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Theorem stoweidlem59 40594
Description: This lemma proves that there exists a function 𝑥 as in the proof in [BrosowskiDeutsh] p. 91, after Lemma 2: xj is in the subalgebra, 0 <= xj <= 1, xj < ε / n on Aj (meaning A in the paper), xj > 1 - \epslon / n on Bj. Here 𝐷 is used to represent A in the paper (because A is used for the subalgebra of functions), 𝐸 is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem59.1 𝑡𝐹
stoweidlem59.2 𝑡𝜑
stoweidlem59.3 𝐾 = (topGen‘ran (,))
stoweidlem59.4 𝑇 = 𝐽
stoweidlem59.5 𝐶 = (𝐽 Cn 𝐾)
stoweidlem59.6 𝐷 = (𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
stoweidlem59.7 𝐵 = (𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
stoweidlem59.8 𝑌 = {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)}
stoweidlem59.9 𝐻 = (𝑗 ∈ (0...𝑁) ↦ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
stoweidlem59.10 (𝜑𝐽 ∈ Comp)
stoweidlem59.11 (𝜑𝐴𝐶)
stoweidlem59.12 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
stoweidlem59.13 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
stoweidlem59.14 ((𝜑𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
stoweidlem59.15 ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))
stoweidlem59.16 (𝜑𝐹𝐶)
stoweidlem59.17 (𝜑𝐸 ∈ ℝ+)
stoweidlem59.18 (𝜑𝐸 < (1 / 3))
stoweidlem59.19 (𝜑𝑁 ∈ ℕ)
Assertion
Ref Expression
stoweidlem59 (𝜑 → ∃𝑥(𝑥:(0...𝑁)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑁)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < ((𝑥𝑗)‘𝑡))))
Distinct variable groups:   𝑡,𝑗,𝑦   𝑦,𝐵   𝑦,𝐷   𝑗,𝑁,𝑡,𝑦   𝑗,𝑌   𝑓,𝑔,𝑗,𝑞,𝑟,𝑡,𝑁   𝑥,𝑓,𝑔,𝑗,𝑡,𝑁   𝑦,𝑓,𝑞,𝑟,𝐴   𝐴,𝑔,𝑞,𝑟,𝑡   𝐵,𝑓,𝑔,𝑞,𝑟   𝐷,𝑓,𝑔,𝑞,𝑟   𝑓,𝐸,𝑔,𝑟,𝑡   𝑓,𝐽,𝑔,𝑟,𝑡   𝑇,𝑓,𝑔,𝑞,𝑟,𝑡   𝜑,𝑓,𝑔,𝑗,𝑞,𝑟   𝑥,𝑦,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝐸,𝑦   𝑥,𝑇,𝑦   𝜑,𝑦   𝑡,𝐾   𝑥,𝐻   𝑥,𝑌   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑡)   𝐴(𝑗)   𝐵(𝑡,𝑗)   𝐶(𝑥,𝑦,𝑡,𝑓,𝑔,𝑗,𝑟,𝑞)   𝐷(𝑡,𝑗)   𝑇(𝑗)   𝐸(𝑗,𝑞)   𝐹(𝑥,𝑦,𝑡,𝑓,𝑔,𝑗,𝑟,𝑞)   𝐻(𝑦,𝑡,𝑓,𝑔,𝑗,𝑟,𝑞)   𝐽(𝑥,𝑦,𝑗,𝑞)   𝐾(𝑥,𝑦,𝑓,𝑔,𝑗,𝑟,𝑞)   𝑌(𝑦,𝑡,𝑓,𝑔,𝑟,𝑞)

Proof of Theorem stoweidlem59
Dummy variables 𝑎 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 stoweidlem59.8 . . . . . . . . . 10 𝑌 = {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)}
2 nfrab1 3152 . . . . . . . . . 10 𝑦{𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)}
31, 2nfcxfr 2791 . . . . . . . . 9 𝑦𝑌
4 nfcv 2793 . . . . . . . . 9 𝑧𝑌
5 nfv 1883 . . . . . . . . 9 𝑧(∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))
6 nfv 1883 . . . . . . . . 9 𝑦(∀𝑡 ∈ (𝐷𝑗)(𝑧𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑧𝑡))
7 fveq1 6228 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑦𝑡) = (𝑧𝑡))
87breq1d 4695 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝑦𝑡) < (𝐸 / 𝑁) ↔ (𝑧𝑡) < (𝐸 / 𝑁)))
98ralbidv 3015 . . . . . . . . . 10 (𝑦 = 𝑧 → (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ↔ ∀𝑡 ∈ (𝐷𝑗)(𝑧𝑡) < (𝐸 / 𝑁)))
107breq2d 4697 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ (1 − (𝐸 / 𝑁)) < (𝑧𝑡)))
1110ralbidv 3015 . . . . . . . . . 10 (𝑦 = 𝑧 → (∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑧𝑡)))
129, 11anbi12d 747 . . . . . . . . 9 (𝑦 = 𝑧 → ((∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡)) ↔ (∀𝑡 ∈ (𝐷𝑗)(𝑧𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑧𝑡))))
133, 4, 5, 6, 12cbvrab 3229 . . . . . . . 8 {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} = {𝑧𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑧𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑧𝑡))}
14 ovexd 6720 . . . . . . . . . 10 (𝜑 → (𝐽 Cn 𝐾) ∈ V)
15 stoweidlem59.11 . . . . . . . . . . 11 (𝜑𝐴𝐶)
16 stoweidlem59.5 . . . . . . . . . . 11 𝐶 = (𝐽 Cn 𝐾)
1715, 16syl6sseq 3684 . . . . . . . . . 10 (𝜑𝐴 ⊆ (𝐽 Cn 𝐾))
1814, 17ssexd 4838 . . . . . . . . 9 (𝜑𝐴 ∈ V)
191, 18rabexd 4846 . . . . . . . 8 (𝜑𝑌 ∈ V)
2013, 19rabexd 4846 . . . . . . 7 (𝜑 → {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} ∈ V)
2120ralrimivw 2996 . . . . . 6 (𝜑 → ∀𝑗 ∈ (0...𝑁){𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} ∈ V)
22 stoweidlem59.9 . . . . . . 7 𝐻 = (𝑗 ∈ (0...𝑁) ↦ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
2322fnmpt 6058 . . . . . 6 (∀𝑗 ∈ (0...𝑁){𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} ∈ V → 𝐻 Fn (0...𝑁))
2421, 23syl 17 . . . . 5 (𝜑𝐻 Fn (0...𝑁))
25 fzfi 12811 . . . . 5 (0...𝑁) ∈ Fin
26 fnfi 8279 . . . . 5 ((𝐻 Fn (0...𝑁) ∧ (0...𝑁) ∈ Fin) → 𝐻 ∈ Fin)
2724, 25, 26sylancl 695 . . . 4 (𝜑𝐻 ∈ Fin)
28 rnfi 8290 . . . 4 (𝐻 ∈ Fin → ran 𝐻 ∈ Fin)
2927, 28syl 17 . . 3 (𝜑 → ran 𝐻 ∈ Fin)
30 fnchoice 39502 . . 3 (ran 𝐻 ∈ Fin → ∃( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)))
3129, 30syl 17 . 2 (𝜑 → ∃( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)))
32 simprl 809 . . . . 5 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → Fn ran 𝐻)
33 ovex 6718 . . . . . . . 8 (0...𝑁) ∈ V
3433mptex 6527 . . . . . . 7 (𝑗 ∈ (0...𝑁) ↦ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))}) ∈ V
3522, 34eqeltri 2726 . . . . . 6 𝐻 ∈ V
3635rnex 7142 . . . . 5 ran 𝐻 ∈ V
37 fnex 6522 . . . . 5 (( Fn ran 𝐻 ∧ ran 𝐻 ∈ V) → ∈ V)
3832, 36, 37sylancl 695 . . . 4 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ∈ V)
39 coexg 7159 . . . 4 (( ∈ V ∧ 𝐻 ∈ V) → (𝐻) ∈ V)
4038, 35, 39sylancl 695 . . 3 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → (𝐻) ∈ V)
41 dffn3 6092 . . . . . . 7 ( Fn ran 𝐻:ran 𝐻⟶ran )
4232, 41sylib 208 . . . . . 6 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → :ran 𝐻⟶ran )
43 nfv 1883 . . . . . . . . . 10 𝑤𝜑
44 nfv 1883 . . . . . . . . . . 11 𝑤 Fn ran 𝐻
45 nfra1 2970 . . . . . . . . . . 11 𝑤𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)
4644, 45nfan 1868 . . . . . . . . . 10 𝑤( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))
4743, 46nfan 1868 . . . . . . . . 9 𝑤(𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)))
48 simplrr 818 . . . . . . . . . . 11 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑤 ∈ ran 𝐻) → ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))
49 simpr 476 . . . . . . . . . . 11 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑤 ∈ ran 𝐻) → 𝑤 ∈ ran 𝐻)
50 fvelrnb 6282 . . . . . . . . . . . . . . . 16 (𝐻 Fn (0...𝑁) → (𝑤 ∈ ran 𝐻 ↔ ∃𝑎 ∈ (0...𝑁)(𝐻𝑎) = 𝑤))
51 nfv 1883 . . . . . . . . . . . . . . . . 17 𝑎(𝐻𝑗) = 𝑤
52 nfmpt1 4780 . . . . . . . . . . . . . . . . . . . 20 𝑗(𝑗 ∈ (0...𝑁) ↦ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
5322, 52nfcxfr 2791 . . . . . . . . . . . . . . . . . . 19 𝑗𝐻
54 nfcv 2793 . . . . . . . . . . . . . . . . . . 19 𝑗𝑎
5553, 54nffv 6236 . . . . . . . . . . . . . . . . . 18 𝑗(𝐻𝑎)
56 nfcv 2793 . . . . . . . . . . . . . . . . . 18 𝑗𝑤
5755, 56nfeq 2805 . . . . . . . . . . . . . . . . 17 𝑗(𝐻𝑎) = 𝑤
58 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑎 → (𝐻𝑗) = (𝐻𝑎))
5958eqeq1d 2653 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑎 → ((𝐻𝑗) = 𝑤 ↔ (𝐻𝑎) = 𝑤))
6051, 57, 59cbvrex 3198 . . . . . . . . . . . . . . . 16 (∃𝑗 ∈ (0...𝑁)(𝐻𝑗) = 𝑤 ↔ ∃𝑎 ∈ (0...𝑁)(𝐻𝑎) = 𝑤)
6150, 60syl6bbr 278 . . . . . . . . . . . . . . 15 (𝐻 Fn (0...𝑁) → (𝑤 ∈ ran 𝐻 ↔ ∃𝑗 ∈ (0...𝑁)(𝐻𝑗) = 𝑤))
6224, 61syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑤 ∈ ran 𝐻 ↔ ∃𝑗 ∈ (0...𝑁)(𝐻𝑗) = 𝑤))
6362biimpa 500 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ran 𝐻) → ∃𝑗 ∈ (0...𝑁)(𝐻𝑗) = 𝑤)
64 simp3 1083 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0...𝑁) ∧ (𝐻𝑗) = 𝑤) → (𝐻𝑗) = 𝑤)
65 simpr 476 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑗 ∈ (0...𝑁))
6620adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0...𝑁)) → {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} ∈ V)
6722fvmpt2 6330 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ (0...𝑁) ∧ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} ∈ V) → (𝐻𝑗) = {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
6865, 66, 67syl2anc 694 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐻𝑗) = {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
69 stoweidlem59.6 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝐷 = (𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
70 nfcv 2793 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑡(0...𝑁)
71 nfrab1 3152 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑡{𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)}
7270, 71nfmpt 4779 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑡(𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
7369, 72nfcxfr 2791 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡𝐷
74 nfcv 2793 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡𝑗
7573, 74nffv 6236 . . . . . . . . . . . . . . . . . . . . . . 23 𝑡(𝐷𝑗)
76 nfcv 2793 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡𝑇
77 stoweidlem59.7 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝐵 = (𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
78 nfrab1 3152 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑡{𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)}
7970, 78nfmpt 4779 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑡(𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
8077, 79nfcxfr 2791 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑡𝐵
8180, 74nffv 6236 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡(𝐵𝑗)
8276, 81nfdif 3764 . . . . . . . . . . . . . . . . . . . . . . 23 𝑡(𝑇 ∖ (𝐵𝑗))
83 stoweidlem59.2 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡𝜑
84 nfv 1883 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡 𝑗 ∈ (0...𝑁)
8583, 84nfan 1868 . . . . . . . . . . . . . . . . . . . . . . 23 𝑡(𝜑𝑗 ∈ (0...𝑁))
86 stoweidlem59.3 . . . . . . . . . . . . . . . . . . . . . . 23 𝐾 = (topGen‘ran (,))
87 stoweidlem59.4 . . . . . . . . . . . . . . . . . . . . . . 23 𝑇 = 𝐽
88 stoweidlem59.10 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐽 ∈ Comp)
8988adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → 𝐽 ∈ Comp)
9015adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → 𝐴𝐶)
91 stoweidlem59.12 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
92913adant1r 1359 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
93 stoweidlem59.13 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
94933adant1r 1359 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
95 stoweidlem59.14 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
9695adantlr 751 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
97 stoweidlem59.15 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))
9897adantlr 751 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (0...𝑁)) ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))
99 uniexg 6997 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐽 ∈ Comp → 𝐽 ∈ V)
10088, 99syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 𝐽 ∈ V)
10187, 100syl5eqel 2734 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝑇 ∈ V)
102101adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑇 ∈ V)
103 rabexg 4844 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑇 ∈ V → {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} ∈ V)
104102, 103syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑁)) → {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} ∈ V)
10577fvmpt2 6330 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑗 ∈ (0...𝑁) ∧ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} ∈ V) → (𝐵𝑗) = {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
10665, 104, 105syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐵𝑗) = {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
107 stoweidlem59.1 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑡𝐹
108 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . . 25 {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} = {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)}
109 elfzelz 12380 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
110109zred 11520 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℝ)
111 3re 11132 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 ∈ ℝ
112 3ne0 11153 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 ≠ 0
113111, 112rereccli 10828 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (1 / 3) ∈ ℝ
114 readdcl 10057 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑗 ∈ ℝ ∧ (1 / 3) ∈ ℝ) → (𝑗 + (1 / 3)) ∈ ℝ)
115110, 113, 114sylancl 695 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑁) → (𝑗 + (1 / 3)) ∈ ℝ)
116115adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑗 + (1 / 3)) ∈ ℝ)
117 stoweidlem59.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐸 ∈ ℝ+)
118117rpred 11910 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐸 ∈ ℝ)
119118adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (0...𝑁)) → 𝐸 ∈ ℝ)
120116, 119remulcld 10108 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑗 + (1 / 3)) · 𝐸) ∈ ℝ)
121 stoweidlem59.16 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐹𝐶)
122121, 16syl6eleq 2740 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
123122adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑁)) → 𝐹 ∈ (𝐽 Cn 𝐾))
124107, 86, 87, 108, 120, 123rfcnpre3 39506 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (0...𝑁)) → {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} ∈ (Clsd‘𝐽))
125106, 124eqeltrd 2730 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐵𝑗) ∈ (Clsd‘𝐽))
126 rabexg 4844 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑇 ∈ V → {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)} ∈ V)
127102, 126syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑁)) → {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)} ∈ V)
12869fvmpt2 6330 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑗 ∈ (0...𝑁) ∧ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)} ∈ V) → (𝐷𝑗) = {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
12965, 127, 128syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐷𝑗) = {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
130 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . . 25 {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)} = {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)}
131 resubcl 10383 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑗 ∈ ℝ ∧ (1 / 3) ∈ ℝ) → (𝑗 − (1 / 3)) ∈ ℝ)
132110, 113, 131sylancl 695 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑁) → (𝑗 − (1 / 3)) ∈ ℝ)
133132adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑗 − (1 / 3)) ∈ ℝ)
134133, 119remulcld 10108 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑗 − (1 / 3)) · 𝐸) ∈ ℝ)
135107, 86, 87, 130, 134, 123rfcnpre4 39507 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (0...𝑁)) → {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)} ∈ (Clsd‘𝐽))
136129, 135eqeltrd 2730 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐷𝑗) ∈ (Clsd‘𝐽))
137134adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ((𝑗 − (1 / 3)) · 𝐸) ∈ ℝ)
138120adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ((𝑗 + (1 / 3)) · 𝐸) ∈ ℝ)
13986, 87, 16, 121fcnre 39498 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝐹:𝑇⟶ℝ)
140139ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → 𝐹:𝑇⟶ℝ)
141 ssrab2 3720 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} ⊆ 𝑇
142106, 141syl6eqss 3688 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐵𝑗) ⊆ 𝑇)
143142sselda 3636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → 𝑡𝑇)
144140, 143ffvelrnd 6400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → (𝐹𝑡) ∈ ℝ)
145113, 131mpan2 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ ℝ → (𝑗 − (1 / 3)) ∈ ℝ)
146 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ ℝ → 𝑗 ∈ ℝ)
147113, 114mpan2 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ ℝ → (𝑗 + (1 / 3)) ∈ ℝ)
148 3pos 11152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 0 < 3
149111, 148recgt0ii 10967 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 0 < (1 / 3)
150113, 149elrpii 11873 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (1 / 3) ∈ ℝ+
151 ltsubrp 11904 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑗 ∈ ℝ ∧ (1 / 3) ∈ ℝ+) → (𝑗 − (1 / 3)) < 𝑗)
152150, 151mpan2 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ ℝ → (𝑗 − (1 / 3)) < 𝑗)
153 ltaddrp 11905 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑗 ∈ ℝ ∧ (1 / 3) ∈ ℝ+) → 𝑗 < (𝑗 + (1 / 3)))
154150, 153mpan2 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ ℝ → 𝑗 < (𝑗 + (1 / 3)))
155145, 146, 147, 152, 154lttrd 10236 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑗 ∈ ℝ → (𝑗 − (1 / 3)) < (𝑗 + (1 / 3)))
156110, 155syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑗 ∈ (0...𝑁) → (𝑗 − (1 / 3)) < (𝑗 + (1 / 3)))
157156adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑗 − (1 / 3)) < (𝑗 + (1 / 3)))
158117rpregt0d 11916 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
159158adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
160 ltmul1 10911 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑗 − (1 / 3)) ∈ ℝ ∧ (𝑗 + (1 / 3)) ∈ ℝ ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → ((𝑗 − (1 / 3)) < (𝑗 + (1 / 3)) ↔ ((𝑗 − (1 / 3)) · 𝐸) < ((𝑗 + (1 / 3)) · 𝐸)))
161133, 116, 159, 160syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑗 − (1 / 3)) < (𝑗 + (1 / 3)) ↔ ((𝑗 − (1 / 3)) · 𝐸) < ((𝑗 + (1 / 3)) · 𝐸)))
162157, 161mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑗 − (1 / 3)) · 𝐸) < ((𝑗 + (1 / 3)) · 𝐸))
163162adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ((𝑗 − (1 / 3)) · 𝐸) < ((𝑗 + (1 / 3)) · 𝐸))
164106eleq2d 2716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑡 ∈ (𝐵𝑗) ↔ 𝑡 ∈ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)}))
165164biimpa 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → 𝑡 ∈ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
166 rabid 3145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 ∈ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} ↔ (𝑡𝑇 ∧ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)))
167165, 166sylib 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → (𝑡𝑇 ∧ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)))
168167simprd 478 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡))
169137, 138, 144, 163, 168ltletrd 10235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ((𝑗 − (1 / 3)) · 𝐸) < (𝐹𝑡))
170137, 144ltnled 10222 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → (((𝑗 − (1 / 3)) · 𝐸) < (𝐹𝑡) ↔ ¬ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)))
171169, 170mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ¬ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸))
172171intnand 982 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ¬ (𝑡𝑇 ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)))
173 rabid 3145 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 ∈ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)} ↔ (𝑡𝑇 ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)))
174172, 173sylnibr 318 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ¬ 𝑡 ∈ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
175129adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → (𝐷𝑗) = {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
176174, 175neleqtrrd 2752 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ¬ 𝑡 ∈ (𝐷𝑗))
177176ex 449 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑡 ∈ (𝐵𝑗) → ¬ 𝑡 ∈ (𝐷𝑗)))
17885, 177ralrimi 2986 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (0...𝑁)) → ∀𝑡 ∈ (𝐵𝑗) ¬ 𝑡 ∈ (𝐷𝑗))
179 disj 4050 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐵𝑗) ∩ (𝐷𝑗)) = ∅ ↔ ∀𝑎 ∈ (𝐵𝑗) ¬ 𝑎 ∈ (𝐷𝑗))
180 nfcv 2793 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑎(𝐵𝑗)
18175nfcri 2787 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑡 𝑎 ∈ (𝐷𝑗)
182181nfn 1824 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑡 ¬ 𝑎 ∈ (𝐷𝑗)
183 nfv 1883 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑎 ¬ 𝑡 ∈ (𝐷𝑗)
184 eleq1 2718 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎 = 𝑡 → (𝑎 ∈ (𝐷𝑗) ↔ 𝑡 ∈ (𝐷𝑗)))
185184notbid 307 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = 𝑡 → (¬ 𝑎 ∈ (𝐷𝑗) ↔ ¬ 𝑡 ∈ (𝐷𝑗)))
186180, 81, 182, 183, 185cbvralf 3195 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∀𝑎 ∈ (𝐵𝑗) ¬ 𝑎 ∈ (𝐷𝑗) ↔ ∀𝑡 ∈ (𝐵𝑗) ¬ 𝑡 ∈ (𝐷𝑗))
187179, 186bitri 264 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐵𝑗) ∩ (𝐷𝑗)) = ∅ ↔ ∀𝑡 ∈ (𝐵𝑗) ¬ 𝑡 ∈ (𝐷𝑗))
188178, 187sylibr 224 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝐵𝑗) ∩ (𝐷𝑗)) = ∅)
189 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑇 ∖ (𝐵𝑗)) = (𝑇 ∖ (𝐵𝑗))
190 stoweidlem59.19 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑁 ∈ ℕ)
191190nnrpd 11908 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑁 ∈ ℝ+)
192117, 191rpdivcld 11927 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐸 / 𝑁) ∈ ℝ+)
193192adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐸 / 𝑁) ∈ ℝ+)
194118, 190nndivred 11107 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐸 / 𝑁) ∈ ℝ)
195113a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (1 / 3) ∈ ℝ)
196190nnge1d 11101 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → 1 ≤ 𝑁)
197 1re 10077 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1 ∈ ℝ
198 0lt1 10588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 0 < 1
199197, 198pm3.2i 470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (1 ∈ ℝ ∧ 0 < 1)
200199a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (1 ∈ ℝ ∧ 0 < 1))
201190nnred 11073 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑁 ∈ ℝ)
202190nngt0d 11102 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → 0 < 𝑁)
203 lediv2 10951 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁) ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → (1 ≤ 𝑁 ↔ (𝐸 / 𝑁) ≤ (𝐸 / 1)))
204200, 201, 202, 158, 203syl121anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (1 ≤ 𝑁 ↔ (𝐸 / 𝑁) ≤ (𝐸 / 1)))
205196, 204mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐸 / 𝑁) ≤ (𝐸 / 1))
206117rpcnd 11912 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐸 ∈ ℂ)
207206div1d 10831 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐸 / 1) = 𝐸)
208205, 207breqtrd 4711 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐸 / 𝑁) ≤ 𝐸)
209 stoweidlem59.18 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐸 < (1 / 3))
210194, 118, 195, 208, 209lelttrd 10233 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐸 / 𝑁) < (1 / 3))
211210adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐸 / 𝑁) < (1 / 3))
21275, 82, 85, 86, 87, 16, 89, 90, 92, 94, 96, 98, 125, 136, 188, 189, 193, 211stoweidlem58 40593 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (0...𝑁)) → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))
213 df-rex 2947 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)) ↔ ∃𝑥(𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡))))
214212, 213sylib 208 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0...𝑁)) → ∃𝑥(𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡))))
215 simprl 809 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ (0...𝑁)) ∧ (𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))) → 𝑥𝐴)
216 simprr1 1129 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ (0...𝑁)) ∧ (𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))) → ∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1))
217 fveq1 6228 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = 𝑥 → (𝑦𝑡) = (𝑥𝑡))
218217breq2d 4697 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = 𝑥 → (0 ≤ (𝑦𝑡) ↔ 0 ≤ (𝑥𝑡)))
219217breq1d 4695 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = 𝑥 → ((𝑦𝑡) ≤ 1 ↔ (𝑥𝑡) ≤ 1))
220218, 219anbi12d 747 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑥 → ((0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1)))
221220ralbidv 3015 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑥 → (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1)))
222221, 1elrab2 3399 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝑌 ↔ (𝑥𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1)))
223215, 216, 222sylanbrc 699 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ (0...𝑁)) ∧ (𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))) → 𝑥𝑌)
224 simprr2 1130 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ (0...𝑁)) ∧ (𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))) → ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁))
225 simprr3 1131 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ (0...𝑁)) ∧ (𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡))
226224, 225jca 553 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ (0...𝑁)) ∧ (𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))) → (∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))
227 nfcv 2793 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦𝑥
228 nfv 1883 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦(∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡))
229217breq1d 4695 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑥 → ((𝑦𝑡) < (𝐸 / 𝑁) ↔ (𝑥𝑡) < (𝐸 / 𝑁)))
230229ralbidv 3015 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑥 → (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ↔ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁)))
231217breq2d 4697 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑥 → ((1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ (1 − (𝐸 / 𝑁)) < (𝑥𝑡)))
232231ralbidv 3015 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑥 → (∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))
233230, 232anbi12d 747 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑥 → ((∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡)) ↔ (∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡))))
234227, 3, 228, 233elrabf 3392 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} ↔ (𝑥𝑌 ∧ (∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡))))
235223, 226, 234sylanbrc 699 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (0...𝑁)) ∧ (𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))) → 𝑥 ∈ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
236235ex 449 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡))) → 𝑥 ∈ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))}))
237236eximdv 1886 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0...𝑁)) → (∃𝑥(𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡))) → ∃𝑥 𝑥 ∈ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))}))
238214, 237mpd 15 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0...𝑁)) → ∃𝑥 𝑥 ∈ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
239 ne0i 3954 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} → {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} ≠ ∅)
240239exlimiv 1898 . . . . . . . . . . . . . . . . . . . 20 (∃𝑥 𝑥 ∈ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} → {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} ≠ ∅)
241238, 240syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0...𝑁)) → {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} ≠ ∅)
24268, 241eqnetrd 2890 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐻𝑗) ≠ ∅)
2432423adant3 1101 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0...𝑁) ∧ (𝐻𝑗) = 𝑤) → (𝐻𝑗) ≠ ∅)
24464, 243eqnetrrd 2891 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0...𝑁) ∧ (𝐻𝑗) = 𝑤) → 𝑤 ≠ ∅)
2452443exp 1283 . . . . . . . . . . . . . . 15 (𝜑 → (𝑗 ∈ (0...𝑁) → ((𝐻𝑗) = 𝑤𝑤 ≠ ∅)))
246245rexlimdv 3059 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑗 ∈ (0...𝑁)(𝐻𝑗) = 𝑤𝑤 ≠ ∅))
247246adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ran 𝐻) → (∃𝑗 ∈ (0...𝑁)(𝐻𝑗) = 𝑤𝑤 ≠ ∅))
24863, 247mpd 15 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ran 𝐻) → 𝑤 ≠ ∅)
249248adantlr 751 . . . . . . . . . . 11 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑤 ∈ ran 𝐻) → 𝑤 ≠ ∅)
250 rsp 2958 . . . . . . . . . . 11 (∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤) → (𝑤 ∈ ran 𝐻 → (𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)))
25148, 49, 249, 250syl3c 66 . . . . . . . . . 10 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑤 ∈ ran 𝐻) → (𝑤) ∈ 𝑤)
252251ex 449 . . . . . . . . 9 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → (𝑤 ∈ ran 𝐻 → (𝑤) ∈ 𝑤))
25347, 252ralrimi 2986 . . . . . . . 8 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ∀𝑤 ∈ ran 𝐻(𝑤) ∈ 𝑤)
254 chfnrn 6368 . . . . . . . 8 (( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤) ∈ 𝑤) → ran ran 𝐻)
25532, 253, 254syl2anc 694 . . . . . . 7 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ran ran 𝐻)
256 nfv 1883 . . . . . . . . . 10 𝑦𝜑
257 nfcv 2793 . . . . . . . . . . . 12 𝑦
258 nfcv 2793 . . . . . . . . . . . . . . 15 𝑦(0...𝑁)
259 nfrab1 3152 . . . . . . . . . . . . . . 15 𝑦{𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))}
260258, 259nfmpt 4779 . . . . . . . . . . . . . 14 𝑦(𝑗 ∈ (0...𝑁) ↦ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
26122, 260nfcxfr 2791 . . . . . . . . . . . . 13 𝑦𝐻
262261nfrn 5400 . . . . . . . . . . . 12 𝑦ran 𝐻
263257, 262nffn 6025 . . . . . . . . . . 11 𝑦 Fn ran 𝐻
264 nfv 1883 . . . . . . . . . . . 12 𝑦(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)
265262, 264nfral 2974 . . . . . . . . . . 11 𝑦𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)
266263, 265nfan 1868 . . . . . . . . . 10 𝑦( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))
267256, 266nfan 1868 . . . . . . . . 9 𝑦(𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)))
268262nfuni 4474 . . . . . . . . 9 𝑦 ran 𝐻
269 fnunirn 6551 . . . . . . . . . . . . . . 15 (𝐻 Fn (0...𝑁) → (𝑦 ran 𝐻 ↔ ∃𝑧 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑧)))
270 nfcv 2793 . . . . . . . . . . . . . . . . . 18 𝑗𝑧
27153, 270nffv 6236 . . . . . . . . . . . . . . . . 17 𝑗(𝐻𝑧)
272271nfcri 2787 . . . . . . . . . . . . . . . 16 𝑗 𝑦 ∈ (𝐻𝑧)
273 nfv 1883 . . . . . . . . . . . . . . . 16 𝑧 𝑦 ∈ (𝐻𝑗)
274 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑗 → (𝐻𝑧) = (𝐻𝑗))
275274eleq2d 2716 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑗 → (𝑦 ∈ (𝐻𝑧) ↔ 𝑦 ∈ (𝐻𝑗)))
276272, 273, 275cbvrex 3198 . . . . . . . . . . . . . . 15 (∃𝑧 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑧) ↔ ∃𝑗 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑗))
277269, 276syl6bb 276 . . . . . . . . . . . . . 14 (𝐻 Fn (0...𝑁) → (𝑦 ran 𝐻 ↔ ∃𝑗 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑗)))
27824, 277syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑦 ran 𝐻 ↔ ∃𝑗 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑗)))
279278biimpa 500 . . . . . . . . . . . 12 ((𝜑𝑦 ran 𝐻) → ∃𝑗 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑗))
280 nfv 1883 . . . . . . . . . . . . . 14 𝑗𝜑
28153nfrn 5400 . . . . . . . . . . . . . . . 16 𝑗ran 𝐻
282281nfuni 4474 . . . . . . . . . . . . . . 15 𝑗 ran 𝐻
283282nfcri 2787 . . . . . . . . . . . . . 14 𝑗 𝑦 ran 𝐻
284280, 283nfan 1868 . . . . . . . . . . . . 13 𝑗(𝜑𝑦 ran 𝐻)
285 nfv 1883 . . . . . . . . . . . . 13 𝑗 𝑦𝑌
286 simp1l 1105 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ran 𝐻) ∧ 𝑗 ∈ (0...𝑁) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝜑)
287 simp2 1082 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ran 𝐻) ∧ 𝑗 ∈ (0...𝑁) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑗 ∈ (0...𝑁))
288 simp3 1083 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ran 𝐻) ∧ 𝑗 ∈ (0...𝑁) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑦 ∈ (𝐻𝑗))
28968eleq2d 2716 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑦 ∈ (𝐻𝑗) ↔ 𝑦 ∈ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))}))
290289biimpa 500 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑦 ∈ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
291 rabid 3145 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} ↔ (𝑦𝑌 ∧ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))))
292290, 291sylib 208 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → (𝑦𝑌 ∧ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))))
293292simpld 474 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑦𝑌)
294286, 287, 288, 293syl21anc 1365 . . . . . . . . . . . . . 14 (((𝜑𝑦 ran 𝐻) ∧ 𝑗 ∈ (0...𝑁) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑦𝑌)
2952943exp 1283 . . . . . . . . . . . . 13 ((𝜑𝑦 ran 𝐻) → (𝑗 ∈ (0...𝑁) → (𝑦 ∈ (𝐻𝑗) → 𝑦𝑌)))
296284, 285, 295rexlimd 3055 . . . . . . . . . . . 12 ((𝜑𝑦 ran 𝐻) → (∃𝑗 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑗) → 𝑦𝑌))
297279, 296mpd 15 . . . . . . . . . . 11 ((𝜑𝑦 ran 𝐻) → 𝑦𝑌)
298297adantlr 751 . . . . . . . . . 10 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑦 ran 𝐻) → 𝑦𝑌)
299298ex 449 . . . . . . . . 9 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → (𝑦 ran 𝐻𝑦𝑌))
300267, 268, 3, 299ssrd 3641 . . . . . . . 8 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ran 𝐻𝑌)
301 ssrab2 3720 . . . . . . . . 9 {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)} ⊆ 𝐴
3021, 301eqsstri 3668 . . . . . . . 8 𝑌𝐴
303300, 302syl6ss 3648 . . . . . . 7 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ran 𝐻𝐴)
304255, 303sstrd 3646 . . . . . 6 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ran 𝐴)
30542, 304fssd 6095 . . . . 5 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → :ran 𝐻𝐴)
306 dffn3 6092 . . . . . . 7 (𝐻 Fn (0...𝑁) ↔ 𝐻:(0...𝑁)⟶ran 𝐻)
30724, 306sylib 208 . . . . . 6 (𝜑𝐻:(0...𝑁)⟶ran 𝐻)
308307adantr 480 . . . . 5 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → 𝐻:(0...𝑁)⟶ran 𝐻)
309 fco 6096 . . . . 5 ((:ran 𝐻𝐴𝐻:(0...𝑁)⟶ran 𝐻) → (𝐻):(0...𝑁)⟶𝐴)
310305, 308, 309syl2anc 694 . . . 4 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → (𝐻):(0...𝑁)⟶𝐴)
311 nfcv 2793 . . . . . . . 8 𝑗
312311, 281nffn 6025 . . . . . . 7 𝑗 Fn ran 𝐻
313 nfv 1883 . . . . . . . 8 𝑗(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)
314281, 313nfral 2974 . . . . . . 7 𝑗𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)
315312, 314nfan 1868 . . . . . 6 𝑗( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))
316280, 315nfan 1868 . . . . 5 𝑗(𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)))
317 simpll 805 . . . . . . . . . 10 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → 𝜑)
318 simpr 476 . . . . . . . . . 10 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑗 ∈ (0...𝑁))
31924ad2antrr 762 . . . . . . . . . . . 12 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → 𝐻 Fn (0...𝑁))
320 fvco2 6312 . . . . . . . . . . . 12 ((𝐻 Fn (0...𝑁) ∧ 𝑗 ∈ (0...𝑁)) → ((𝐻)‘𝑗) = (‘(𝐻𝑗)))
321319, 320sylancom 702 . . . . . . . . . . 11 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝐻)‘𝑗) = (‘(𝐻𝑗)))
322 simplrr 818 . . . . . . . . . . . . 13 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))
323 fnfun 6026 . . . . . . . . . . . . . . . 16 (𝐻 Fn (0...𝑁) → Fun 𝐻)
32424, 323syl 17 . . . . . . . . . . . . . . 15 (𝜑 → Fun 𝐻)
325324ad2antrr 762 . . . . . . . . . . . . . 14 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → Fun 𝐻)
326 fndm 6028 . . . . . . . . . . . . . . . . . 18 (𝐻 Fn (0...𝑁) → dom 𝐻 = (0...𝑁))
32724, 326syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → dom 𝐻 = (0...𝑁))
328327adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0...𝑁)) → dom 𝐻 = (0...𝑁))
32965, 328eleqtrrd 2733 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑗 ∈ dom 𝐻)
330329adantlr 751 . . . . . . . . . . . . . 14 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑗 ∈ dom 𝐻)
331 fvelrn 6392 . . . . . . . . . . . . . 14 ((Fun 𝐻𝑗 ∈ dom 𝐻) → (𝐻𝑗) ∈ ran 𝐻)
332325, 330, 331syl2anc 694 . . . . . . . . . . . . 13 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → (𝐻𝑗) ∈ ran 𝐻)
333322, 332jca 553 . . . . . . . . . . . 12 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → (∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤) ∧ (𝐻𝑗) ∈ ran 𝐻))
334242adantlr 751 . . . . . . . . . . . 12 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → (𝐻𝑗) ≠ ∅)
335 neeq1 2885 . . . . . . . . . . . . . 14 (𝑤 = (𝐻𝑗) → (𝑤 ≠ ∅ ↔ (𝐻𝑗) ≠ ∅))
336 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑤 = (𝐻𝑗) → (𝑤) = (‘(𝐻𝑗)))
337 id 22 . . . . . . . . . . . . . . 15 (𝑤 = (𝐻𝑗) → 𝑤 = (𝐻𝑗))
338336, 337eleq12d 2724 . . . . . . . . . . . . . 14 (𝑤 = (𝐻𝑗) → ((𝑤) ∈ 𝑤 ↔ (‘(𝐻𝑗)) ∈ (𝐻𝑗)))
339335, 338imbi12d 333 . . . . . . . . . . . . 13 (𝑤 = (𝐻𝑗) → ((𝑤 ≠ ∅ → (𝑤) ∈ 𝑤) ↔ ((𝐻𝑗) ≠ ∅ → (‘(𝐻𝑗)) ∈ (𝐻𝑗))))
340339rspccva 3339 . . . . . . . . . . . 12 ((∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤) ∧ (𝐻𝑗) ∈ ran 𝐻) → ((𝐻𝑗) ≠ ∅ → (‘(𝐻𝑗)) ∈ (𝐻𝑗)))
341333, 334, 340sylc 65 . . . . . . . . . . 11 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → (‘(𝐻𝑗)) ∈ (𝐻𝑗))
342321, 341eqeltrd 2730 . . . . . . . . . 10 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝐻)‘𝑗) ∈ (𝐻𝑗))
343257, 261nfco 5320 . . . . . . . . . . . . 13 𝑦(𝐻)
344 nfcv 2793 . . . . . . . . . . . . 13 𝑦𝑗
345343, 344nffv 6236 . . . . . . . . . . . 12 𝑦((𝐻)‘𝑗)
346 nfv 1883 . . . . . . . . . . . . . 14 𝑦(𝜑𝑗 ∈ (0...𝑁))
347261, 344nffv 6236 . . . . . . . . . . . . . . 15 𝑦(𝐻𝑗)
348345, 347nfel 2806 . . . . . . . . . . . . . 14 𝑦((𝐻)‘𝑗) ∈ (𝐻𝑗)
349346, 348nfan 1868 . . . . . . . . . . . . 13 𝑦((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗))
350345, 3nfel 2806 . . . . . . . . . . . . 13 𝑦((𝐻)‘𝑗) ∈ 𝑌
351349, 350nfim 1865 . . . . . . . . . . . 12 𝑦(((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ((𝐻)‘𝑗) ∈ 𝑌)
352 eleq1 2718 . . . . . . . . . . . . . 14 (𝑦 = ((𝐻)‘𝑗) → (𝑦 ∈ (𝐻𝑗) ↔ ((𝐻)‘𝑗) ∈ (𝐻𝑗)))
353352anbi2d 740 . . . . . . . . . . . . 13 (𝑦 = ((𝐻)‘𝑗) → (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) ↔ ((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗))))
354 eleq1 2718 . . . . . . . . . . . . 13 (𝑦 = ((𝐻)‘𝑗) → (𝑦𝑌 ↔ ((𝐻)‘𝑗) ∈ 𝑌))
355353, 354imbi12d 333 . . . . . . . . . . . 12 (𝑦 = ((𝐻)‘𝑗) → ((((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑦𝑌) ↔ (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ((𝐻)‘𝑗) ∈ 𝑌)))
356345, 351, 355, 293vtoclgf 3295 . . . . . . . . . . 11 (((𝐻)‘𝑗) ∈ (𝐻𝑗) → (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ((𝐻)‘𝑗) ∈ 𝑌))
357356anabsi7 877 . . . . . . . . . 10 (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ((𝐻)‘𝑗) ∈ 𝑌)
358317, 318, 342, 357syl21anc 1365 . . . . . . . . 9 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝐻)‘𝑗) ∈ 𝑌)
3591eleq2i 2722 . . . . . . . . . 10 (((𝐻)‘𝑗) ∈ 𝑌 ↔ ((𝐻)‘𝑗) ∈ {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)})
360 nfcv 2793 . . . . . . . . . . 11 𝑦𝐴
361 nfcv 2793 . . . . . . . . . . . 12 𝑦𝑇
362 nfcv 2793 . . . . . . . . . . . . . 14 𝑦0
363 nfcv 2793 . . . . . . . . . . . . . 14 𝑦
364 nfcv 2793 . . . . . . . . . . . . . . 15 𝑦𝑡
365345, 364nffv 6236 . . . . . . . . . . . . . 14 𝑦(((𝐻)‘𝑗)‘𝑡)
366362, 363, 365nfbr 4732 . . . . . . . . . . . . 13 𝑦0 ≤ (((𝐻)‘𝑗)‘𝑡)
367 nfcv 2793 . . . . . . . . . . . . . 14 𝑦1
368365, 363, 367nfbr 4732 . . . . . . . . . . . . 13 𝑦(((𝐻)‘𝑗)‘𝑡) ≤ 1
369366, 368nfan 1868 . . . . . . . . . . . 12 𝑦(0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)
370361, 369nfral 2974 . . . . . . . . . . 11 𝑦𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)
371 nfcv 2793 . . . . . . . . . . . . 13 𝑡𝑦
372 nfcv 2793 . . . . . . . . . . . . . . 15 𝑡
373 nfra1 2970 . . . . . . . . . . . . . . . . . . 19 𝑡𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁)
374 nfra1 2970 . . . . . . . . . . . . . . . . . . 19 𝑡𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡)
375373, 374nfan 1868 . . . . . . . . . . . . . . . . . 18 𝑡(∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))
376 nfra1 2970 . . . . . . . . . . . . . . . . . . . 20 𝑡𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)
377 nfcv 2793 . . . . . . . . . . . . . . . . . . . 20 𝑡𝐴
378376, 377nfrab 3153 . . . . . . . . . . . . . . . . . . 19 𝑡{𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)}
3791, 378nfcxfr 2791 . . . . . . . . . . . . . . . . . 18 𝑡𝑌
380375, 379nfrab 3153 . . . . . . . . . . . . . . . . 17 𝑡{𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))}
38170, 380nfmpt 4779 . . . . . . . . . . . . . . . 16 𝑡(𝑗 ∈ (0...𝑁) ↦ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
38222, 381nfcxfr 2791 . . . . . . . . . . . . . . 15 𝑡𝐻
383372, 382nfco 5320 . . . . . . . . . . . . . 14 𝑡(𝐻)
384383, 74nffv 6236 . . . . . . . . . . . . 13 𝑡((𝐻)‘𝑗)
385371, 384nfeq 2805 . . . . . . . . . . . 12 𝑡 𝑦 = ((𝐻)‘𝑗)
386 fveq1 6228 . . . . . . . . . . . . . 14 (𝑦 = ((𝐻)‘𝑗) → (𝑦𝑡) = (((𝐻)‘𝑗)‘𝑡))
387386breq2d 4697 . . . . . . . . . . . . 13 (𝑦 = ((𝐻)‘𝑗) → (0 ≤ (𝑦𝑡) ↔ 0 ≤ (((𝐻)‘𝑗)‘𝑡)))
388386breq1d 4695 . . . . . . . . . . . . 13 (𝑦 = ((𝐻)‘𝑗) → ((𝑦𝑡) ≤ 1 ↔ (((𝐻)‘𝑗)‘𝑡) ≤ 1))
389387, 388anbi12d 747 . . . . . . . . . . . 12 (𝑦 = ((𝐻)‘𝑗) → ((0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
390385, 389ralbid 3012 . . . . . . . . . . 11 (𝑦 = ((𝐻)‘𝑗) → (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
391345, 360, 370, 390elrabf 3392 . . . . . . . . . 10 (((𝐻)‘𝑗) ∈ {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)} ↔ (((𝐻)‘𝑗) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
392359, 391bitri 264 . . . . . . . . 9 (((𝐻)‘𝑗) ∈ 𝑌 ↔ (((𝐻)‘𝑗) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
393358, 392sylib 208 . . . . . . . 8 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → (((𝐻)‘𝑗) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
394393simprd 478 . . . . . . 7 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1))
395 nfcv 2793 . . . . . . . . . . . 12 𝑦(𝐷𝑗)
396 nfcv 2793 . . . . . . . . . . . . 13 𝑦 <
397 nfcv 2793 . . . . . . . . . . . . 13 𝑦(𝐸 / 𝑁)
398365, 396, 397nfbr 4732 . . . . . . . . . . . 12 𝑦(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)
399395, 398nfral 2974 . . . . . . . . . . 11 𝑦𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)
400349, 399nfim 1865 . . . . . . . . . 10 𝑦(((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁))
401386breq1d 4695 . . . . . . . . . . . 12 (𝑦 = ((𝐻)‘𝑗) → ((𝑦𝑡) < (𝐸 / 𝑁) ↔ (((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)))
402385, 401ralbid 3012 . . . . . . . . . . 11 (𝑦 = ((𝐻)‘𝑗) → (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ↔ ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)))
403353, 402imbi12d 333 . . . . . . . . . 10 (𝑦 = ((𝐻)‘𝑗) → ((((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁)) ↔ (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁))))
404292simprd 478 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡)))
405404simpld 474 . . . . . . . . . 10 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁))
406345, 400, 403, 405vtoclgf 3295 . . . . . . . . 9 (((𝐻)‘𝑗) ∈ (𝐻𝑗) → (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)))
407406anabsi7 877 . . . . . . . 8 (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁))
408317, 318, 342, 407syl21anc 1365 . . . . . . 7 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁))
409 nfcv 2793 . . . . . . . . . . . 12 𝑦(𝐵𝑗)
410 nfcv 2793 . . . . . . . . . . . . 13 𝑦(1 − (𝐸 / 𝑁))
411410, 396, 365nfbr 4732 . . . . . . . . . . . 12 𝑦(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)
412409, 411nfral 2974 . . . . . . . . . . 11 𝑦𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)
413349, 412nfim 1865 . . . . . . . . . 10 𝑦(((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡))
414386breq2d 4697 . . . . . . . . . . . 12 (𝑦 = ((𝐻)‘𝑗) → ((1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ (1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))
415385, 414ralbid 3012 . . . . . . . . . . 11 (𝑦 = ((𝐻)‘𝑗) → (∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))
416353, 415imbi12d 333 . . . . . . . . . 10 (𝑦 = ((𝐻)‘𝑗) → ((((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡)) ↔ (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡))))
417404simprd 478 . . . . . . . . . 10 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))
418345, 413, 416, 417vtoclgf 3295 . . . . . . . . 9 (((𝐻)‘𝑗) ∈ (𝐻𝑗) → (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))
419418anabsi7 877 . . . . . . . 8 (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡))
420317, 318, 342, 419syl21anc 1365 . . . . . . 7 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡))
421394, 408, 4203jca 1261 . . . . . 6 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → (∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))
422421ex 449 . . . . 5 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → (𝑗 ∈ (0...𝑁) → (∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡))))
423316, 422ralrimi 2986 . . . 4 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ∀𝑗 ∈ (0...𝑁)(∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))
424310, 423jca 553 . . 3 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ((𝐻):(0...𝑁)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑁)(∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡))))
425 feq1 6064 . . . . 5 (𝑥 = (𝐻) → (𝑥:(0...𝑁)⟶𝐴 ↔ (𝐻):(0...𝑁)⟶𝐴))
426 nfcv 2793 . . . . . . 7 𝑗𝑥
427311, 53nfco 5320 . . . . . . 7 𝑗(𝐻)
428426, 427nfeq 2805 . . . . . 6 𝑗 𝑥 = (𝐻)
429 nfcv 2793 . . . . . . . . 9 𝑡𝑥
430429, 383nfeq 2805 . . . . . . . 8 𝑡 𝑥 = (𝐻)
431 fveq1 6228 . . . . . . . . . . 11 (𝑥 = (𝐻) → (𝑥𝑗) = ((𝐻)‘𝑗))
432431fveq1d 6231 . . . . . . . . . 10 (𝑥 = (𝐻) → ((𝑥𝑗)‘𝑡) = (((𝐻)‘𝑗)‘𝑡))
433432breq2d 4697 . . . . . . . . 9 (𝑥 = (𝐻) → (0 ≤ ((𝑥𝑗)‘𝑡) ↔ 0 ≤ (((𝐻)‘𝑗)‘𝑡)))
434432breq1d 4695 . . . . . . . . 9 (𝑥 = (𝐻) → (((𝑥𝑗)‘𝑡) ≤ 1 ↔ (((𝐻)‘𝑗)‘𝑡) ≤ 1))
435433, 434anbi12d 747 . . . . . . . 8 (𝑥 = (𝐻) → ((0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ↔ (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
436430, 435ralbid 3012 . . . . . . 7 (𝑥 = (𝐻) → (∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
437432breq1d 4695 . . . . . . . 8 (𝑥 = (𝐻) → (((𝑥𝑗)‘𝑡) < (𝐸 / 𝑁) ↔ (((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)))
438430, 437ralbid 3012 . . . . . . 7 (𝑥 = (𝐻) → (∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑁) ↔ ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)))
439432breq2d 4697 . . . . . . . 8 (𝑥 = (𝐻) → ((1 − (𝐸 / 𝑁)) < ((𝑥𝑗)‘𝑡) ↔ (1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))
440430, 439ralbid 3012 . . . . . . 7 (𝑥 = (𝐻) → (∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < ((𝑥𝑗)‘𝑡) ↔ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))
441436, 438, 4403anbi123d 1439 . . . . . 6 (𝑥 = (𝐻) → ((∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < ((𝑥𝑗)‘𝑡)) ↔ (∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡))))
442428, 441ralbid 3012 . . . . 5 (𝑥 = (𝐻) → (∀𝑗 ∈ (0...𝑁)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < ((𝑥𝑗)‘𝑡)) ↔ ∀𝑗 ∈ (0...𝑁)(∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡))))
443425, 442anbi12d 747 . . . 4 (𝑥 = (𝐻) → ((𝑥:(0...𝑁)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑁)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < ((𝑥𝑗)‘𝑡))) ↔ ((𝐻):(0...𝑁)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑁)(∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))))
444443spcegv 3325 . . 3 ((𝐻) ∈ V → (((𝐻):(0...𝑁)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑁)(∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡))) → ∃𝑥(𝑥:(0...𝑁)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑁)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < ((𝑥𝑗)‘𝑡)))))
44540, 424, 444sylc 65 . 2 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ∃𝑥(𝑥:(0...𝑁)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑁)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < ((𝑥𝑗)‘𝑡))))
44631, 445exlimddv 1903 1 (𝜑 → ∃𝑥(𝑥:(0...𝑁)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑁)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < ((𝑥𝑗)‘𝑡))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wnf 1748  wcel 2030  wnfc 2780  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  cin 3606  wss 3607  c0 3948   cuni 4468   class class class wbr 4685  cmpt 4762  dom cdm 5143  ran crn 5144  ccom 5147  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  Fincfn 7997  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  3c3 11109  +crp 11870  (,)cioo 12213  ...cfz 12364  topGenctg 16145  Clsdccld 20868   Cn ccn 21076  Compccmp 21237
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-inf2 8576  ax-cnex 10030  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  ax-pre-sup 10052  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  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-int 4508  df-iun 4554  df-iin 4555  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-se 5103  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-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  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-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-cn 21079  df-cnp 21080  df-cmp 21238  df-tx 21413  df-hmeo 21606  df-xms 22172  df-ms 22173  df-tms 22174
This theorem is referenced by:  stoweidlem60  40595
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