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Theorem stoweidlem31 44262
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 ≤ xi ≤ 1, xi < ε / m on V(ti), and xi > 1 - ε / m on 𝐵. Here M is used to represent m in the paper, 𝐸 is used to represent ε in the paper, vi is used to represent V(ti). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem31.1 𝜑
stoweidlem31.2 𝑡𝜑
stoweidlem31.3 𝑤𝜑
stoweidlem31.4 𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
stoweidlem31.5 𝑉 = {𝑤𝐽 ∣ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))}
stoweidlem31.6 𝐺 = (𝑤𝑅 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
stoweidlem31.7 (𝜑𝑅𝑉)
stoweidlem31.8 (𝜑𝑀 ∈ ℕ)
stoweidlem31.9 (𝜑𝑣:(1...𝑀)–1-1-onto𝑅)
stoweidlem31.10 (𝜑𝐸 ∈ ℝ+)
stoweidlem31.11 (𝜑𝐵 ⊆ (𝑇𝑈))
stoweidlem31.12 (𝜑𝑉 ∈ V)
stoweidlem31.13 (𝜑𝐴 ∈ V)
stoweidlem31.14 (𝜑 → ran 𝐺 ∈ Fin)
Assertion
Ref Expression
stoweidlem31 (𝜑 → ∃𝑥(𝑥:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥𝑖)‘𝑡))))
Distinct variable groups:   ,𝑖,𝑡,𝑣,𝑤   𝑖,𝐺   𝑤,𝑌   𝜑,𝑖   𝑒,,𝑡,𝑤,𝐴   𝑒,𝐸,,𝑡,𝑤   𝑒,𝑀,,𝑡,𝑤   𝑇,𝑒,,𝑤   𝑈,𝑒,,𝑤   𝑅,,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,)   𝐴(𝑥,𝑣,𝑖)   𝐵(𝑤,𝑣,𝑡,𝑒,,𝑖)   𝑅(𝑥,𝑣,𝑒,𝑖)   𝑇(𝑥,𝑣,𝑡,𝑖)   𝑈(𝑥,𝑣,𝑡,𝑖)   𝐸(𝑣,𝑖)   𝐺(𝑤,𝑣,𝑡,𝑒,)   𝐽(𝑥,𝑤,𝑣,𝑡,𝑒,,𝑖)   𝑀(𝑣)   𝑉(𝑥,𝑤,𝑣,𝑡,𝑒,,𝑖)   𝑌(𝑣,𝑡,𝑒,,𝑖)

Proof of Theorem stoweidlem31
Dummy variables 𝑏 𝑙 𝑢 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 stoweidlem31.14 . . 3 (𝜑 → ran 𝐺 ∈ Fin)
2 fnchoice 43224 . . 3 (ran 𝐺 ∈ Fin → ∃𝑙(𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)))
31, 2syl 17 . 2 (𝜑 → ∃𝑙(𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)))
4 vex 3449 . . . . 5 𝑙 ∈ V
5 stoweidlem31.6 . . . . . . 7 𝐺 = (𝑤𝑅 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
6 stoweidlem31.12 . . . . . . . . 9 (𝜑𝑉 ∈ V)
7 stoweidlem31.7 . . . . . . . . 9 (𝜑𝑅𝑉)
86, 7ssexd 5281 . . . . . . . 8 (𝜑𝑅 ∈ V)
9 mptexg 7171 . . . . . . . 8 (𝑅 ∈ V → (𝑤𝑅 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}) ∈ V)
108, 9syl 17 . . . . . . 7 (𝜑 → (𝑤𝑅 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}) ∈ V)
115, 10eqeltrid 2842 . . . . . 6 (𝜑𝐺 ∈ V)
12 vex 3449 . . . . . 6 𝑣 ∈ V
13 coexg 7866 . . . . . 6 ((𝐺 ∈ V ∧ 𝑣 ∈ V) → (𝐺𝑣) ∈ V)
1411, 12, 13sylancl 586 . . . . 5 (𝜑 → (𝐺𝑣) ∈ V)
15 coexg 7866 . . . . 5 ((𝑙 ∈ V ∧ (𝐺𝑣) ∈ V) → (𝑙 ∘ (𝐺𝑣)) ∈ V)
164, 14, 15sylancr 587 . . . 4 (𝜑 → (𝑙 ∘ (𝐺𝑣)) ∈ V)
1716adantr 481 . . 3 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → (𝑙 ∘ (𝐺𝑣)) ∈ V)
18 simprl 769 . . . . . 6 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → 𝑙 Fn ran 𝐺)
19 stoweidlem31.1 . . . . . . . . 9 𝜑
20 nfcv 2907 . . . . . . . . . . 11 𝑙
21 nfcv 2907 . . . . . . . . . . . . . 14 𝑅
22 nfrab1 3426 . . . . . . . . . . . . . 14 {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}
2321, 22nfmpt 5212 . . . . . . . . . . . . 13 (𝑤𝑅 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
245, 23nfcxfr 2905 . . . . . . . . . . . 12 𝐺
2524nfrn 5907 . . . . . . . . . . 11 ran 𝐺
2620, 25nffn 6601 . . . . . . . . . 10 𝑙 Fn ran 𝐺
27 nfv 1917 . . . . . . . . . . 11 (𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)
2825, 27nfralw 3294 . . . . . . . . . 10 𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)
2926, 28nfan 1902 . . . . . . . . 9 (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
3019, 29nfan 1902 . . . . . . . 8 (𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)))
31 fvelrnb 6903 . . . . . . . . . . . . 13 (𝑙 Fn ran 𝐺 → ( ∈ ran 𝑙 ↔ ∃𝑏 ∈ ran 𝐺(𝑙𝑏) = ))
3218, 31syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → ( ∈ ran 𝑙 ↔ ∃𝑏 ∈ ran 𝐺(𝑙𝑏) = ))
3332biimpa 477 . . . . . . . . . . 11 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) → ∃𝑏 ∈ ran 𝐺(𝑙𝑏) = )
34 nfv 1917 . . . . . . . . . . . . . 14 𝑏𝜑
35 nfv 1917 . . . . . . . . . . . . . . 15 𝑏 𝑙 Fn ran 𝐺
36 nfra1 3267 . . . . . . . . . . . . . . 15 𝑏𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)
3735, 36nfan 1902 . . . . . . . . . . . . . 14 𝑏(𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
3834, 37nfan 1902 . . . . . . . . . . . . 13 𝑏(𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)))
39 nfv 1917 . . . . . . . . . . . . 13 𝑏 ∈ ran 𝑙
4038, 39nfan 1902 . . . . . . . . . . . 12 𝑏((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙)
41 simp3 1138 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) = ) → (𝑙𝑏) = )
42 simp1ll 1236 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) = ) → 𝜑)
43 simplrr 776 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) → ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
44433ad2ant1 1133 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) = ) → ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
45 simp2 1137 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) = ) → 𝑏 ∈ ran 𝐺)
46 simp3 1138 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → 𝑏 ∈ ran 𝐺)
47 3simpc 1150 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → (∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺))
48 simpr 485 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑏 ∈ ran 𝐺) → 𝑏 ∈ ran 𝐺)
49 stoweidlem31.3 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑤𝜑
50 stoweidlem31.13 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐴 ∈ V)
51 rabexg 5288 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴 ∈ V → {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ∈ V)
5250, 51syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ∈ V)
5352a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑤𝑅 → {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ∈ V))
5449, 53ralrimi 3240 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ∀𝑤𝑅 {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ∈ V)
555fnmpt 6641 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑤𝑅 {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ∈ V → 𝐺 Fn 𝑅)
5654, 55syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐺 Fn 𝑅)
5756adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑏 ∈ ran 𝐺) → 𝐺 Fn 𝑅)
58 fvelrnb 6903 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 Fn 𝑅 → (𝑏 ∈ ran 𝐺 ↔ ∃𝑢𝑅 (𝐺𝑢) = 𝑏))
59 nfmpt1 5213 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑤(𝑤𝑅 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
605, 59nfcxfr 2905 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑤𝐺
61 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑤𝑢
6260, 61nffv 6852 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑤(𝐺𝑢)
63 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑤𝑏
6462, 63nfeq 2920 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑤(𝐺𝑢) = 𝑏
65 nfv 1917 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑢(𝐺𝑤) = 𝑏
66 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑢 = 𝑤 → (𝐺𝑢) = (𝐺𝑤))
6766eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢 = 𝑤 → ((𝐺𝑢) = 𝑏 ↔ (𝐺𝑤) = 𝑏))
6864, 65, 67cbvrexw 3290 . . . . . . . . . . . . . . . . . . . . . . 23 (∃𝑢𝑅 (𝐺𝑢) = 𝑏 ↔ ∃𝑤𝑅 (𝐺𝑤) = 𝑏)
6958, 68bitrdi 286 . . . . . . . . . . . . . . . . . . . . . 22 (𝐺 Fn 𝑅 → (𝑏 ∈ ran 𝐺 ↔ ∃𝑤𝑅 (𝐺𝑤) = 𝑏))
7057, 69syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑏 ∈ ran 𝐺) → (𝑏 ∈ ran 𝐺 ↔ ∃𝑤𝑅 (𝐺𝑤) = 𝑏))
7148, 70mpbid 231 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑏 ∈ ran 𝐺) → ∃𝑤𝑅 (𝐺𝑤) = 𝑏)
7260nfrn 5907 . . . . . . . . . . . . . . . . . . . . . . 23 𝑤ran 𝐺
7372nfcri 2894 . . . . . . . . . . . . . . . . . . . . . 22 𝑤 𝑏 ∈ ran 𝐺
7449, 73nfan 1902 . . . . . . . . . . . . . . . . . . . . 21 𝑤(𝜑𝑏 ∈ ran 𝐺)
75 nfv 1917 . . . . . . . . . . . . . . . . . . . . 21 𝑤 𝑏 ≠ ∅
76 simp3 1138 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑤𝑅 ∧ (𝐺𝑤) = 𝑏) → (𝐺𝑤) = 𝑏)
77 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑤𝑅) → 𝑤𝑅)
7850adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑤𝑅) → 𝐴 ∈ V)
7978, 51syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑤𝑅) → {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ∈ V)
805fvmpt2 6959 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑤𝑅 ∧ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ∈ V) → (𝐺𝑤) = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
8177, 79, 80syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑤𝑅) → (𝐺𝑤) = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
827sselda 3944 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑤𝑅) → 𝑤𝑉)
83 stoweidlem31.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 𝑉 = {𝑤𝐽 ∣ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))}
8483reqabi 3429 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤𝑉 ↔ (𝑤𝐽 ∧ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))))
8582, 84sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑤𝑅) → (𝑤𝐽 ∧ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))))
8685simprd 496 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑤𝑅) → ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡)))
87 stoweidlem31.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑𝐸 ∈ ℝ+)
88 stoweidlem31.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑𝑀 ∈ ℕ)
8988nnrpd 12955 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑𝑀 ∈ ℝ+)
9087, 89rpdivcld 12974 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (𝐸 / 𝑀) ∈ ℝ+)
9190adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑤𝑅) → (𝐸 / 𝑀) ∈ ℝ+)
92 breq2 5109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 = (𝐸 / 𝑀) → ((𝑡) < 𝑒 ↔ (𝑡) < (𝐸 / 𝑀)))
9392ralbidv 3174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑒 = (𝐸 / 𝑀) → (∀𝑡𝑤 (𝑡) < 𝑒 ↔ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀)))
94 oveq2 7365 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = (𝐸 / 𝑀) → (1 − 𝑒) = (1 − (𝐸 / 𝑀)))
9594breq1d 5115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 = (𝐸 / 𝑀) → ((1 − 𝑒) < (𝑡) ↔ (1 − (𝐸 / 𝑀)) < (𝑡)))
9695ralbidv 3174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑒 = (𝐸 / 𝑀) → (∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡)))
9793, 963anbi23d 1439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑒 = (𝐸 / 𝑀) → ((∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡)) ↔ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))))
9897rexbidv 3175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑒 = (𝐸 / 𝑀) → (∃𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡)) ↔ ∃𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))))
9998rspccva 3580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡)) ∧ (𝐸 / 𝑀) ∈ ℝ+) → ∃𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡)))
10086, 91, 99syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑤𝑅) → ∃𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡)))
101 nfv 1917 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑤𝑅
10219, 101nfan 1902 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑤𝑅)
103 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
10422, 103nfne 3045 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ≠ ∅
105 3simpc 1150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑤𝑅) ∧ 𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))) → (𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))))
106 rabid 3427 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ( ∈ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ↔ (𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))))
107105, 106sylibr 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑤𝑅) ∧ 𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))) → ∈ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
108 ne0i 4294 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ( ∈ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} → {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ≠ ∅)
109107, 108syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑤𝑅) ∧ 𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))) → {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ≠ ∅)
1101093exp 1119 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑤𝑅) → (𝐴 → ((∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡)) → {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ≠ ∅)))
111102, 104, 110rexlimd 3249 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑤𝑅) → (∃𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡)) → {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ≠ ∅))
112100, 111mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑤𝑅) → {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ≠ ∅)
11381, 112eqnetrd 3011 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑤𝑅) → (𝐺𝑤) ≠ ∅)
1141133adant3 1132 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑤𝑅 ∧ (𝐺𝑤) = 𝑏) → (𝐺𝑤) ≠ ∅)
11576, 114eqnetrrd 3012 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤𝑅 ∧ (𝐺𝑤) = 𝑏) → 𝑏 ≠ ∅)
1161153adant1r 1177 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑏 ∈ ran 𝐺) ∧ 𝑤𝑅 ∧ (𝐺𝑤) = 𝑏) → 𝑏 ≠ ∅)
1171163exp 1119 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑏 ∈ ran 𝐺) → (𝑤𝑅 → ((𝐺𝑤) = 𝑏𝑏 ≠ ∅)))
11874, 75, 117rexlimd 3249 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑏 ∈ ran 𝐺) → (∃𝑤𝑅 (𝐺𝑤) = 𝑏𝑏 ≠ ∅))
11971, 118mpd 15 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑏 ∈ ran 𝐺) → 𝑏 ≠ ∅)
1201193adant2 1131 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → 𝑏 ≠ ∅)
121 rspa 3231 . . . . . . . . . . . . . . . . . 18 ((∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → (𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
12247, 120, 121sylc 65 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → (𝑙𝑏) ∈ 𝑏)
12346, 122jca 512 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → (𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) ∈ 𝑏))
124 vex 3449 . . . . . . . . . . . . . . . . . 18 𝑏 ∈ V
1255elrnmpt 5911 . . . . . . . . . . . . . . . . . 18 (𝑏 ∈ V → (𝑏 ∈ ran 𝐺 ↔ ∃𝑤𝑅 𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}))
126124, 125ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ ran 𝐺 ↔ ∃𝑤𝑅 𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
12746, 126sylib 217 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → ∃𝑤𝑅 𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
128 nfv 1917 . . . . . . . . . . . . . . . . . 18 𝑤(𝑙𝑏) ∈ 𝑏
12973, 128nfan 1902 . . . . . . . . . . . . . . . . 17 𝑤(𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) ∈ 𝑏)
130 nfv 1917 . . . . . . . . . . . . . . . . 17 𝑤(𝑙𝑏) ∈ 𝑌
131 simp1r 1198 . . . . . . . . . . . . . . . . . . 19 (((𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) ∈ 𝑏) ∧ 𝑤𝑅𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}) → (𝑙𝑏) ∈ 𝑏)
132 simp3 1138 . . . . . . . . . . . . . . . . . . 19 (((𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) ∈ 𝑏) ∧ 𝑤𝑅𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}) → 𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
133 simpl 483 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑙𝑏) ∈ 𝑏𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}) → (𝑙𝑏) ∈ 𝑏)
134 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑙𝑏) ∈ 𝑏𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}) → 𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
135133, 134eleqtrd 2840 . . . . . . . . . . . . . . . . . . . . 21 (((𝑙𝑏) ∈ 𝑏𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}) → (𝑙𝑏) ∈ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
136 elrabi 3639 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑙𝑏) ∈ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} → (𝑙𝑏) ∈ 𝐴)
137 fveq1 6841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ( = (𝑙𝑏) → (𝑡) = ((𝑙𝑏)‘𝑡))
138137breq2d 5117 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( = (𝑙𝑏) → (0 ≤ (𝑡) ↔ 0 ≤ ((𝑙𝑏)‘𝑡)))
139137breq1d 5115 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( = (𝑙𝑏) → ((𝑡) ≤ 1 ↔ ((𝑙𝑏)‘𝑡) ≤ 1))
140138, 139anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ( = (𝑙𝑏) → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ ((𝑙𝑏)‘𝑡) ∧ ((𝑙𝑏)‘𝑡) ≤ 1)))
141140ralbidv 3174 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( = (𝑙𝑏) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝑙𝑏)‘𝑡) ∧ ((𝑙𝑏)‘𝑡) ≤ 1)))
142137breq1d 5115 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ( = (𝑙𝑏) → ((𝑡) < (𝐸 / 𝑀) ↔ ((𝑙𝑏)‘𝑡) < (𝐸 / 𝑀)))
143142ralbidv 3174 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( = (𝑙𝑏) → (∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ↔ ∀𝑡𝑤 ((𝑙𝑏)‘𝑡) < (𝐸 / 𝑀)))
144137breq2d 5117 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ( = (𝑙𝑏) → ((1 − (𝐸 / 𝑀)) < (𝑡) ↔ (1 − (𝐸 / 𝑀)) < ((𝑙𝑏)‘𝑡)))
145144ralbidv 3174 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( = (𝑙𝑏) → (∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < ((𝑙𝑏)‘𝑡)))
146141, 143, 1453anbi123d 1436 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( = (𝑙𝑏) → ((∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡)) ↔ (∀𝑡𝑇 (0 ≤ ((𝑙𝑏)‘𝑡) ∧ ((𝑙𝑏)‘𝑡) ≤ 1) ∧ ∀𝑡𝑤 ((𝑙𝑏)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < ((𝑙𝑏)‘𝑡))))
147146elrab 3645 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑙𝑏) ∈ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ↔ ((𝑙𝑏) ∈ 𝐴 ∧ (∀𝑡𝑇 (0 ≤ ((𝑙𝑏)‘𝑡) ∧ ((𝑙𝑏)‘𝑡) ≤ 1) ∧ ∀𝑡𝑤 ((𝑙𝑏)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < ((𝑙𝑏)‘𝑡))))
148147simprbi 497 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑙𝑏) ∈ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} → (∀𝑡𝑇 (0 ≤ ((𝑙𝑏)‘𝑡) ∧ ((𝑙𝑏)‘𝑡) ≤ 1) ∧ ∀𝑡𝑤 ((𝑙𝑏)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < ((𝑙𝑏)‘𝑡)))
149148simp1d 1142 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑙𝑏) ∈ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} → ∀𝑡𝑇 (0 ≤ ((𝑙𝑏)‘𝑡) ∧ ((𝑙𝑏)‘𝑡) ≤ 1))
150141elrab 3645 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑙𝑏) ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)} ↔ ((𝑙𝑏) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ ((𝑙𝑏)‘𝑡) ∧ ((𝑙𝑏)‘𝑡) ≤ 1)))
151136, 149, 150sylanbrc 583 . . . . . . . . . . . . . . . . . . . . 21 ((𝑙𝑏) ∈ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} → (𝑙𝑏) ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)})
152135, 151syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑙𝑏) ∈ 𝑏𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}) → (𝑙𝑏) ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)})
153 stoweidlem31.4 . . . . . . . . . . . . . . . . . . . 20 𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
154152, 153eleqtrrdi 2849 . . . . . . . . . . . . . . . . . . 19 (((𝑙𝑏) ∈ 𝑏𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}) → (𝑙𝑏) ∈ 𝑌)
155131, 132, 154syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) ∈ 𝑏) ∧ 𝑤𝑅𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}) → (𝑙𝑏) ∈ 𝑌)
1561553exp 1119 . . . . . . . . . . . . . . . . 17 ((𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) ∈ 𝑏) → (𝑤𝑅 → (𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} → (𝑙𝑏) ∈ 𝑌)))
157129, 130, 156rexlimd 3249 . . . . . . . . . . . . . . . 16 ((𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) ∈ 𝑏) → (∃𝑤𝑅 𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} → (𝑙𝑏) ∈ 𝑌))
158123, 127, 157sylc 65 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → (𝑙𝑏) ∈ 𝑌)
15942, 44, 45, 158syl3anc 1371 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) = ) → (𝑙𝑏) ∈ 𝑌)
16041, 159eqeltrrd 2839 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) = ) → 𝑌)
1611603exp 1119 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) → (𝑏 ∈ ran 𝐺 → ((𝑙𝑏) = 𝑌)))
16240, 161reximdai 3244 . . . . . . . . . . 11 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) → (∃𝑏 ∈ ran 𝐺(𝑙𝑏) = → ∃𝑏 ∈ ran 𝐺 𝑌))
16333, 162mpd 15 . . . . . . . . . 10 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) → ∃𝑏 ∈ ran 𝐺 𝑌)
164 nfv 1917 . . . . . . . . . . 11 𝑏 𝑌
165 idd 24 . . . . . . . . . . . 12 (𝑏 ∈ ran 𝐺 → (𝑌𝑌))
166165a1i 11 . . . . . . . . . . 11 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) → (𝑏 ∈ ran 𝐺 → (𝑌𝑌)))
16740, 164, 166rexlimd 3249 . . . . . . . . . 10 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) → (∃𝑏 ∈ ran 𝐺 𝑌𝑌))
168163, 167mpd 15 . . . . . . . . 9 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) → 𝑌)
169168ex 413 . . . . . . . 8 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → ( ∈ ran 𝑙𝑌))
17030, 169ralrimi 3240 . . . . . . 7 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → ∀ ∈ ran 𝑙 𝑌)
171 dfss3 3932 . . . . . . . 8 (ran 𝑙𝑌 ↔ ∀𝑧 ∈ ran 𝑙 𝑧𝑌)
172 nfrab1 3426 . . . . . . . . . . 11 {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
173153, 172nfcxfr 2905 . . . . . . . . . 10 𝑌
174173nfcri 2894 . . . . . . . . 9 𝑧𝑌
175 nfv 1917 . . . . . . . . 9 𝑧 𝑌
176 eleq1 2825 . . . . . . . . 9 (𝑧 = → (𝑧𝑌𝑌))
177174, 175, 176cbvralw 3289 . . . . . . . 8 (∀𝑧 ∈ ran 𝑙 𝑧𝑌 ↔ ∀ ∈ ran 𝑙 𝑌)
178171, 177bitri 274 . . . . . . 7 (ran 𝑙𝑌 ↔ ∀ ∈ ran 𝑙 𝑌)
179170, 178sylibr 233 . . . . . 6 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → ran 𝑙𝑌)
180 df-f 6500 . . . . . 6 (𝑙:ran 𝐺𝑌 ↔ (𝑙 Fn ran 𝐺 ∧ ran 𝑙𝑌))
18118, 179, 180sylanbrc 583 . . . . 5 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → 𝑙:ran 𝐺𝑌)
182 dffn3 6681 . . . . . . . 8 (𝐺 Fn 𝑅𝐺:𝑅⟶ran 𝐺)
18356, 182sylib 217 . . . . . . 7 (𝜑𝐺:𝑅⟶ran 𝐺)
184183adantr 481 . . . . . 6 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → 𝐺:𝑅⟶ran 𝐺)
185 stoweidlem31.9 . . . . . . . 8 (𝜑𝑣:(1...𝑀)–1-1-onto𝑅)
186 f1of 6784 . . . . . . . 8 (𝑣:(1...𝑀)–1-1-onto𝑅𝑣:(1...𝑀)⟶𝑅)
187185, 186syl 17 . . . . . . 7 (𝜑𝑣:(1...𝑀)⟶𝑅)
188187adantr 481 . . . . . 6 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → 𝑣:(1...𝑀)⟶𝑅)
189 fco 6692 . . . . . 6 ((𝐺:𝑅⟶ran 𝐺𝑣:(1...𝑀)⟶𝑅) → (𝐺𝑣):(1...𝑀)⟶ran 𝐺)
190184, 188, 189syl2anc 584 . . . . 5 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → (𝐺𝑣):(1...𝑀)⟶ran 𝐺)
191 fco 6692 . . . . 5 ((𝑙:ran 𝐺𝑌 ∧ (𝐺𝑣):(1...𝑀)⟶ran 𝐺) → (𝑙 ∘ (𝐺𝑣)):(1...𝑀)⟶𝑌)
192181, 190, 191syl2anc 584 . . . 4 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → (𝑙 ∘ (𝐺𝑣)):(1...𝑀)⟶𝑌)
193 fvco3 6940 . . . . . . . . 9 (((𝐺𝑣):(1...𝑀)⟶ran 𝐺𝑖 ∈ (1...𝑀)) → ((𝑙 ∘ (𝐺𝑣))‘𝑖) = (𝑙‘((𝐺𝑣)‘𝑖)))
194190, 193sylan 580 . . . . . . . 8 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑙 ∘ (𝐺𝑣))‘𝑖) = (𝑙‘((𝐺𝑣)‘𝑖)))
195 simpll 765 . . . . . . . . 9 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
196 simplrr 776 . . . . . . . . 9 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
197190ffvelcdmda 7035 . . . . . . . . 9 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺𝑣)‘𝑖) ∈ ran 𝐺)
198 simp3 1138 . . . . . . . . . 10 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ ((𝐺𝑣)‘𝑖) ∈ ran 𝐺) → ((𝐺𝑣)‘𝑖) ∈ ran 𝐺)
199 nfv 1917 . . . . . . . . . . . . 13 𝑏((𝐺𝑣)‘𝑖) ∈ ran 𝐺
20034, 36, 199nf3an 1904 . . . . . . . . . . . 12 𝑏(𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ ((𝐺𝑣)‘𝑖) ∈ ran 𝐺)
201 nfv 1917 . . . . . . . . . . . 12 𝑏(𝑙‘((𝐺𝑣)‘𝑖)) ∈ ((𝐺𝑣)‘𝑖)
202200, 201nfim 1899 . . . . . . . . . . 11 𝑏((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ ((𝐺𝑣)‘𝑖) ∈ ran 𝐺) → (𝑙‘((𝐺𝑣)‘𝑖)) ∈ ((𝐺𝑣)‘𝑖))
203 eleq1 2825 . . . . . . . . . . . . 13 (𝑏 = ((𝐺𝑣)‘𝑖) → (𝑏 ∈ ran 𝐺 ↔ ((𝐺𝑣)‘𝑖) ∈ ran 𝐺))
2042033anbi3d 1442 . . . . . . . . . . . 12 (𝑏 = ((𝐺𝑣)‘𝑖) → ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) ↔ (𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ ((𝐺𝑣)‘𝑖) ∈ ran 𝐺)))
205 fveq2 6842 . . . . . . . . . . . . 13 (𝑏 = ((𝐺𝑣)‘𝑖) → (𝑙𝑏) = (𝑙‘((𝐺𝑣)‘𝑖)))
206 id 22 . . . . . . . . . . . . 13 (𝑏 = ((𝐺𝑣)‘𝑖) → 𝑏 = ((𝐺𝑣)‘𝑖))
207205, 206eleq12d 2832 . . . . . . . . . . . 12 (𝑏 = ((𝐺𝑣)‘𝑖) → ((𝑙𝑏) ∈ 𝑏 ↔ (𝑙‘((𝐺𝑣)‘𝑖)) ∈ ((𝐺𝑣)‘𝑖)))
208204, 207imbi12d 344 . . . . . . . . . . 11 (𝑏 = ((𝐺𝑣)‘𝑖) → (((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → (𝑙𝑏) ∈ 𝑏) ↔ ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ ((𝐺𝑣)‘𝑖) ∈ ran 𝐺) → (𝑙‘((𝐺𝑣)‘𝑖)) ∈ ((𝐺𝑣)‘𝑖))))
209202, 208, 122vtoclg1f 3524 . . . . . . . . . 10 (((𝐺𝑣)‘𝑖) ∈ ran 𝐺 → ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ ((𝐺𝑣)‘𝑖) ∈ ran 𝐺) → (𝑙‘((𝐺𝑣)‘𝑖)) ∈ ((𝐺𝑣)‘𝑖)))
210198, 209mpcom 38 . . . . . . . . 9 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ ((𝐺𝑣)‘𝑖) ∈ ran 𝐺) → (𝑙‘((𝐺𝑣)‘𝑖)) ∈ ((𝐺𝑣)‘𝑖))
211195, 196, 197, 210syl3anc 1371 . . . . . . . 8 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → (𝑙‘((𝐺𝑣)‘𝑖)) ∈ ((𝐺𝑣)‘𝑖))
212194, 211eqeltrd 2838 . . . . . . 7 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑙 ∘ (𝐺𝑣))‘𝑖) ∈ ((𝐺𝑣)‘𝑖))
213 fvco3 6940 . . . . . . . . . . . 12 ((𝑣:(1...𝑀)⟶𝑅𝑖 ∈ (1...𝑀)) → ((𝐺𝑣)‘𝑖) = (𝐺‘(𝑣𝑖)))
214187, 213sylan 580 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐺𝑣)‘𝑖) = (𝐺‘(𝑣𝑖)))
215 raleq 3309 . . . . . . . . . . . . . 14 (𝑤 = (𝑣𝑖) → (∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ↔ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀)))
2162153anbi2d 1441 . . . . . . . . . . . . 13 (𝑤 = (𝑣𝑖) → ((∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡)) ↔ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))))
217216rabbidv 3415 . . . . . . . . . . . 12 (𝑤 = (𝑣𝑖) → {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
218187ffvelcdmda 7035 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑣𝑖) ∈ 𝑅)
21950adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐴 ∈ V)
220 rabexg 5288 . . . . . . . . . . . . 13 (𝐴 ∈ V → {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ∈ V)
221219, 220syl 17 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ∈ V)
2225, 217, 218, 221fvmptd3 6971 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘(𝑣𝑖)) = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
223214, 222eqtrd 2776 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐺𝑣)‘𝑖) = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
224223adantlr 713 . . . . . . . . 9 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺𝑣)‘𝑖) = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
225224eleq2d 2823 . . . . . . . 8 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → (((𝑙 ∘ (𝐺𝑣))‘𝑖) ∈ ((𝐺𝑣)‘𝑖) ↔ ((𝑙 ∘ (𝐺𝑣))‘𝑖) ∈ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}))
226 nfcv 2907 . . . . . . . . . . . . . 14 𝑣
22724, 226nfco 5821 . . . . . . . . . . . . 13 (𝐺𝑣)
22820, 227nfco 5821 . . . . . . . . . . . 12 (𝑙 ∘ (𝐺𝑣))
229 nfcv 2907 . . . . . . . . . . . 12 𝑖
230228, 229nffv 6852 . . . . . . . . . . 11 ((𝑙 ∘ (𝐺𝑣))‘𝑖)
231 nfcv 2907 . . . . . . . . . . 11 𝐴
232 nfcv 2907 . . . . . . . . . . . . 13 𝑇
233 nfcv 2907 . . . . . . . . . . . . . . 15 0
234 nfcv 2907 . . . . . . . . . . . . . . 15
235 nfcv 2907 . . . . . . . . . . . . . . . 16 𝑡
236230, 235nffv 6852 . . . . . . . . . . . . . . 15 (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)
237233, 234, 236nfbr 5152 . . . . . . . . . . . . . 14 0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)
238 nfcv 2907 . . . . . . . . . . . . . . 15 1
239236, 234, 238nfbr 5152 . . . . . . . . . . . . . 14 (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1
240237, 239nfan 1902 . . . . . . . . . . . . 13 (0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1)
241232, 240nfralw 3294 . . . . . . . . . . . 12 𝑡𝑇 (0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1)
242 nfcv 2907 . . . . . . . . . . . . 13 (𝑣𝑖)
243 nfcv 2907 . . . . . . . . . . . . . 14 <
244 nfcv 2907 . . . . . . . . . . . . . 14 (𝐸 / 𝑀)
245236, 243, 244nfbr 5152 . . . . . . . . . . . . 13 (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)
246242, 245nfralw 3294 . . . . . . . . . . . 12 𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)
247 nfcv 2907 . . . . . . . . . . . . 13 (𝑇𝑈)
248 nfcv 2907 . . . . . . . . . . . . . 14 (1 − (𝐸 / 𝑀))
249248, 243, 236nfbr 5152 . . . . . . . . . . . . 13 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)
250247, 249nfralw 3294 . . . . . . . . . . . 12 𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)
251241, 246, 250nf3an 1904 . . . . . . . . . . 11 (∀𝑡𝑇 (0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))
252 nfcv 2907 . . . . . . . . . . . . . 14 𝑡
253 nfcv 2907 . . . . . . . . . . . . . . . 16 𝑡𝑙
254 nfcv 2907 . . . . . . . . . . . . . . . . . . 19 𝑡𝑅
255 nfra1 3267 . . . . . . . . . . . . . . . . . . . . 21 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
256 nfra1 3267 . . . . . . . . . . . . . . . . . . . . 21 𝑡𝑡𝑤 (𝑡) < (𝐸 / 𝑀)
257 nfra1 3267 . . . . . . . . . . . . . . . . . . . . 21 𝑡𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡)
258255, 256, 257nf3an 1904 . . . . . . . . . . . . . . . . . . . 20 𝑡(∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))
259 nfcv 2907 . . . . . . . . . . . . . . . . . . . 20 𝑡𝐴
260258, 259nfrabw 3440 . . . . . . . . . . . . . . . . . . 19 𝑡{𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}
261254, 260nfmpt 5212 . . . . . . . . . . . . . . . . . 18 𝑡(𝑤𝑅 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
2625, 261nfcxfr 2905 . . . . . . . . . . . . . . . . 17 𝑡𝐺
263 nfcv 2907 . . . . . . . . . . . . . . . . 17 𝑡𝑣
264262, 263nfco 5821 . . . . . . . . . . . . . . . 16 𝑡(𝐺𝑣)
265253, 264nfco 5821 . . . . . . . . . . . . . . 15 𝑡(𝑙 ∘ (𝐺𝑣))
266 nfcv 2907 . . . . . . . . . . . . . . 15 𝑡𝑖
267265, 266nffv 6852 . . . . . . . . . . . . . 14 𝑡((𝑙 ∘ (𝐺𝑣))‘𝑖)
268252, 267nfeq 2920 . . . . . . . . . . . . 13 𝑡 = ((𝑙 ∘ (𝐺𝑣))‘𝑖)
269 fveq1 6841 . . . . . . . . . . . . . . 15 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → (𝑡) = (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))
270269breq2d 5117 . . . . . . . . . . . . . 14 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → (0 ≤ (𝑡) ↔ 0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
271269breq1d 5115 . . . . . . . . . . . . . 14 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → ((𝑡) ≤ 1 ↔ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1))
272270, 271anbi12d 631 . . . . . . . . . . . . 13 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1)))
273268, 272ralbid 3256 . . . . . . . . . . . 12 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1)))
274269breq1d 5115 . . . . . . . . . . . . 13 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → ((𝑡) < (𝐸 / 𝑀) ↔ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
275268, 274ralbid 3256 . . . . . . . . . . . 12 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → (∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ↔ ∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
276269breq2d 5117 . . . . . . . . . . . . 13 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → ((1 − (𝐸 / 𝑀)) < (𝑡) ↔ (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
277268, 276ralbid 3256 . . . . . . . . . . . 12 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → (∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
278273, 275, 2773anbi123d 1436 . . . . . . . . . . 11 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → ((∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡)) ↔ (∀𝑡𝑇 (0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))))
279230, 231, 251, 278elrabf 3641 . . . . . . . . . 10 (((𝑙 ∘ (𝐺𝑣))‘𝑖) ∈ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ↔ (((𝑙 ∘ (𝐺𝑣))‘𝑖) ∈ 𝐴 ∧ (∀𝑡𝑇 (0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))))
280279simprbi 497 . . . . . . . . 9 (((𝑙 ∘ (𝐺𝑣))‘𝑖) ∈ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} → (∀𝑡𝑇 (0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
281280simp2d 1143 . . . . . . . 8 (((𝑙 ∘ (𝐺𝑣))‘𝑖) ∈ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} → ∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀))
282225, 281syl6bi 252 . . . . . . 7 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → (((𝑙 ∘ (𝐺𝑣))‘𝑖) ∈ ((𝐺𝑣)‘𝑖) → ∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
283212, 282mpd 15 . . . . . 6 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀))
284 stoweidlem31.2 . . . . . . . . 9 𝑡𝜑
285262nfrn 5907 . . . . . . . . . . 11 𝑡ran 𝐺
286253, 285nffn 6601 . . . . . . . . . 10 𝑡 𝑙 Fn ran 𝐺
287 nfv 1917 . . . . . . . . . . 11 𝑡(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)
288285, 287nfralw 3294 . . . . . . . . . 10 𝑡𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)
289286, 288nfan 1902 . . . . . . . . 9 𝑡(𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
290284, 289nfan 1902 . . . . . . . 8 𝑡(𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)))
291 nfv 1917 . . . . . . . 8 𝑡 𝑖 ∈ (1...𝑀)
292290, 291nfan 1902 . . . . . . 7 𝑡((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀))
293 stoweidlem31.11 . . . . . . . . . . 11 (𝜑𝐵 ⊆ (𝑇𝑈))
294293ad3antrrr 728 . . . . . . . . . 10 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡𝐵) → 𝐵 ⊆ (𝑇𝑈))
295 simpr 485 . . . . . . . . . 10 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡𝐵) → 𝑡𝐵)
296294, 295sseldd 3945 . . . . . . . . 9 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡𝐵) → 𝑡 ∈ (𝑇𝑈))
297280simp3d 1144 . . . . . . . . . . . 12 (((𝑙 ∘ (𝐺𝑣))‘𝑖) ∈ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} → ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))
298225, 297syl6bi 252 . . . . . . . . . . 11 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → (((𝑙 ∘ (𝐺𝑣))‘𝑖) ∈ ((𝐺𝑣)‘𝑖) → ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
299212, 298mpd 15 . . . . . . . . . 10 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))
300299r19.21bi 3234 . . . . . . . . 9 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑇𝑈)) → (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))
301296, 300syldan 591 . . . . . . . 8 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡𝐵) → (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))
302301ex 413 . . . . . . 7 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → (𝑡𝐵 → (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
303292, 302ralrimi 3240 . . . . . 6 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))
304283, 303jca 512 . . . . 5 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → (∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
305304ralrimiva 3143 . . . 4 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
306192, 305jca 512 . . 3 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → ((𝑙 ∘ (𝐺𝑣)):(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))))
307 feq1 6649 . . . . 5 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → (𝑥:(1...𝑀)⟶𝑌 ↔ (𝑙 ∘ (𝐺𝑣)):(1...𝑀)⟶𝑌))
308 nfcv 2907 . . . . . . . . 9 𝑡𝑥
309308, 265nfeq 2920 . . . . . . . 8 𝑡 𝑥 = (𝑙 ∘ (𝐺𝑣))
310 fveq1 6841 . . . . . . . . . 10 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → (𝑥𝑖) = ((𝑙 ∘ (𝐺𝑣))‘𝑖))
311310fveq1d 6844 . . . . . . . . 9 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → ((𝑥𝑖)‘𝑡) = (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))
312311breq1d 5115 . . . . . . . 8 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → (((𝑥𝑖)‘𝑡) < (𝐸 / 𝑀) ↔ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
313309, 312ralbid 3256 . . . . . . 7 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → (∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑀) ↔ ∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
314311breq2d 5117 . . . . . . . 8 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → ((1 − (𝐸 / 𝑀)) < ((𝑥𝑖)‘𝑡) ↔ (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
315309, 314ralbid 3256 . . . . . . 7 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → (∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥𝑖)‘𝑡) ↔ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
316313, 315anbi12d 631 . . . . . 6 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → ((∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥𝑖)‘𝑡)) ↔ (∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))))
317316ralbidv 3174 . . . . 5 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → (∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥𝑖)‘𝑡)) ↔ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))))
318307, 317anbi12d 631 . . . 4 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → ((𝑥:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥𝑖)‘𝑡))) ↔ ((𝑙 ∘ (𝐺𝑣)):(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))))
319318spcegv 3556 . . 3 ((𝑙 ∘ (𝐺𝑣)) ∈ V → (((𝑙 ∘ (𝐺𝑣)):(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))) → ∃𝑥(𝑥:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥𝑖)‘𝑡)))))
32017, 306, 319sylc 65 . 2 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → ∃𝑥(𝑥:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥𝑖)‘𝑡))))
3213, 320exlimddv 1938 1 (𝜑 → ∃𝑥(𝑥:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥𝑖)‘𝑡))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wnf 1785  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cdif 3907  wss 3910  c0 4282   class class class wbr 5105  cmpt 5188  ran crn 5634  ccom 5637   Fn wfn 6491  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  Fincfn 8883  0cc0 11051  1c1 11052   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  cn 12153  +crp 12915  ...cfz 13424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-rp 12916
This theorem is referenced by:  stoweidlem39  44270
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