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Theorem stoweidlem31 43543
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 42542 . . 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 3435 . . . . 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 5252 . . . . . . . 8 (𝜑𝑅 ∈ V)
9 mptexg 7094 . . . . . . . 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 2845 . . . . . 6 (𝜑𝐺 ∈ V)
12 vex 3435 . . . . . 6 𝑣 ∈ V
13 coexg 7770 . . . . . 6 ((𝐺 ∈ V ∧ 𝑣 ∈ V) → (𝐺𝑣) ∈ V)
1411, 12, 13sylancl 586 . . . . 5 (𝜑 → (𝐺𝑣) ∈ V)
15 coexg 7770 . . . . 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 768 . . . . . 6 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → 𝑙 Fn ran 𝐺)
19 stoweidlem31.1 . . . . . . . . 9 𝜑
20 nfcv 2909 . . . . . . . . . . 11 𝑙
21 nfcv 2909 . . . . . . . . . . . . . 14 𝑅
22 nfrab1 3316 . . . . . . . . . . . . . 14 {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}
2321, 22nfmpt 5186 . . . . . . . . . . . . 13 (𝑤𝑅 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
245, 23nfcxfr 2907 . . . . . . . . . . . 12 𝐺
2524nfrn 5860 . . . . . . . . . . 11 ran 𝐺
2620, 25nffn 6530 . . . . . . . . . 10 𝑙 Fn ran 𝐺
27 nfv 1921 . . . . . . . . . . 11 (𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)
2825, 27nfralw 3152 . . . . . . . . . 10 𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)
2926, 28nfan 1906 . . . . . . . . 9 (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
3019, 29nfan 1906 . . . . . . . 8 (𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)))
31 fvelrnb 6827 . . . . . . . . . . . . 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 1921 . . . . . . . . . . . . . 14 𝑏𝜑
35 nfv 1921 . . . . . . . . . . . . . . 15 𝑏 𝑙 Fn ran 𝐺
36 nfra1 3145 . . . . . . . . . . . . . . 15 𝑏𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)
3735, 36nfan 1906 . . . . . . . . . . . . . 14 𝑏(𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
3834, 37nfan 1906 . . . . . . . . . . . . 13 𝑏(𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)))
39 nfv 1921 . . . . . . . . . . . . 13 𝑏 ∈ ran 𝑙
4038, 39nfan 1906 . . . . . . . . . . . 12 𝑏((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙)
41 simp3 1137 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) = ) → (𝑙𝑏) = )
42 simp1ll 1235 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) = ) → 𝜑)
43 simplrr 775 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) → ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
44433ad2ant1 1132 . . . . . . . . . . . . . . 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 1136 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) = ) → 𝑏 ∈ ran 𝐺)
46 simp3 1137 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → 𝑏 ∈ ran 𝐺)
47 3simpc 1149 . . . . . . . . . . . . . . . . . 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 5259 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3142 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ∀𝑤𝑅 {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ∈ V)
555fnmpt 6571 . . . . . . . . . . . . . . . . . . . . . . . 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 6827 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 Fn 𝑅 → (𝑏 ∈ ran 𝐺 ↔ ∃𝑢𝑅 (𝐺𝑢) = 𝑏))
59 nfmpt1 5187 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑤(𝑤𝑅 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
605, 59nfcxfr 2907 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑤𝐺
61 nfcv 2909 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑤𝑢
6260, 61nffv 6781 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑤(𝐺𝑢)
63 nfcv 2909 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑤𝑏
6462, 63nfeq 2922 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑤(𝐺𝑢) = 𝑏
65 nfv 1921 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑢(𝐺𝑤) = 𝑏
66 fveq2 6771 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑢 = 𝑤 → (𝐺𝑢) = (𝐺𝑤))
6766eqeq1d 2742 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢 = 𝑤 → ((𝐺𝑢) = 𝑏 ↔ (𝐺𝑤) = 𝑏))
6864, 65, 67cbvrexw 3373 . . . . . . . . . . . . . . . . . . . . . . 23 (∃𝑢𝑅 (𝐺𝑢) = 𝑏 ↔ ∃𝑤𝑅 (𝐺𝑤) = 𝑏)
6958, 68bitrdi 287 . . . . . . . . . . . . . . . . . . . . . 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 5860 . . . . . . . . . . . . . . . . . . . . . . 23 𝑤ran 𝐺
7372nfcri 2896 . . . . . . . . . . . . . . . . . . . . . 22 𝑤 𝑏 ∈ ran 𝐺
7449, 73nfan 1906 . . . . . . . . . . . . . . . . . . . . 21 𝑤(𝜑𝑏 ∈ ran 𝐺)
75 nfv 1921 . . . . . . . . . . . . . . . . . . . . 21 𝑤 𝑏 ≠ ∅
76 simp3 1137 . . . . . . . . . . . . . . . . . . . . . . . 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 6883 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3926 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))}
8483rabeq2i 3421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑𝑀 ∈ ℝ+)
9087, 89rpdivcld 12788 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (𝐸 / 𝑀) ∈ ℝ+)
9190adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑤𝑅) → (𝐸 / 𝑀) ∈ ℝ+)
92 breq2 5083 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 = (𝐸 / 𝑀) → ((𝑡) < 𝑒 ↔ (𝑡) < (𝐸 / 𝑀)))
9392ralbidv 3123 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑒 = (𝐸 / 𝑀) → (∀𝑡𝑤 (𝑡) < 𝑒 ↔ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀)))
94 oveq2 7279 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = (𝐸 / 𝑀) → (1 − 𝑒) = (1 − (𝐸 / 𝑀)))
9594breq1d 5089 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 = (𝐸 / 𝑀) → ((1 − 𝑒) < (𝑡) ↔ (1 − (𝐸 / 𝑀)) < (𝑡)))
9695ralbidv 3123 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑒 = (𝐸 / 𝑀) → (∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡)))
9793, 963anbi23d 1438 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3228 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3560 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1921 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑤𝑅
10219, 101nfan 1906 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑤𝑅)
103 nfcv 2909 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
10422, 103nfne 3047 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ≠ ∅
105 3simpc 1149 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3309 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4274 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1118 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3248 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3013 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑤𝑅) → (𝐺𝑤) ≠ ∅)
1141133adant3 1131 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑤𝑅 ∧ (𝐺𝑤) = 𝑏) → (𝐺𝑤) ≠ ∅)
11576, 114eqnetrrd 3014 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤𝑅 ∧ (𝐺𝑤) = 𝑏) → 𝑏 ≠ ∅)
1161153adant1r 1176 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑏 ∈ ran 𝐺) ∧ 𝑤𝑅 ∧ (𝐺𝑤) = 𝑏) → 𝑏 ≠ ∅)
1171163exp 1118 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑏 ∈ ran 𝐺) → (𝑤𝑅 → ((𝐺𝑤) = 𝑏𝑏 ≠ ∅)))
11874, 75, 117rexlimd 3248 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑏 ∈ ran 𝐺) → (∃𝑤𝑅 (𝐺𝑤) = 𝑏𝑏 ≠ ∅))
11971, 118mpd 15 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑏 ∈ ran 𝐺) → 𝑏 ≠ ∅)
1201193adant2 1130 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → 𝑏 ≠ ∅)
121 rspa 3133 . . . . . . . . . . . . . . . . . 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 3435 . . . . . . . . . . . . . . . . . 18 𝑏 ∈ V
1255elrnmpt 5864 . . . . . . . . . . . . . . . . . 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 1921 . . . . . . . . . . . . . . . . . 18 𝑤(𝑙𝑏) ∈ 𝑏
12973, 128nfan 1906 . . . . . . . . . . . . . . . . 17 𝑤(𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) ∈ 𝑏)
130 nfv 1921 . . . . . . . . . . . . . . . . 17 𝑤(𝑙𝑏) ∈ 𝑌
131 simp1r 1197 . . . . . . . . . . . . . . . . . . 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 1137 . . . . . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . . . . . . . . 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 3620 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑙𝑏) ∈ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} → (𝑙𝑏) ∈ 𝐴)
137 fveq1 6770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ( = (𝑙𝑏) → (𝑡) = ((𝑙𝑏)‘𝑡))
138137breq2d 5091 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( = (𝑙𝑏) → (0 ≤ (𝑡) ↔ 0 ≤ ((𝑙𝑏)‘𝑡)))
139137breq1d 5089 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( = (𝑙𝑏) → ((𝑡) ≤ 1 ↔ ((𝑙𝑏)‘𝑡) ≤ 1))
140138, 139anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ( = (𝑙𝑏) → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ ((𝑙𝑏)‘𝑡) ∧ ((𝑙𝑏)‘𝑡) ≤ 1)))
141140ralbidv 3123 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( = (𝑙𝑏) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝑙𝑏)‘𝑡) ∧ ((𝑙𝑏)‘𝑡) ≤ 1)))
142137breq1d 5089 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ( = (𝑙𝑏) → ((𝑡) < (𝐸 / 𝑀) ↔ ((𝑙𝑏)‘𝑡) < (𝐸 / 𝑀)))
143142ralbidv 3123 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( = (𝑙𝑏) → (∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ↔ ∀𝑡𝑤 ((𝑙𝑏)‘𝑡) < (𝐸 / 𝑀)))
144137breq2d 5091 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ( = (𝑙𝑏) → ((1 − (𝐸 / 𝑀)) < (𝑡) ↔ (1 − (𝐸 / 𝑀)) < ((𝑙𝑏)‘𝑡)))
145144ralbidv 3123 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( = (𝑙𝑏) → (∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < ((𝑙𝑏)‘𝑡)))
146141, 143, 1453anbi123d 1435 . . . . . . . . . . . . . . . . . . . . . . . . 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 3626 . . . . . . . . . . . . . . . . . . . . . . . 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 1141 . . . . . . . . . . . . . . . . . . . . . 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 3626 . . . . . . . . . . . . . . . . . . . . . 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 2852 . . . . . . . . . . . . . . . . . . 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 1118 . . . . . . . . . . . . . . . . 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 3248 . . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) = ) → (𝑙𝑏) ∈ 𝑌)
16041, 159eqeltrrd 2842 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) = ) → 𝑌)
1611603exp 1118 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) → (𝑏 ∈ ran 𝐺 → ((𝑙𝑏) = 𝑌)))
16240, 161reximdai 3242 . . . . . . . . . . 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 1921 . . . . . . . . . . 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 3248 . . . . . . . . . 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 3142 . . . . . . 7 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → ∀ ∈ ran 𝑙 𝑌)
171 dfss3 3914 . . . . . . . 8 (ran 𝑙𝑌 ↔ ∀𝑧 ∈ ran 𝑙 𝑧𝑌)
172 nfrab1 3316 . . . . . . . . . . 11 {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
173153, 172nfcxfr 2907 . . . . . . . . . 10 𝑌
174173nfcri 2896 . . . . . . . . 9 𝑧𝑌
175 nfv 1921 . . . . . . . . 9 𝑧 𝑌
176 eleq1 2828 . . . . . . . . 9 (𝑧 = → (𝑧𝑌𝑌))
177174, 175, 176cbvralw 3372 . . . . . . . 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 6436 . . . . . 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 6611 . . . . . . . 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 6714 . . . . . . . 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 6622 . . . . . 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 6622 . . . . 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 6864 . . . . . . . . 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 764 . . . . . . . . 9 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
196 simplrr 775 . . . . . . . . 9 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
197190ffvelrnda 6958 . . . . . . . . 9 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺𝑣)‘𝑖) ∈ ran 𝐺)
198 simp3 1137 . . . . . . . . . 10 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ ((𝐺𝑣)‘𝑖) ∈ ran 𝐺) → ((𝐺𝑣)‘𝑖) ∈ ran 𝐺)
199 nfv 1921 . . . . . . . . . . . . 13 𝑏((𝐺𝑣)‘𝑖) ∈ ran 𝐺
20034, 36, 199nf3an 1908 . . . . . . . . . . . 12 𝑏(𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ ((𝐺𝑣)‘𝑖) ∈ ran 𝐺)
201 nfv 1921 . . . . . . . . . . . 12 𝑏(𝑙‘((𝐺𝑣)‘𝑖)) ∈ ((𝐺𝑣)‘𝑖)
202200, 201nfim 1903 . . . . . . . . . . 11 𝑏((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ ((𝐺𝑣)‘𝑖) ∈ ran 𝐺) → (𝑙‘((𝐺𝑣)‘𝑖)) ∈ ((𝐺𝑣)‘𝑖))
203 eleq1 2828 . . . . . . . . . . . . 13 (𝑏 = ((𝐺𝑣)‘𝑖) → (𝑏 ∈ ran 𝐺 ↔ ((𝐺𝑣)‘𝑖) ∈ ran 𝐺))
2042033anbi3d 1441 . . . . . . . . . . . 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 6771 . . . . . . . . . . . . 13 (𝑏 = ((𝐺𝑣)‘𝑖) → (𝑙𝑏) = (𝑙‘((𝐺𝑣)‘𝑖)))
206 id 22 . . . . . . . . . . . . 13 (𝑏 = ((𝐺𝑣)‘𝑖) → 𝑏 = ((𝐺𝑣)‘𝑖))
207205, 206eleq12d 2835 . . . . . . . . . . . 12 (𝑏 = ((𝐺𝑣)‘𝑖) → ((𝑙𝑏) ∈ 𝑏 ↔ (𝑙‘((𝐺𝑣)‘𝑖)) ∈ ((𝐺𝑣)‘𝑖)))
208204, 207imbi12d 345 . . . . . . . . . . 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 3503 . . . . . . . . . 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 1370 . . . . . . . 8 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → (𝑙‘((𝐺𝑣)‘𝑖)) ∈ ((𝐺𝑣)‘𝑖))
212194, 211eqeltrd 2841 . . . . . . 7 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑙 ∘ (𝐺𝑣))‘𝑖) ∈ ((𝐺𝑣)‘𝑖))
213 fvco3 6864 . . . . . . . . . . . 12 ((𝑣:(1...𝑀)⟶𝑅𝑖 ∈ (1...𝑀)) → ((𝐺𝑣)‘𝑖) = (𝐺‘(𝑣𝑖)))
214187, 213sylan 580 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐺𝑣)‘𝑖) = (𝐺‘(𝑣𝑖)))
215 raleq 3341 . . . . . . . . . . . . . 14 (𝑤 = (𝑣𝑖) → (∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ↔ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀)))
2162153anbi2d 1440 . . . . . . . . . . . . 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 3413 . . . . . . . . . . . 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))})
218187ffvelrnda 6958 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑣𝑖) ∈ 𝑅)
21950adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐴 ∈ V)
220 rabexg 5259 . . . . . . . . . . . . 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 6895 . . . . . . . . . . 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 2780 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐺𝑣)‘𝑖) = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
224223adantlr 712 . . . . . . . . 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 2826 . . . . . . . 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 2909 . . . . . . . . . . . . . 14 𝑣
22724, 226nfco 5773 . . . . . . . . . . . . 13 (𝐺𝑣)
22820, 227nfco 5773 . . . . . . . . . . . 12 (𝑙 ∘ (𝐺𝑣))
229 nfcv 2909 . . . . . . . . . . . 12 𝑖
230228, 229nffv 6781 . . . . . . . . . . 11 ((𝑙 ∘ (𝐺𝑣))‘𝑖)
231 nfcv 2909 . . . . . . . . . . 11 𝐴
232 nfcv 2909 . . . . . . . . . . . . 13 𝑇
233 nfcv 2909 . . . . . . . . . . . . . . 15 0
234 nfcv 2909 . . . . . . . . . . . . . . 15
235 nfcv 2909 . . . . . . . . . . . . . . . 16 𝑡
236230, 235nffv 6781 . . . . . . . . . . . . . . 15 (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)
237233, 234, 236nfbr 5126 . . . . . . . . . . . . . 14 0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)
238 nfcv 2909 . . . . . . . . . . . . . . 15 1
239236, 234, 238nfbr 5126 . . . . . . . . . . . . . 14 (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1
240237, 239nfan 1906 . . . . . . . . . . . . 13 (0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1)
241232, 240nfralw 3152 . . . . . . . . . . . 12 𝑡𝑇 (0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1)
242 nfcv 2909 . . . . . . . . . . . . 13 (𝑣𝑖)
243 nfcv 2909 . . . . . . . . . . . . . 14 <
244 nfcv 2909 . . . . . . . . . . . . . 14 (𝐸 / 𝑀)
245236, 243, 244nfbr 5126 . . . . . . . . . . . . 13 (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)
246242, 245nfralw 3152 . . . . . . . . . . . 12 𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)
247 nfcv 2909 . . . . . . . . . . . . 13 (𝑇𝑈)
248 nfcv 2909 . . . . . . . . . . . . . 14 (1 − (𝐸 / 𝑀))
249248, 243, 236nfbr 5126 . . . . . . . . . . . . 13 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)
250247, 249nfralw 3152 . . . . . . . . . . . 12 𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)
251241, 246, 250nf3an 1908 . . . . . . . . . . 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 2909 . . . . . . . . . . . . . 14 𝑡
253 nfcv 2909 . . . . . . . . . . . . . . . 16 𝑡𝑙
254 nfcv 2909 . . . . . . . . . . . . . . . . . . 19 𝑡𝑅
255 nfra1 3145 . . . . . . . . . . . . . . . . . . . . 21 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
256 nfra1 3145 . . . . . . . . . . . . . . . . . . . . 21 𝑡𝑡𝑤 (𝑡) < (𝐸 / 𝑀)
257 nfra1 3145 . . . . . . . . . . . . . . . . . . . . 21 𝑡𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡)
258255, 256, 257nf3an 1908 . . . . . . . . . . . . . . . . . . . 20 𝑡(∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))
259 nfcv 2909 . . . . . . . . . . . . . . . . . . . 20 𝑡𝐴
260258, 259nfrabw 3317 . . . . . . . . . . . . . . . . . . 19 𝑡{𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}
261254, 260nfmpt 5186 . . . . . . . . . . . . . . . . . 18 𝑡(𝑤𝑅 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
2625, 261nfcxfr 2907 . . . . . . . . . . . . . . . . 17 𝑡𝐺
263 nfcv 2909 . . . . . . . . . . . . . . . . 17 𝑡𝑣
264262, 263nfco 5773 . . . . . . . . . . . . . . . 16 𝑡(𝐺𝑣)
265253, 264nfco 5773 . . . . . . . . . . . . . . 15 𝑡(𝑙 ∘ (𝐺𝑣))
266 nfcv 2909 . . . . . . . . . . . . . . 15 𝑡𝑖
267265, 266nffv 6781 . . . . . . . . . . . . . 14 𝑡((𝑙 ∘ (𝐺𝑣))‘𝑖)
268252, 267nfeq 2922 . . . . . . . . . . . . 13 𝑡 = ((𝑙 ∘ (𝐺𝑣))‘𝑖)
269 fveq1 6770 . . . . . . . . . . . . . . 15 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → (𝑡) = (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))
270269breq2d 5091 . . . . . . . . . . . . . 14 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → (0 ≤ (𝑡) ↔ 0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
271269breq1d 5089 . . . . . . . . . . . . . 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 3161 . . . . . . . . . . . 12 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1)))
274269breq1d 5089 . . . . . . . . . . . . 13 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → ((𝑡) < (𝐸 / 𝑀) ↔ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
275268, 274ralbid 3161 . . . . . . . . . . . 12 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → (∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ↔ ∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
276269breq2d 5091 . . . . . . . . . . . . 13 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → ((1 − (𝐸 / 𝑀)) < (𝑡) ↔ (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
277268, 276ralbid 3161 . . . . . . . . . . . 12 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → (∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
278273, 275, 2773anbi123d 1435 . . . . . . . . . . 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 3622 . . . . . . . . . 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 1142 . . . . . . . 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 5860 . . . . . . . . . . 11 𝑡ran 𝐺
286253, 285nffn 6530 . . . . . . . . . 10 𝑡 𝑙 Fn ran 𝐺
287 nfv 1921 . . . . . . . . . . 11 𝑡(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)
288285, 287nfralw 3152 . . . . . . . . . 10 𝑡𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)
289286, 288nfan 1906 . . . . . . . . 9 𝑡(𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
290284, 289nfan 1906 . . . . . . . 8 𝑡(𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)))
291 nfv 1921 . . . . . . . 8 𝑡 𝑖 ∈ (1...𝑀)
292290, 291nfan 1906 . . . . . . 7 𝑡((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀))
293 stoweidlem31.11 . . . . . . . . . . 11 (𝜑𝐵 ⊆ (𝑇𝑈))
294293ad3antrrr 727 . . . . . . . . . 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 3927 . . . . . . . . 9 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡𝐵) → 𝑡 ∈ (𝑇𝑈))
297280simp3d 1143 . . . . . . . . . . . 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 3135 . . . . . . . . 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 3142 . . . . . 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 3110 . . . 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 6579 . . . . 5 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → (𝑥:(1...𝑀)⟶𝑌 ↔ (𝑙 ∘ (𝐺𝑣)):(1...𝑀)⟶𝑌))
308 nfcv 2909 . . . . . . . . 9 𝑡𝑥
309308, 265nfeq 2922 . . . . . . . 8 𝑡 𝑥 = (𝑙 ∘ (𝐺𝑣))
310 fveq1 6770 . . . . . . . . . 10 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → (𝑥𝑖) = ((𝑙 ∘ (𝐺𝑣))‘𝑖))
311310fveq1d 6773 . . . . . . . . 9 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → ((𝑥𝑖)‘𝑡) = (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))
312311breq1d 5089 . . . . . . . 8 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → (((𝑥𝑖)‘𝑡) < (𝐸 / 𝑀) ↔ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
313309, 312ralbid 3161 . . . . . . 7 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → (∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑀) ↔ ∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
314311breq2d 5091 . . . . . . . 8 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → ((1 − (𝐸 / 𝑀)) < ((𝑥𝑖)‘𝑡) ↔ (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
315309, 314ralbid 3161 . . . . . . 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 3123 . . . . 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 3535 . . 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 1942 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 1086   = wceq 1542  wex 1786  wnf 1790  wcel 2110  wne 2945  wral 3066  wrex 3067  {crab 3070  Vcvv 3431  cdif 3889  wss 3892  c0 4262   class class class wbr 5079  cmpt 5162  ran crn 5591  ccom 5594   Fn wfn 6427  wf 6428  1-1-ontowf1o 6431  cfv 6432  (class class class)co 7271  Fincfn 8716  0cc0 10872  1c1 10873   < clt 11010  cle 11011  cmin 11205   / cdiv 11632  cn 11973  +crp 12729  ...cfz 13238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-er 8481  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-rp 12730
This theorem is referenced by:  stoweidlem39  43551
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