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Theorem stoweidlem31 40566
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 39502 . . 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 3234 . . . . 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 4838 . . . . . . . 8 (𝜑𝑅 ∈ V)
9 mptexg 6525 . . . . . . . 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, 10syl5eqel 2734 . . . . . 6 (𝜑𝐺 ∈ V)
12 vex 3234 . . . . . 6 𝑣 ∈ V
13 coexg 7159 . . . . . 6 ((𝐺 ∈ V ∧ 𝑣 ∈ V) → (𝐺𝑣) ∈ V)
1411, 12, 13sylancl 695 . . . . 5 (𝜑 → (𝐺𝑣) ∈ V)
15 coexg 7159 . . . . 5 ((𝑙 ∈ V ∧ (𝐺𝑣) ∈ V) → (𝑙 ∘ (𝐺𝑣)) ∈ V)
164, 14, 15sylancr 696 . . . 4 (𝜑 → (𝑙 ∘ (𝐺𝑣)) ∈ V)
1716adantr 480 . . 3 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → (𝑙 ∘ (𝐺𝑣)) ∈ V)
18 simprl 809 . . . . . 6 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → 𝑙 Fn ran 𝐺)
19 stoweidlem31.1 . . . . . . . . 9 𝜑
20 nfcv 2793 . . . . . . . . . . 11 𝑙
21 nfcv 2793 . . . . . . . . . . . . . 14 𝑅
22 nfrab1 3152 . . . . . . . . . . . . . 14 {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}
2321, 22nfmpt 4779 . . . . . . . . . . . . 13 (𝑤𝑅 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
245, 23nfcxfr 2791 . . . . . . . . . . . 12 𝐺
2524nfrn 5400 . . . . . . . . . . 11 ran 𝐺
2620, 25nffn 6025 . . . . . . . . . 10 𝑙 Fn ran 𝐺
27 nfv 1883 . . . . . . . . . . 11 (𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)
2825, 27nfral 2974 . . . . . . . . . 10 𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)
2926, 28nfan 1868 . . . . . . . . 9 (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
3019, 29nfan 1868 . . . . . . . 8 (𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)))
31 fvelrnb 6282 . . . . . . . . . . . . 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 500 . . . . . . . . . . 11 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) → ∃𝑏 ∈ ran 𝐺(𝑙𝑏) = )
34 nfv 1883 . . . . . . . . . . . . . 14 𝑏𝜑
35 nfv 1883 . . . . . . . . . . . . . . 15 𝑏 𝑙 Fn ran 𝐺
36 nfra1 2970 . . . . . . . . . . . . . . 15 𝑏𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)
3735, 36nfan 1868 . . . . . . . . . . . . . 14 𝑏(𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
3834, 37nfan 1868 . . . . . . . . . . . . 13 𝑏(𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)))
39 nfv 1883 . . . . . . . . . . . . 13 𝑏 ∈ ran 𝑙
4038, 39nfan 1868 . . . . . . . . . . . 12 𝑏((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙)
41 simp3 1083 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) = ) → (𝑙𝑏) = )
42 simp1ll 1144 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) = ) → 𝜑)
43 simplrr 818 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) → ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
44433ad2ant1 1102 . . . . . . . . . . . . . . 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 1082 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) = ) → 𝑏 ∈ ran 𝐺)
46 simp3 1083 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → 𝑏 ∈ ran 𝐺)
47 3simpc 1080 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → (∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺))
48 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑏 ∈ ran 𝐺) → 𝑏 ∈ ran 𝐺)
49 stoweidlem31.3 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑤𝜑
50 stoweidlem31.13 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐴 ∈ V)
51 rabexg 4844 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2986 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ∀𝑤𝑅 {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ∈ V)
555fnmpt 6058 . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑏 ∈ ran 𝐺) → 𝐺 Fn 𝑅)
58 fvelrnb 6282 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 Fn 𝑅 → (𝑏 ∈ ran 𝐺 ↔ ∃𝑢𝑅 (𝐺𝑢) = 𝑏))
59 nfmpt1 4780 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑤(𝑤𝑅 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
605, 59nfcxfr 2791 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑤𝐺
61 nfcv 2793 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑤𝑢
6260, 61nffv 6236 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑤(𝐺𝑢)
63 nfcv 2793 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑤𝑏
6462, 63nfeq 2805 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑤(𝐺𝑢) = 𝑏
65 nfv 1883 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑢(𝐺𝑤) = 𝑏
66 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑢 = 𝑤 → (𝐺𝑢) = (𝐺𝑤))
6766eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢 = 𝑤 → ((𝐺𝑢) = 𝑏 ↔ (𝐺𝑤) = 𝑏))
6864, 65, 67cbvrex 3198 . . . . . . . . . . . . . . . . . . . . . . 23 (∃𝑢𝑅 (𝐺𝑢) = 𝑏 ↔ ∃𝑤𝑅 (𝐺𝑤) = 𝑏)
6958, 68syl6bb 276 . . . . . . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑏 ∈ ran 𝐺) → ∃𝑤𝑅 (𝐺𝑤) = 𝑏)
7260nfrn 5400 . . . . . . . . . . . . . . . . . . . . . . 23 𝑤ran 𝐺
7372nfcri 2787 . . . . . . . . . . . . . . . . . . . . . 22 𝑤 𝑏 ∈ ran 𝐺
7449, 73nfan 1868 . . . . . . . . . . . . . . . . . . . . 21 𝑤(𝜑𝑏 ∈ ran 𝐺)
75 nfv 1883 . . . . . . . . . . . . . . . . . . . . 21 𝑤 𝑏 ≠ ∅
76 simp3 1083 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑤𝑅 ∧ (𝐺𝑤) = 𝑏) → (𝐺𝑤) = 𝑏)
77 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑤𝑅) → 𝑤𝑅)
7850adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6330 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑤𝑅) → (𝐺𝑤) = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
827sselda 3636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3228 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤𝑉 ↔ (𝑤𝐽 ∧ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))))
8582, 84sylib 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑤𝑅) → (𝑤𝐽 ∧ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))))
8685simprd 478 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11908 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑𝑀 ∈ ℝ+)
9087, 89rpdivcld 11927 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (𝐸 / 𝑀) ∈ ℝ+)
9190adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑤𝑅) → (𝐸 / 𝑀) ∈ ℝ+)
92 breq2 4689 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 = (𝐸 / 𝑀) → ((𝑡) < 𝑒 ↔ (𝑡) < (𝐸 / 𝑀)))
9392ralbidv 3015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑒 = (𝐸 / 𝑀) → (∀𝑡𝑤 (𝑡) < 𝑒 ↔ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀)))
94 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = (𝐸 / 𝑀) → (1 − 𝑒) = (1 − (𝐸 / 𝑀)))
9594breq1d 4695 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 = (𝐸 / 𝑀) → ((1 − 𝑒) < (𝑡) ↔ (1 − (𝐸 / 𝑀)) < (𝑡)))
9695ralbidv 3015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑒 = (𝐸 / 𝑀) → (∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡)))
9793, 963anbi23d 1442 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3081 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3339 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑤𝑅) → ∃𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡)))
101 nfv 1883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑤𝑅
10219, 101nfan 1868 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑤𝑅)
103 nfcv 2793 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
10422, 103nfne 2923 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ≠ ∅
105 3simpc 1080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3954 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1283 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3055 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2890 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑤𝑅) → (𝐺𝑤) ≠ ∅)
1141133adant3 1101 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑤𝑅 ∧ (𝐺𝑤) = 𝑏) → (𝐺𝑤) ≠ ∅)
11576, 114eqnetrrd 2891 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤𝑅 ∧ (𝐺𝑤) = 𝑏) → 𝑏 ≠ ∅)
1161153adant1r 1359 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑏 ∈ ran 𝐺) ∧ 𝑤𝑅 ∧ (𝐺𝑤) = 𝑏) → 𝑏 ≠ ∅)
1171163exp 1283 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑏 ∈ ran 𝐺) → (𝑤𝑅 → ((𝐺𝑤) = 𝑏𝑏 ≠ ∅)))
11874, 75, 117rexlimd 3055 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑏 ∈ ran 𝐺) → (∃𝑤𝑅 (𝐺𝑤) = 𝑏𝑏 ≠ ∅))
11971, 118mpd 15 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑏 ∈ ran 𝐺) → 𝑏 ≠ ∅)
1201193adant2 1100 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → 𝑏 ≠ ∅)
121 rspa 2959 . . . . . . . . . . . . . . . . . 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 553 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ 𝑏 ∈ ran 𝐺) → (𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) ∈ 𝑏))
124 vex 3234 . . . . . . . . . . . . . . . . . 18 𝑏 ∈ V
1255elrnmpt 5404 . . . . . . . . . . . . . . . . . 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 208 . . . . . . . . . . . . . . . 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 1883 . . . . . . . . . . . . . . . . . 18 𝑤(𝑙𝑏) ∈ 𝑏
12973, 128nfan 1868 . . . . . . . . . . . . . . . . 17 𝑤(𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) ∈ 𝑏)
130 nfv 1883 . . . . . . . . . . . . . . . . 17 𝑤(𝑙𝑏) ∈ 𝑌
131 simp1r 1106 . . . . . . . . . . . . . . . . . . 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 1083 . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑙𝑏) ∈ 𝑏𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}) → (𝑙𝑏) ∈ 𝑏)
134 simpr 476 . . . . . . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . . . . 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 3391 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑙𝑏) ∈ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} → (𝑙𝑏) ∈ 𝐴)
137 fveq1 6228 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ( = (𝑙𝑏) → (𝑡) = ((𝑙𝑏)‘𝑡))
138137breq2d 4697 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( = (𝑙𝑏) → (0 ≤ (𝑡) ↔ 0 ≤ ((𝑙𝑏)‘𝑡)))
139137breq1d 4695 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( = (𝑙𝑏) → ((𝑡) ≤ 1 ↔ ((𝑙𝑏)‘𝑡) ≤ 1))
140138, 139anbi12d 747 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ( = (𝑙𝑏) → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ ((𝑙𝑏)‘𝑡) ∧ ((𝑙𝑏)‘𝑡) ≤ 1)))
141140ralbidv 3015 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( = (𝑙𝑏) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝑙𝑏)‘𝑡) ∧ ((𝑙𝑏)‘𝑡) ≤ 1)))
142137breq1d 4695 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ( = (𝑙𝑏) → ((𝑡) < (𝐸 / 𝑀) ↔ ((𝑙𝑏)‘𝑡) < (𝐸 / 𝑀)))
143142ralbidv 3015 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( = (𝑙𝑏) → (∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ↔ ∀𝑡𝑤 ((𝑙𝑏)‘𝑡) < (𝐸 / 𝑀)))
144137breq2d 4697 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ( = (𝑙𝑏) → ((1 − (𝐸 / 𝑀)) < (𝑡) ↔ (1 − (𝐸 / 𝑀)) < ((𝑙𝑏)‘𝑡)))
145144ralbidv 3015 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( = (𝑙𝑏) → (∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < ((𝑙𝑏)‘𝑡)))
146141, 143, 1453anbi123d 1439 . . . . . . . . . . . . . . . . . . . . . . . . 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 3396 . . . . . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . . . . 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 1093 . . . . . . . . . . . . . . . . . . . . . 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 3396 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑙𝑏) ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)} ↔ ((𝑙𝑏) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ ((𝑙𝑏)‘𝑡) ∧ ((𝑙𝑏)‘𝑡) ≤ 1)))
151136, 149, 150sylanbrc 699 . . . . . . . . . . . . . . . . . . . . 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, 153syl6eleqr 2741 . . . . . . . . . . . . . . . . . . 19 (((𝑙𝑏) ∈ 𝑏𝑏 = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}) → (𝑙𝑏) ∈ 𝑌)
155131, 132, 154syl2anc 694 . . . . . . . . . . . . . . . . . 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 1283 . . . . . . . . . . . . . . . . 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 3055 . . . . . . . . . . . . . . . 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 1366 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) = ) → (𝑙𝑏) ∈ 𝑌)
16041, 159eqeltrrd 2731 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) ∧ 𝑏 ∈ ran 𝐺 ∧ (𝑙𝑏) = ) → 𝑌)
1611603exp 1283 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ ∈ ran 𝑙) → (𝑏 ∈ ran 𝐺 → ((𝑙𝑏) = 𝑌)))
16240, 161reximdai 3041 . . . . . . . . . . 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 1883 . . . . . . . . . . 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 3055 . . . . . . . . . 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 449 . . . . . . . 8 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → ( ∈ ran 𝑙𝑌))
17030, 169ralrimi 2986 . . . . . . 7 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → ∀ ∈ ran 𝑙 𝑌)
171 dfss3 3625 . . . . . . . 8 (ran 𝑙𝑌 ↔ ∀𝑧 ∈ ran 𝑙 𝑧𝑌)
172 nfrab1 3152 . . . . . . . . . . 11 {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
173153, 172nfcxfr 2791 . . . . . . . . . 10 𝑌
174173nfcri 2787 . . . . . . . . 9 𝑧𝑌
175 nfv 1883 . . . . . . . . 9 𝑧 𝑌
176 eleq1 2718 . . . . . . . . 9 (𝑧 = → (𝑧𝑌𝑌))
177174, 175, 176cbvral 3197 . . . . . . . 8 (∀𝑧 ∈ ran 𝑙 𝑧𝑌 ↔ ∀ ∈ ran 𝑙 𝑌)
178171, 177bitri 264 . . . . . . 7 (ran 𝑙𝑌 ↔ ∀ ∈ ran 𝑙 𝑌)
179170, 178sylibr 224 . . . . . 6 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → ran 𝑙𝑌)
180 df-f 5930 . . . . . 6 (𝑙:ran 𝐺𝑌 ↔ (𝑙 Fn ran 𝐺 ∧ ran 𝑙𝑌))
18118, 179, 180sylanbrc 699 . . . . 5 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → 𝑙:ran 𝐺𝑌)
182 dffn3 6092 . . . . . . . 8 (𝐺 Fn 𝑅𝐺:𝑅⟶ran 𝐺)
18356, 182sylib 208 . . . . . . 7 (𝜑𝐺:𝑅⟶ran 𝐺)
184183adantr 480 . . . . . 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 6175 . . . . . . . 8 (𝑣:(1...𝑀)–1-1-onto𝑅𝑣:(1...𝑀)⟶𝑅)
187185, 186syl 17 . . . . . . 7 (𝜑𝑣:(1...𝑀)⟶𝑅)
188187adantr 480 . . . . . 6 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → 𝑣:(1...𝑀)⟶𝑅)
189 fco 6096 . . . . . 6 ((𝐺:𝑅⟶ran 𝐺𝑣:(1...𝑀)⟶𝑅) → (𝐺𝑣):(1...𝑀)⟶ran 𝐺)
190184, 188, 189syl2anc 694 . . . . 5 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → (𝐺𝑣):(1...𝑀)⟶ran 𝐺)
191 fco 6096 . . . . 5 ((𝑙:ran 𝐺𝑌 ∧ (𝐺𝑣):(1...𝑀)⟶ran 𝐺) → (𝑙 ∘ (𝐺𝑣)):(1...𝑀)⟶𝑌)
192181, 190, 191syl2anc 694 . . . 4 ((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) → (𝑙 ∘ (𝐺𝑣)):(1...𝑀)⟶𝑌)
193 fvco3 6314 . . . . . . . . 9 (((𝐺𝑣):(1...𝑀)⟶ran 𝐺𝑖 ∈ (1...𝑀)) → ((𝑙 ∘ (𝐺𝑣))‘𝑖) = (𝑙‘((𝐺𝑣)‘𝑖)))
194190, 193sylan 487 . . . . . . . 8 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑙 ∘ (𝐺𝑣))‘𝑖) = (𝑙‘((𝐺𝑣)‘𝑖)))
195 simpll 805 . . . . . . . . 9 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
196 simplrr 818 . . . . . . . . 9 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
197190ffvelrnda 6399 . . . . . . . . 9 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺𝑣)‘𝑖) ∈ ran 𝐺)
198 simp3 1083 . . . . . . . . . 10 ((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ ((𝐺𝑣)‘𝑖) ∈ ran 𝐺) → ((𝐺𝑣)‘𝑖) ∈ ran 𝐺)
199 nfv 1883 . . . . . . . . . . . . 13 𝑏((𝐺𝑣)‘𝑖) ∈ ran 𝐺
20034, 36, 199nf3an 1871 . . . . . . . . . . . 12 𝑏(𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ ((𝐺𝑣)‘𝑖) ∈ ran 𝐺)
201 nfv 1883 . . . . . . . . . . . 12 𝑏(𝑙‘((𝐺𝑣)‘𝑖)) ∈ ((𝐺𝑣)‘𝑖)
202200, 201nfim 1865 . . . . . . . . . . 11 𝑏((𝜑 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏) ∧ ((𝐺𝑣)‘𝑖) ∈ ran 𝐺) → (𝑙‘((𝐺𝑣)‘𝑖)) ∈ ((𝐺𝑣)‘𝑖))
203 eleq1 2718 . . . . . . . . . . . . 13 (𝑏 = ((𝐺𝑣)‘𝑖) → (𝑏 ∈ ran 𝐺 ↔ ((𝐺𝑣)‘𝑖) ∈ ran 𝐺))
2042033anbi3d 1445 . . . . . . . . . . . 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 6229 . . . . . . . . . . . . 13 (𝑏 = ((𝐺𝑣)‘𝑖) → (𝑙𝑏) = (𝑙‘((𝐺𝑣)‘𝑖)))
206 id 22 . . . . . . . . . . . . 13 (𝑏 = ((𝐺𝑣)‘𝑖) → 𝑏 = ((𝐺𝑣)‘𝑖))
207205, 206eleq12d 2724 . . . . . . . . . . . 12 (𝑏 = ((𝐺𝑣)‘𝑖) → ((𝑙𝑏) ∈ 𝑏 ↔ (𝑙‘((𝐺𝑣)‘𝑖)) ∈ ((𝐺𝑣)‘𝑖)))
208204, 207imbi12d 333 . . . . . . . . . . 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 3296 . . . . . . . . . 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 1366 . . . . . . . 8 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → (𝑙‘((𝐺𝑣)‘𝑖)) ∈ ((𝐺𝑣)‘𝑖))
212194, 211eqeltrd 2730 . . . . . . 7 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑙 ∘ (𝐺𝑣))‘𝑖) ∈ ((𝐺𝑣)‘𝑖))
213 fvco3 6314 . . . . . . . . . . . 12 ((𝑣:(1...𝑀)⟶𝑅𝑖 ∈ (1...𝑀)) → ((𝐺𝑣)‘𝑖) = (𝐺‘(𝑣𝑖)))
214187, 213sylan 487 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐺𝑣)‘𝑖) = (𝐺‘(𝑣𝑖)))
215187ffvelrnda 6399 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑣𝑖) ∈ 𝑅)
21650adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐴 ∈ V)
217 rabexg 4844 . . . . . . . . . . . . 13 (𝐴 ∈ V → {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ∈ V)
218216, 217syl 17 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ∈ V)
219 raleq 3168 . . . . . . . . . . . . . . 15 (𝑤 = (𝑣𝑖) → (∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ↔ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀)))
2202193anbi2d 1444 . . . . . . . . . . . . . 14 (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))))
221220rabbidv 3220 . . . . . . . . . . . . 13 (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))})
222221, 5fvmptg 6319 . . . . . . . . . . . 12 (((𝑣𝑖) ∈ 𝑅 ∧ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))} ∈ V) → (𝐺‘(𝑣𝑖)) = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
223215, 218, 222syl2anc 694 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘(𝑣𝑖)) = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
224214, 223eqtrd 2685 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐺𝑣)‘𝑖) = {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
225224adantlr 751 . . . . . . . . 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))})
226225eleq2d 2716 . . . . . . . 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))}))
227 nfcv 2793 . . . . . . . . . . . . . 14 𝑣
22824, 227nfco 5320 . . . . . . . . . . . . 13 (𝐺𝑣)
22920, 228nfco 5320 . . . . . . . . . . . 12 (𝑙 ∘ (𝐺𝑣))
230 nfcv 2793 . . . . . . . . . . . 12 𝑖
231229, 230nffv 6236 . . . . . . . . . . 11 ((𝑙 ∘ (𝐺𝑣))‘𝑖)
232 nfcv 2793 . . . . . . . . . . 11 𝐴
233 nfcv 2793 . . . . . . . . . . . . 13 𝑇
234 nfcv 2793 . . . . . . . . . . . . . . 15 0
235 nfcv 2793 . . . . . . . . . . . . . . 15
236 nfcv 2793 . . . . . . . . . . . . . . . 16 𝑡
237231, 236nffv 6236 . . . . . . . . . . . . . . 15 (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)
238234, 235, 237nfbr 4732 . . . . . . . . . . . . . 14 0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)
239 nfcv 2793 . . . . . . . . . . . . . . 15 1
240237, 235, 239nfbr 4732 . . . . . . . . . . . . . 14 (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1
241238, 240nfan 1868 . . . . . . . . . . . . 13 (0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1)
242233, 241nfral 2974 . . . . . . . . . . . 12 𝑡𝑇 (0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1)
243 nfcv 2793 . . . . . . . . . . . . 13 (𝑣𝑖)
244 nfcv 2793 . . . . . . . . . . . . . 14 <
245 nfcv 2793 . . . . . . . . . . . . . 14 (𝐸 / 𝑀)
246237, 244, 245nfbr 4732 . . . . . . . . . . . . 13 (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)
247243, 246nfral 2974 . . . . . . . . . . . 12 𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)
248 nfcv 2793 . . . . . . . . . . . . 13 (𝑇𝑈)
249 nfcv 2793 . . . . . . . . . . . . . 14 (1 − (𝐸 / 𝑀))
250249, 244, 237nfbr 4732 . . . . . . . . . . . . 13 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)
251248, 250nfral 2974 . . . . . . . . . . . 12 𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)
252242, 247, 251nf3an 1871 . . . . . . . . . . 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))
253 nfcv 2793 . . . . . . . . . . . . . 14 𝑡
254 nfcv 2793 . . . . . . . . . . . . . . . 16 𝑡𝑙
255 nfcv 2793 . . . . . . . . . . . . . . . . . . 19 𝑡𝑅
256 nfra1 2970 . . . . . . . . . . . . . . . . . . . . 21 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
257 nfra1 2970 . . . . . . . . . . . . . . . . . . . . 21 𝑡𝑡𝑤 (𝑡) < (𝐸 / 𝑀)
258 nfra1 2970 . . . . . . . . . . . . . . . . . . . . 21 𝑡𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡)
259256, 257, 258nf3an 1871 . . . . . . . . . . . . . . . . . . . 20 𝑡(∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))
260 nfcv 2793 . . . . . . . . . . . . . . . . . . . 20 𝑡𝐴
261259, 260nfrab 3153 . . . . . . . . . . . . . . . . . . 19 𝑡{𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))}
262255, 261nfmpt 4779 . . . . . . . . . . . . . . . . . 18 𝑡(𝑤𝑅 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})
2635, 262nfcxfr 2791 . . . . . . . . . . . . . . . . 17 𝑡𝐺
264 nfcv 2793 . . . . . . . . . . . . . . . . 17 𝑡𝑣
265263, 264nfco 5320 . . . . . . . . . . . . . . . 16 𝑡(𝐺𝑣)
266254, 265nfco 5320 . . . . . . . . . . . . . . 15 𝑡(𝑙 ∘ (𝐺𝑣))
267 nfcv 2793 . . . . . . . . . . . . . . 15 𝑡𝑖
268266, 267nffv 6236 . . . . . . . . . . . . . 14 𝑡((𝑙 ∘ (𝐺𝑣))‘𝑖)
269253, 268nfeq 2805 . . . . . . . . . . . . 13 𝑡 = ((𝑙 ∘ (𝐺𝑣))‘𝑖)
270 fveq1 6228 . . . . . . . . . . . . . . 15 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → (𝑡) = (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))
271270breq2d 4697 . . . . . . . . . . . . . 14 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → (0 ≤ (𝑡) ↔ 0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
272270breq1d 4695 . . . . . . . . . . . . . 14 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → ((𝑡) ≤ 1 ↔ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1))
273271, 272anbi12d 747 . . . . . . . . . . . . 13 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1)))
274269, 273ralbid 3012 . . . . . . . . . . . 12 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ∧ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) ≤ 1)))
275270breq1d 4695 . . . . . . . . . . . . 13 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → ((𝑡) < (𝐸 / 𝑀) ↔ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
276269, 275ralbid 3012 . . . . . . . . . . . 12 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → (∀𝑡 ∈ (𝑣𝑖)(𝑡) < (𝐸 / 𝑀) ↔ ∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
277270breq2d 4697 . . . . . . . . . . . . 13 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → ((1 − (𝐸 / 𝑀)) < (𝑡) ↔ (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
278269, 277ralbid 3012 . . . . . . . . . . . 12 ( = ((𝑙 ∘ (𝐺𝑣))‘𝑖) → (∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
279274, 276, 2783anbi123d 1439 . . . . . . . . . . 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))))
280231, 232, 252, 279elrabf 3392 . . . . . . . . . 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))))
281280simprbi 479 . . . . . . . . 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)))
282281simp2d 1094 . . . . . . . 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))
283226, 282syl6bi 243 . . . . . . 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)))
284212, 283mpd 15 . . . . . 6 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀))
285 stoweidlem31.2 . . . . . . . . 9 𝑡𝜑
286263nfrn 5400 . . . . . . . . . . 11 𝑡ran 𝐺
287254, 286nffn 6025 . . . . . . . . . 10 𝑡 𝑙 Fn ran 𝐺
288 nfv 1883 . . . . . . . . . . 11 𝑡(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)
289286, 288nfral 2974 . . . . . . . . . 10 𝑡𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)
290287, 289nfan 1868 . . . . . . . . 9 𝑡(𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))
291285, 290nfan 1868 . . . . . . . 8 𝑡(𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏)))
292 nfv 1883 . . . . . . . 8 𝑡 𝑖 ∈ (1...𝑀)
293291, 292nfan 1868 . . . . . . 7 𝑡((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀))
294 stoweidlem31.11 . . . . . . . . . . 11 (𝜑𝐵 ⊆ (𝑇𝑈))
295294ad3antrrr 766 . . . . . . . . . 10 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡𝐵) → 𝐵 ⊆ (𝑇𝑈))
296 simpr 476 . . . . . . . . . 10 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡𝐵) → 𝑡𝐵)
297295, 296sseldd 3637 . . . . . . . . 9 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡𝐵) → 𝑡 ∈ (𝑇𝑈))
298281simp3d 1095 . . . . . . . . . . . 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))
299226, 298syl6bi 243 . . . . . . . . . . 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)))
300212, 299mpd 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))
301300r19.21bi 2961 . . . . . . . . 9 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑇𝑈)) → (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))
302297, 301syldan 486 . . . . . . . 8 ((((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡𝐵) → (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))
303302ex 449 . . . . . . 7 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → (𝑡𝐵 → (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
304293, 303ralrimi 2986 . . . . . 6 (((𝜑 ∧ (𝑙 Fn ran 𝐺 ∧ ∀𝑏 ∈ ran 𝐺(𝑏 ≠ ∅ → (𝑙𝑏) ∈ 𝑏))) ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))
305284, 304jca 553 . . . . 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)))
306305ralrimiva 2995 . . . 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)))
307192, 306jca 553 . . 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))))
308 feq1 6064 . . . . 5 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → (𝑥:(1...𝑀)⟶𝑌 ↔ (𝑙 ∘ (𝐺𝑣)):(1...𝑀)⟶𝑌))
309 nfcv 2793 . . . . . . . . 9 𝑡𝑥
310309, 266nfeq 2805 . . . . . . . 8 𝑡 𝑥 = (𝑙 ∘ (𝐺𝑣))
311 fveq1 6228 . . . . . . . . . 10 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → (𝑥𝑖) = ((𝑙 ∘ (𝐺𝑣))‘𝑖))
312311fveq1d 6231 . . . . . . . . 9 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → ((𝑥𝑖)‘𝑡) = (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡))
313312breq1d 4695 . . . . . . . 8 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → (((𝑥𝑖)‘𝑡) < (𝐸 / 𝑀) ↔ (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
314310, 313ralbid 3012 . . . . . . 7 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → (∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑀) ↔ ∀𝑡 ∈ (𝑣𝑖)(((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡) < (𝐸 / 𝑀)))
315312breq2d 4697 . . . . . . . 8 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → ((1 − (𝐸 / 𝑀)) < ((𝑥𝑖)‘𝑡) ↔ (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
316310, 315ralbid 3012 . . . . . . 7 (𝑥 = (𝑙 ∘ (𝐺𝑣)) → (∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥𝑖)‘𝑡) ↔ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < (((𝑙 ∘ (𝐺𝑣))‘𝑖)‘𝑡)))
317314, 316anbi12d 747 . . . . . 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))))
318317ralbidv 3015 . . . . 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))))
319308, 318anbi12d 747 . . . 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)))))
320319spcegv 3325 . . 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)))))
32117, 307, 320sylc 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))))
3223, 321exlimddv 1903 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 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wnf 1748  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  wss 3607  c0 3948   class class class wbr 4685  cmpt 4762  ran crn 5144  ccom 5147   Fn wfn 5921  wf 5922  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  Fincfn 7997  0cc0 9974  1c1 9975   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  +crp 11870  ...cfz 12364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-rp 11871
This theorem is referenced by:  stoweidlem39  40574
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