⊆ (𝑇 ∖ 𝑈))
296		simpr 476	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝐵) → 𝑡 ∈ 𝐵)
297	295, 296	sseldd 3569	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝐵) → 𝑡 ∈ (𝑇 ∖ 𝑈))
298	281	simp3d 1068	. . . . . . . . . . . 12 ⊢ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) ∈ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣‘𝑖)(ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))} → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡))
299	226, 298	syl6bi 242	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖) ∈ ((𝐺 ∘ 𝑣)‘𝑖) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡)))
300	212, 299	mpd 15	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡))
301	300	r19.21bi 2916	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡))
302	297, 301	syldan 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝐵) → (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡))
303	302	ex 449	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → (𝑡 ∈ 𝐵 → (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡)))
304	293, 303	ralrimi 2940	. . . . . 6 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡))
305	284, 304	jca 553	. . . . 5 ⊢ (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → (∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡)))
306	305	ralrimiva 2949	. . . 4 ⊢ ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) → ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡)))
307	192, 306	jca 553	. . 3 ⊢ ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) → ((𝑙 ∘ (𝐺 ∘ 𝑣)):(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡))))
308		feq1 5939	. . . . 5 ⊢ (𝑥 = (𝑙 ∘ (𝐺 ∘ 𝑣)) → (𝑥:(1...𝑀)⟶𝑌 ↔ (𝑙 ∘ (𝐺 ∘ 𝑣)):(1...𝑀)⟶𝑌))
309		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑥
310	309, 266	nfeq 2762	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑥 = (𝑙 ∘ (𝐺 ∘ 𝑣))
311		fveq1 6102	. . . . . . . . . 10 ⊢ (𝑥 = (𝑙 ∘ (𝐺 ∘ 𝑣)) → (𝑥‘𝑖) = ((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖))
312	311	fveq1d 6105	. . . . . . . . 9 ⊢ (𝑥 = (𝑙 ∘ (𝐺 ∘ 𝑣)) → ((𝑥‘𝑖)‘𝑡) = (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡))
313	312	breq1d 4593	. . . . . . . 8 ⊢ (𝑥 = (𝑙 ∘ (𝐺 ∘ 𝑣)) → (((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑀) ↔ (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
314	310, 313	ralbid 2966	. . . . . . 7 ⊢ (𝑥 = (𝑙 ∘ (𝐺 ∘ 𝑣)) → (∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑀) ↔ ∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
315	312	breq2d 4595	. . . . . . . 8 ⊢ (𝑥 = (𝑙 ∘ (𝐺 ∘ 𝑣)) → ((1 − (𝐸 / 𝑀)) < ((𝑥‘𝑖)‘𝑡) ↔ (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡)))
316	310, 315	ralbid 2966	. . . . . . 7 ⊢ (𝑥 = (𝑙 ∘ (𝐺 ∘ 𝑣)) → (∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥‘𝑖)‘𝑡) ↔ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡)))
317	314, 316	anbi12d 743	. . . . . 6 ⊢ (𝑥 = (𝑙 ∘ (𝐺 ∘ 𝑣)) → ((∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥‘𝑖)‘𝑡)) ↔ (∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡))))
318	317	ralbidv 2969	. . . . 5 ⊢ (𝑥 = (𝑙 ∘ (𝐺 ∘ 𝑣)) → (∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥‘𝑖)‘𝑡)) ↔ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡))))
319	308, 318	anbi12d 743	. . . 4 ⊢ (𝑥 = (𝑙 ∘ (𝐺 ∘ 𝑣)) → ((𝑥:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥‘𝑖)‘𝑡))) ↔ ((𝑙 ∘ (𝐺 ∘ 𝑣)):(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡)))))
320	319	spcegv 3267	. . 3 ⊢ ((𝑙 ∘ (𝐺 ∘ 𝑣)) ∈ V → (((𝑙 ∘ (𝐺 ∘ 𝑣)):(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣‘𝑖)(((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺 ∘ 𝑣))‘𝑖)‘𝑡))) → ∃𝑥(𝑥:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥‘𝑖)‘𝑡)))))
321	17, 307, 320	sylc 63	. 2 ⊢ ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙‘𝑏) ∈ 𝑏))) → ∃𝑥(𝑥:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥‘𝑖)‘𝑡))))
322	3, 321	exlimddv 1850	1 ⊢ (𝜑 → ∃𝑥(𝑥:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥‘𝑖)‘𝑡))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 Ⅎwnf 1699 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 0cc0 9815 1c1 9816 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 ℝ⁺crp 11708 ...cfz 12197

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-rp 11709

This theorem is referenced by: stoweidlem39 38932

W3C validator