Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem59 Structured version   Visualization version   GIF version

Theorem stoweidlem59 44290
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 - \epsilon / 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 3426 . . . . . . . . . 10 𝑦{𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)}
31, 2nfcxfr 2905 . . . . . . . . 9 𝑦𝑌
4 nfcv 2907 . . . . . . . . 9 𝑧𝑌
5 nfv 1917 . . . . . . . . 9 𝑧(∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))
6 nfv 1917 . . . . . . . . 9 𝑦(∀𝑡 ∈ (𝐷𝑗)(𝑧𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑧𝑡))
7 fveq1 6841 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑦𝑡) = (𝑧𝑡))
87breq1d 5115 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝑦𝑡) < (𝐸 / 𝑁) ↔ (𝑧𝑡) < (𝐸 / 𝑁)))
98ralbidv 3174 . . . . . . . . . 10 (𝑦 = 𝑧 → (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ↔ ∀𝑡 ∈ (𝐷𝑗)(𝑧𝑡) < (𝐸 / 𝑁)))
107breq2d 5117 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ (1 − (𝐸 / 𝑁)) < (𝑧𝑡)))
1110ralbidv 3174 . . . . . . . . . 10 (𝑦 = 𝑧 → (∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑧𝑡)))
129, 11anbi12d 631 . . . . . . . . 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, 12cbvrabw 3439 . . . . . . . 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 7392 . . . . . . . . . 10 (𝜑 → (𝐽 Cn 𝐾) ∈ V)
15 stoweidlem59.11 . . . . . . . . . . 11 (𝜑𝐴𝐶)
16 stoweidlem59.5 . . . . . . . . . . 11 𝐶 = (𝐽 Cn 𝐾)
1715, 16sseqtrdi 3994 . . . . . . . . . 10 (𝜑𝐴 ⊆ (𝐽 Cn 𝐾))
1814, 17ssexd 5281 . . . . . . . . 9 (𝜑𝐴 ∈ V)
191, 18rabexd 5290 . . . . . . . 8 (𝜑𝑌 ∈ V)
2013, 19rabexd 5290 . . . . . . 7 (𝜑 → {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} ∈ V)
2120ralrimivw 3147 . . . . . 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 6641 . . . . . 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 13877 . . . . 5 (0...𝑁) ∈ Fin
26 fnfi 9125 . . . . 5 ((𝐻 Fn (0...𝑁) ∧ (0...𝑁) ∈ Fin) → 𝐻 ∈ Fin)
2724, 25, 26sylancl 586 . . . 4 (𝜑𝐻 ∈ Fin)
28 rnfi 9279 . . . 4 (𝐻 ∈ Fin → ran 𝐻 ∈ Fin)
2927, 28syl 17 . . 3 (𝜑 → ran 𝐻 ∈ Fin)
30 fnchoice 43224 . . 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 769 . . . . 5 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → Fn ran 𝐻)
33 ovex 7390 . . . . . . . 8 (0...𝑁) ∈ V
3433mptex 7173 . . . . . . 7 (𝑗 ∈ (0...𝑁) ↦ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))}) ∈ V
3522, 34eqeltri 2834 . . . . . 6 𝐻 ∈ V
3635rnex 7849 . . . . 5 ran 𝐻 ∈ V
37 fnex 7167 . . . . 5 (( Fn ran 𝐻 ∧ ran 𝐻 ∈ V) → ∈ V)
3832, 36, 37sylancl 586 . . . 4 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ∈ V)
39 coexg 7866 . . . 4 (( ∈ V ∧ 𝐻 ∈ V) → (𝐻) ∈ V)
4038, 35, 39sylancl 586 . . 3 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → (𝐻) ∈ V)
41 dffn3 6681 . . . . . . 7 ( Fn ran 𝐻:ran 𝐻⟶ran )
4232, 41sylib 217 . . . . . 6 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → :ran 𝐻⟶ran )
43 nfv 1917 . . . . . . . . . 10 𝑤𝜑
44 nfv 1917 . . . . . . . . . . 11 𝑤 Fn ran 𝐻
45 nfra1 3267 . . . . . . . . . . 11 𝑤𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)
4644, 45nfan 1902 . . . . . . . . . 10 𝑤( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))
4743, 46nfan 1902 . . . . . . . . 9 𝑤(𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)))
48 simplrr 776 . . . . . . . . . . 11 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑤 ∈ ran 𝐻) → ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))
49 simpr 485 . . . . . . . . . . 11 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑤 ∈ ran 𝐻) → 𝑤 ∈ ran 𝐻)
50 fvelrnb 6903 . . . . . . . . . . . . . . . 16 (𝐻 Fn (0...𝑁) → (𝑤 ∈ ran 𝐻 ↔ ∃𝑎 ∈ (0...𝑁)(𝐻𝑎) = 𝑤))
51 nfv 1917 . . . . . . . . . . . . . . . . 17 𝑎(𝐻𝑗) = 𝑤
52 nfmpt1 5213 . . . . . . . . . . . . . . . . . . . 20 𝑗(𝑗 ∈ (0...𝑁) ↦ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
5322, 52nfcxfr 2905 . . . . . . . . . . . . . . . . . . 19 𝑗𝐻
54 nfcv 2907 . . . . . . . . . . . . . . . . . . 19 𝑗𝑎
5553, 54nffv 6852 . . . . . . . . . . . . . . . . . 18 𝑗(𝐻𝑎)
56 nfcv 2907 . . . . . . . . . . . . . . . . . 18 𝑗𝑤
5755, 56nfeq 2920 . . . . . . . . . . . . . . . . 17 𝑗(𝐻𝑎) = 𝑤
58 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑎 → (𝐻𝑗) = (𝐻𝑎))
5958eqeq1d 2738 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑎 → ((𝐻𝑗) = 𝑤 ↔ (𝐻𝑎) = 𝑤))
6051, 57, 59cbvrexw 3290 . . . . . . . . . . . . . . . 16 (∃𝑗 ∈ (0...𝑁)(𝐻𝑗) = 𝑤 ↔ ∃𝑎 ∈ (0...𝑁)(𝐻𝑎) = 𝑤)
6150, 60bitr4di 288 . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ran 𝐻) → ∃𝑗 ∈ (0...𝑁)(𝐻𝑗) = 𝑤)
64 simp3 1138 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0...𝑁) ∧ (𝐻𝑗) = 𝑤) → (𝐻𝑗) = 𝑤)
65 simpr 485 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑗 ∈ (0...𝑁))
6620adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0...𝑁)) → {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} ∈ V)
6722fvmpt2 6959 . . . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . . . 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 2907 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑡(0...𝑁)
71 nfrab1 3426 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑡{𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)}
7270, 71nfmpt 5212 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑡(𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
7369, 72nfcxfr 2905 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡𝐷
74 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡𝑗
7573, 74nffv 6852 . . . . . . . . . . . . . . . . . . . . . . 23 𝑡(𝐷𝑗)
76 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡𝑇
77 stoweidlem59.7 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝐵 = (𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
78 nfrab1 3426 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑡{𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)}
7970, 78nfmpt 5212 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑡(𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
8077, 79nfcxfr 2905 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑡𝐵
8180, 74nffv 6852 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡(𝐵𝑗)
8276, 81nfdif 4085 . . . . . . . . . . . . . . . . . . . . . . 23 𝑡(𝑇 ∖ (𝐵𝑗))
83 stoweidlem59.2 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡𝜑
84 nfv 1917 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡 𝑗 ∈ (0...𝑁)
8583, 84nfan 1902 . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → 𝐽 ∈ Comp)
9015adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → 𝐴𝐶)
91 stoweidlem59.12 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
92913adant1r 1177 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
93 stoweidlem59.13 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
94933adant1r 1177 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
95 stoweidlem59.14 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
9695adantlr 713 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
97 stoweidlem59.15 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))
9897adantlr 713 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (0...𝑁)) ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))
9988uniexd 7679 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 𝐽 ∈ V)
10087, 99eqeltrid 2842 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝑇 ∈ V)
101100adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑇 ∈ V)
102 rabexg 5288 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑇 ∈ V → {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} ∈ V)
103101, 102syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑁)) → {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} ∈ V)
10477fvmpt2 6959 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑗 ∈ (0...𝑁) ∧ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} ∈ V) → (𝐵𝑗) = {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
10565, 103, 104syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐵𝑗) = {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
106 stoweidlem59.1 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑡𝐹
107 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . 25 {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} = {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)}
108 elfzelz 13441 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
109108zred 12607 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℝ)
110 3re 12233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 ∈ ℝ
111 3ne0 12259 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 ≠ 0
112110, 111rereccli 11920 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (1 / 3) ∈ ℝ
113 readdcl 11134 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑗 ∈ ℝ ∧ (1 / 3) ∈ ℝ) → (𝑗 + (1 / 3)) ∈ ℝ)
114109, 112, 113sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑁) → (𝑗 + (1 / 3)) ∈ ℝ)
115114adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑗 + (1 / 3)) ∈ ℝ)
116 stoweidlem59.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐸 ∈ ℝ+)
117116rpred 12957 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐸 ∈ ℝ)
118117adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (0...𝑁)) → 𝐸 ∈ ℝ)
119115, 118remulcld 11185 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑗 + (1 / 3)) · 𝐸) ∈ ℝ)
120 stoweidlem59.16 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐹𝐶)
121120, 16eleqtrdi 2848 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
122121adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑁)) → 𝐹 ∈ (𝐽 Cn 𝐾))
123106, 86, 87, 107, 119, 122rfcnpre3 43228 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (0...𝑁)) → {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} ∈ (Clsd‘𝐽))
124105, 123eqeltrd 2838 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐵𝑗) ∈ (Clsd‘𝐽))
125 rabexg 5288 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑇 ∈ V → {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)} ∈ V)
126101, 125syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑁)) → {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)} ∈ V)
12769fvmpt2 6959 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑗 ∈ (0...𝑁) ∧ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)} ∈ V) → (𝐷𝑗) = {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
12865, 126, 127syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐷𝑗) = {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
129 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . 25 {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)} = {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)}
130 resubcl 11465 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑗 ∈ ℝ ∧ (1 / 3) ∈ ℝ) → (𝑗 − (1 / 3)) ∈ ℝ)
131109, 112, 130sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑁) → (𝑗 − (1 / 3)) ∈ ℝ)
132131adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑗 − (1 / 3)) ∈ ℝ)
133132, 118remulcld 11185 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑗 − (1 / 3)) · 𝐸) ∈ ℝ)
134106, 86, 87, 129, 133, 122rfcnpre4 43229 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (0...𝑁)) → {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)} ∈ (Clsd‘𝐽))
135128, 134eqeltrd 2838 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐷𝑗) ∈ (Clsd‘𝐽))
136133adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ((𝑗 − (1 / 3)) · 𝐸) ∈ ℝ)
137119adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ((𝑗 + (1 / 3)) · 𝐸) ∈ ℝ)
13886, 87, 16, 120fcnre 43220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝐹:𝑇⟶ℝ)
139138ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → 𝐹:𝑇⟶ℝ)
140 ssrab2 4037 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} ⊆ 𝑇
141105, 140eqsstrdi 3998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐵𝑗) ⊆ 𝑇)
142141sselda 3944 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → 𝑡𝑇)
143139, 142ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → (𝐹𝑡) ∈ ℝ)
144112, 130mpan2 689 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ ℝ → (𝑗 − (1 / 3)) ∈ ℝ)
145 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ ℝ → 𝑗 ∈ ℝ)
146112, 113mpan2 689 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ ℝ → (𝑗 + (1 / 3)) ∈ ℝ)
147 3pos 12258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 0 < 3
148110, 147recgt0ii 12061 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 0 < (1 / 3)
149112, 148elrpii 12918 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (1 / 3) ∈ ℝ+
150 ltsubrp 12951 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑗 ∈ ℝ ∧ (1 / 3) ∈ ℝ+) → (𝑗 − (1 / 3)) < 𝑗)
151149, 150mpan2 689 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ ℝ → (𝑗 − (1 / 3)) < 𝑗)
152 ltaddrp 12952 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑗 ∈ ℝ ∧ (1 / 3) ∈ ℝ+) → 𝑗 < (𝑗 + (1 / 3)))
153149, 152mpan2 689 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ ℝ → 𝑗 < (𝑗 + (1 / 3)))
154144, 145, 146, 151, 153lttrd 11316 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑗 ∈ ℝ → (𝑗 − (1 / 3)) < (𝑗 + (1 / 3)))
155109, 154syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑗 ∈ (0...𝑁) → (𝑗 − (1 / 3)) < (𝑗 + (1 / 3)))
156155adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑗 − (1 / 3)) < (𝑗 + (1 / 3)))
157116rpregt0d 12963 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
158157adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
159 ltmul1 12005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
160132, 115, 158, 159syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑗 − (1 / 3)) < (𝑗 + (1 / 3)) ↔ ((𝑗 − (1 / 3)) · 𝐸) < ((𝑗 + (1 / 3)) · 𝐸)))
161156, 160mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑗 − (1 / 3)) · 𝐸) < ((𝑗 + (1 / 3)) · 𝐸))
162161adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ((𝑗 − (1 / 3)) · 𝐸) < ((𝑗 + (1 / 3)) · 𝐸))
163105eleq2d 2823 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑡 ∈ (𝐵𝑗) ↔ 𝑡 ∈ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)}))
164163biimpa 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → 𝑡 ∈ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
165 rabid 3427 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 ∈ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} ↔ (𝑡𝑇 ∧ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)))
166164, 165sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → (𝑡𝑇 ∧ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)))
167166simprd 496 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡))
168136, 137, 143, 162, 167ltletrd 11315 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ((𝑗 − (1 / 3)) · 𝐸) < (𝐹𝑡))
169136, 143ltnled 11302 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → (((𝑗 − (1 / 3)) · 𝐸) < (𝐹𝑡) ↔ ¬ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)))
170168, 169mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ¬ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸))
171170intnand 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ¬ (𝑡𝑇 ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)))
172 rabid 3427 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 ∈ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)} ↔ (𝑡𝑇 ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)))
173171, 172sylnibr 328 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ¬ 𝑡 ∈ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
174128adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → (𝐷𝑗) = {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
175173, 174neleqtrrd 2860 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ¬ 𝑡 ∈ (𝐷𝑗))
176175ex 413 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑡 ∈ (𝐵𝑗) → ¬ 𝑡 ∈ (𝐷𝑗)))
17785, 176ralrimi 3240 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (0...𝑁)) → ∀𝑡 ∈ (𝐵𝑗) ¬ 𝑡 ∈ (𝐷𝑗))
178 disj 4407 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐵𝑗) ∩ (𝐷𝑗)) = ∅ ↔ ∀𝑎 ∈ (𝐵𝑗) ¬ 𝑎 ∈ (𝐷𝑗))
179 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑎(𝐵𝑗)
18075nfcri 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑡 𝑎 ∈ (𝐷𝑗)
181180nfn 1860 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑡 ¬ 𝑎 ∈ (𝐷𝑗)
182 nfv 1917 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑎 ¬ 𝑡 ∈ (𝐷𝑗)
183 eleq1 2825 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎 = 𝑡 → (𝑎 ∈ (𝐷𝑗) ↔ 𝑡 ∈ (𝐷𝑗)))
184183notbid 317 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = 𝑡 → (¬ 𝑎 ∈ (𝐷𝑗) ↔ ¬ 𝑡 ∈ (𝐷𝑗)))
185179, 81, 181, 182, 184cbvralfw 3287 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∀𝑎 ∈ (𝐵𝑗) ¬ 𝑎 ∈ (𝐷𝑗) ↔ ∀𝑡 ∈ (𝐵𝑗) ¬ 𝑡 ∈ (𝐷𝑗))
186178, 185bitri 274 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐵𝑗) ∩ (𝐷𝑗)) = ∅ ↔ ∀𝑡 ∈ (𝐵𝑗) ¬ 𝑡 ∈ (𝐷𝑗))
187177, 186sylibr 233 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝐵𝑗) ∩ (𝐷𝑗)) = ∅)
188 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑇 ∖ (𝐵𝑗)) = (𝑇 ∖ (𝐵𝑗))
189 stoweidlem59.19 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑁 ∈ ℕ)
190189nnrpd 12955 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑁 ∈ ℝ+)
191116, 190rpdivcld 12974 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐸 / 𝑁) ∈ ℝ+)
192191adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐸 / 𝑁) ∈ ℝ+)
193117, 189nndivred 12207 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐸 / 𝑁) ∈ ℝ)
194112a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (1 / 3) ∈ ℝ)
195189nnge1d 12201 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → 1 ≤ 𝑁)
196 1re 11155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1 ∈ ℝ
197 0lt1 11677 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 0 < 1
198196, 197pm3.2i 471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (1 ∈ ℝ ∧ 0 < 1)
199198a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (1 ∈ ℝ ∧ 0 < 1))
200189nnred 12168 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑁 ∈ ℝ)
201189nngt0d 12202 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → 0 < 𝑁)
202 lediv2 12045 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁) ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → (1 ≤ 𝑁 ↔ (𝐸 / 𝑁) ≤ (𝐸 / 1)))
203199, 200, 201, 157, 202syl121anc 1375 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (1 ≤ 𝑁 ↔ (𝐸 / 𝑁) ≤ (𝐸 / 1)))
204195, 203mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐸 / 𝑁) ≤ (𝐸 / 1))
205116rpcnd 12959 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐸 ∈ ℂ)
206205div1d 11923 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐸 / 1) = 𝐸)
207204, 206breqtrd 5131 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐸 / 𝑁) ≤ 𝐸)
208 stoweidlem59.18 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐸 < (1 / 3))
209193, 117, 194, 207, 208lelttrd 11313 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐸 / 𝑁) < (1 / 3))
210209adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐸 / 𝑁) < (1 / 3))
21175, 82, 85, 86, 87, 16, 89, 90, 92, 94, 96, 98, 124, 135, 187, 188, 192, 210stoweidlem58 44289 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (0...𝑁)) → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))
212 df-rex 3074 . . . . . . . . . . . . . . . . . . . . . 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))))
213211, 212sylib 217 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0...𝑁)) → ∃𝑥(𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡))))
214 simprl 769 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ (0...𝑁)) ∧ (𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))) → 𝑥𝐴)
215 simprr1 1221 . . . . . . . . . . . . . . . . . . . . . . . . 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))
216 fveq1 6841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = 𝑥 → (𝑦𝑡) = (𝑥𝑡))
217216breq2d 5117 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = 𝑥 → (0 ≤ (𝑦𝑡) ↔ 0 ≤ (𝑥𝑡)))
218216breq1d 5115 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = 𝑥 → ((𝑦𝑡) ≤ 1 ↔ (𝑥𝑡) ≤ 1))
219217, 218anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑥 → ((0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1)))
220219ralbidv 3174 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑥 → (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1)))
221220, 1elrab2 3648 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝑌 ↔ (𝑥𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1)))
222214, 215, 221sylanbrc 583 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ (0...𝑁)) ∧ (𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))) → 𝑥𝑌)
223 simprr2 1222 . . . . . . . . . . . . . . . . . . . . . . . . 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))
224 simprr3 1223 . . . . . . . . . . . . . . . . . . . . . . . . 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))
225223, 224jca 512 . . . . . . . . . . . . . . . . . . . . . . . 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)))
226 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦𝑥
227 nfv 1917 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦(∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡))
228216breq1d 5115 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑥 → ((𝑦𝑡) < (𝐸 / 𝑁) ↔ (𝑥𝑡) < (𝐸 / 𝑁)))
229228ralbidv 3174 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑥 → (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ↔ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁)))
230216breq2d 5117 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑥 → ((1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ (1 − (𝐸 / 𝑁)) < (𝑥𝑡)))
231230ralbidv 3174 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑥 → (∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))
232229, 231anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
233226, 3, 227, 232elrabf 3641 . . . . . . . . . . . . . . . . . . . . . . . 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))))
234222, 225, 233sylanbrc 583 . . . . . . . . . . . . . . . . . . . . . . 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))})
235234ex 413 . . . . . . . . . . . . . . . . . . . . . 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))}))
236235eximdv 1920 . . . . . . . . . . . . . . . . . . . . 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))}))
237213, 236mpd 15 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0...𝑁)) → ∃𝑥 𝑥 ∈ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
238 ne0i 4294 . . . . . . . . . . . . . . . . . . . . 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))} ≠ ∅)
239238exlimiv 1933 . . . . . . . . . . . . . . . . . . . 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))} ≠ ∅)
240237, 239syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0...𝑁)) → {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} ≠ ∅)
24168, 240eqnetrd 3011 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐻𝑗) ≠ ∅)
2422413adant3 1132 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0...𝑁) ∧ (𝐻𝑗) = 𝑤) → (𝐻𝑗) ≠ ∅)
24364, 242eqnetrrd 3012 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0...𝑁) ∧ (𝐻𝑗) = 𝑤) → 𝑤 ≠ ∅)
2442433exp 1119 . . . . . . . . . . . . . . 15 (𝜑 → (𝑗 ∈ (0...𝑁) → ((𝐻𝑗) = 𝑤𝑤 ≠ ∅)))
245244rexlimdv 3150 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑗 ∈ (0...𝑁)(𝐻𝑗) = 𝑤𝑤 ≠ ∅))
246245adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ran 𝐻) → (∃𝑗 ∈ (0...𝑁)(𝐻𝑗) = 𝑤𝑤 ≠ ∅))
24763, 246mpd 15 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ran 𝐻) → 𝑤 ≠ ∅)
248247adantlr 713 . . . . . . . . . . 11 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑤 ∈ ran 𝐻) → 𝑤 ≠ ∅)
249 rsp 3230 . . . . . . . . . . 11 (∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤) → (𝑤 ∈ ran 𝐻 → (𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)))
25048, 49, 248, 249syl3c 66 . . . . . . . . . 10 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑤 ∈ ran 𝐻) → (𝑤) ∈ 𝑤)
251250ex 413 . . . . . . . . 9 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → (𝑤 ∈ ran 𝐻 → (𝑤) ∈ 𝑤))
25247, 251ralrimi 3240 . . . . . . . 8 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ∀𝑤 ∈ ran 𝐻(𝑤) ∈ 𝑤)
253 chfnrn 6999 . . . . . . . 8 (( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤) ∈ 𝑤) → ran ran 𝐻)
25432, 252, 253syl2anc 584 . . . . . . 7 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ran ran 𝐻)
255 nfv 1917 . . . . . . . . . 10 𝑦𝜑
256 nfcv 2907 . . . . . . . . . . . 12 𝑦
257 nfcv 2907 . . . . . . . . . . . . . . 15 𝑦(0...𝑁)
258 nfrab1 3426 . . . . . . . . . . . . . . 15 𝑦{𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))}
259257, 258nfmpt 5212 . . . . . . . . . . . . . 14 𝑦(𝑗 ∈ (0...𝑁) ↦ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
26022, 259nfcxfr 2905 . . . . . . . . . . . . 13 𝑦𝐻
261260nfrn 5907 . . . . . . . . . . . 12 𝑦ran 𝐻
262256, 261nffn 6601 . . . . . . . . . . 11 𝑦 Fn ran 𝐻
263 nfv 1917 . . . . . . . . . . . 12 𝑦(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)
264261, 263nfralw 3294 . . . . . . . . . . 11 𝑦𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)
265262, 264nfan 1902 . . . . . . . . . 10 𝑦( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))
266255, 265nfan 1902 . . . . . . . . 9 𝑦(𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)))
267261nfuni 4872 . . . . . . . . 9 𝑦 ran 𝐻
268 fnunirn 7201 . . . . . . . . . . . . . . 15 (𝐻 Fn (0...𝑁) → (𝑦 ran 𝐻 ↔ ∃𝑧 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑧)))
269 nfcv 2907 . . . . . . . . . . . . . . . . . 18 𝑗𝑧
27053, 269nffv 6852 . . . . . . . . . . . . . . . . 17 𝑗(𝐻𝑧)
271270nfcri 2894 . . . . . . . . . . . . . . . 16 𝑗 𝑦 ∈ (𝐻𝑧)
272 nfv 1917 . . . . . . . . . . . . . . . 16 𝑧 𝑦 ∈ (𝐻𝑗)
273 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑗 → (𝐻𝑧) = (𝐻𝑗))
274273eleq2d 2823 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑗 → (𝑦 ∈ (𝐻𝑧) ↔ 𝑦 ∈ (𝐻𝑗)))
275271, 272, 274cbvrexw 3290 . . . . . . . . . . . . . . 15 (∃𝑧 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑧) ↔ ∃𝑗 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑗))
276268, 275bitrdi 286 . . . . . . . . . . . . . 14 (𝐻 Fn (0...𝑁) → (𝑦 ran 𝐻 ↔ ∃𝑗 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑗)))
27724, 276syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑦 ran 𝐻 ↔ ∃𝑗 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑗)))
278277biimpa 477 . . . . . . . . . . . 12 ((𝜑𝑦 ran 𝐻) → ∃𝑗 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑗))
279 nfv 1917 . . . . . . . . . . . . . 14 𝑗𝜑
28053nfrn 5907 . . . . . . . . . . . . . . . 16 𝑗ran 𝐻
281280nfuni 4872 . . . . . . . . . . . . . . 15 𝑗 ran 𝐻
282281nfcri 2894 . . . . . . . . . . . . . 14 𝑗 𝑦 ran 𝐻
283279, 282nfan 1902 . . . . . . . . . . . . 13 𝑗(𝜑𝑦 ran 𝐻)
284 nfv 1917 . . . . . . . . . . . . 13 𝑗 𝑦𝑌
285 simp1l 1197 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ran 𝐻) ∧ 𝑗 ∈ (0...𝑁) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝜑)
286 simp2 1137 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ran 𝐻) ∧ 𝑗 ∈ (0...𝑁) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑗 ∈ (0...𝑁))
287 simp3 1138 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ran 𝐻) ∧ 𝑗 ∈ (0...𝑁) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑦 ∈ (𝐻𝑗))
28868eleq2d 2823 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑦 ∈ (𝐻𝑗) ↔ 𝑦 ∈ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))}))
289288biimpa 477 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑦 ∈ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
290 rabid 3427 . . . . . . . . . . . . . . . . 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))))
291289, 290sylib 217 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → (𝑦𝑌 ∧ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))))
292291simpld 495 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑦𝑌)
293285, 286, 287, 292syl21anc 836 . . . . . . . . . . . . . 14 (((𝜑𝑦 ran 𝐻) ∧ 𝑗 ∈ (0...𝑁) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑦𝑌)
2942933exp 1119 . . . . . . . . . . . . 13 ((𝜑𝑦 ran 𝐻) → (𝑗 ∈ (0...𝑁) → (𝑦 ∈ (𝐻𝑗) → 𝑦𝑌)))
295283, 284, 294rexlimd 3249 . . . . . . . . . . . 12 ((𝜑𝑦 ran 𝐻) → (∃𝑗 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑗) → 𝑦𝑌))
296278, 295mpd 15 . . . . . . . . . . 11 ((𝜑𝑦 ran 𝐻) → 𝑦𝑌)
297296adantlr 713 . . . . . . . . . 10 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑦 ran 𝐻) → 𝑦𝑌)
298297ex 413 . . . . . . . . 9 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → (𝑦 ran 𝐻𝑦𝑌))
299266, 267, 3, 298ssrd 3949 . . . . . . . 8 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ran 𝐻𝑌)
300 ssrab2 4037 . . . . . . . . 9 {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)} ⊆ 𝐴
3011, 300eqsstri 3978 . . . . . . . 8 𝑌𝐴
302299, 301sstrdi 3956 . . . . . . 7 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ran 𝐻𝐴)
303254, 302sstrd 3954 . . . . . 6 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ran 𝐴)
30442, 303fssd 6686 . . . . 5 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → :ran 𝐻𝐴)
305 dffn3 6681 . . . . . . 7 (𝐻 Fn (0...𝑁) ↔ 𝐻:(0...𝑁)⟶ran 𝐻)
30624, 305sylib 217 . . . . . 6 (𝜑𝐻:(0...𝑁)⟶ran 𝐻)
307306adantr 481 . . . . 5 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → 𝐻:(0...𝑁)⟶ran 𝐻)
308 fco 6692 . . . . 5 ((:ran 𝐻𝐴𝐻:(0...𝑁)⟶ran 𝐻) → (𝐻):(0...𝑁)⟶𝐴)
309304, 307, 308syl2anc 584 . . . 4 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → (𝐻):(0...𝑁)⟶𝐴)
310 nfcv 2907 . . . . . . . 8 𝑗
311310, 280nffn 6601 . . . . . . 7 𝑗 Fn ran 𝐻
312 nfv 1917 . . . . . . . 8 𝑗(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)
313280, 312nfralw 3294 . . . . . . 7 𝑗𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)
314311, 313nfan 1902 . . . . . 6 𝑗( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))
315279, 314nfan 1902 . . . . 5 𝑗(𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)))
316 simpll 765 . . . . . . . . . 10 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → 𝜑)
317 simpr 485 . . . . . . . . . 10 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑗 ∈ (0...𝑁))
31824ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → 𝐻 Fn (0...𝑁))
319 fvco2 6938 . . . . . . . . . . . 12 ((𝐻 Fn (0...𝑁) ∧ 𝑗 ∈ (0...𝑁)) → ((𝐻)‘𝑗) = (‘(𝐻𝑗)))
320318, 319sylancom 588 . . . . . . . . . . 11 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝐻)‘𝑗) = (‘(𝐻𝑗)))
321 simplrr 776 . . . . . . . . . . . . 13 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))
322 fnfun 6602 . . . . . . . . . . . . . . . 16 (𝐻 Fn (0...𝑁) → Fun 𝐻)
32324, 322syl 17 . . . . . . . . . . . . . . 15 (𝜑 → Fun 𝐻)
324323ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → Fun 𝐻)
32524fndmd 6607 . . . . . . . . . . . . . . . . 17 (𝜑 → dom 𝐻 = (0...𝑁))
326325adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0...𝑁)) → dom 𝐻 = (0...𝑁))
32765, 326eleqtrrd 2841 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑗 ∈ dom 𝐻)
328327adantlr 713 . . . . . . . . . . . . . 14 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑗 ∈ dom 𝐻)
329 fvelrn 7027 . . . . . . . . . . . . . 14 ((Fun 𝐻𝑗 ∈ dom 𝐻) → (𝐻𝑗) ∈ ran 𝐻)
330324, 328, 329syl2anc 584 . . . . . . . . . . . . 13 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → (𝐻𝑗) ∈ ran 𝐻)
331321, 330jca 512 . . . . . . . . . . . 12 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → (∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤) ∧ (𝐻𝑗) ∈ ran 𝐻))
332241adantlr 713 . . . . . . . . . . . 12 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → (𝐻𝑗) ≠ ∅)
333 neeq1 3006 . . . . . . . . . . . . . 14 (𝑤 = (𝐻𝑗) → (𝑤 ≠ ∅ ↔ (𝐻𝑗) ≠ ∅))
334 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑤 = (𝐻𝑗) → (𝑤) = (‘(𝐻𝑗)))
335 id 22 . . . . . . . . . . . . . . 15 (𝑤 = (𝐻𝑗) → 𝑤 = (𝐻𝑗))
336334, 335eleq12d 2832 . . . . . . . . . . . . . 14 (𝑤 = (𝐻𝑗) → ((𝑤) ∈ 𝑤 ↔ (‘(𝐻𝑗)) ∈ (𝐻𝑗)))
337333, 336imbi12d 344 . . . . . . . . . . . . 13 (𝑤 = (𝐻𝑗) → ((𝑤 ≠ ∅ → (𝑤) ∈ 𝑤) ↔ ((𝐻𝑗) ≠ ∅ → (‘(𝐻𝑗)) ∈ (𝐻𝑗))))
338337rspccva 3580 . . . . . . . . . . . 12 ((∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤) ∧ (𝐻𝑗) ∈ ran 𝐻) → ((𝐻𝑗) ≠ ∅ → (‘(𝐻𝑗)) ∈ (𝐻𝑗)))
339331, 332, 338sylc 65 . . . . . . . . . . 11 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → (‘(𝐻𝑗)) ∈ (𝐻𝑗))
340320, 339eqeltrd 2838 . . . . . . . . . 10 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝐻)‘𝑗) ∈ (𝐻𝑗))
341256, 260nfco 5821 . . . . . . . . . . . . 13 𝑦(𝐻)
342 nfcv 2907 . . . . . . . . . . . . 13 𝑦𝑗
343341, 342nffv 6852 . . . . . . . . . . . 12 𝑦((𝐻)‘𝑗)
344 nfv 1917 . . . . . . . . . . . . . 14 𝑦(𝜑𝑗 ∈ (0...𝑁))
345260, 342nffv 6852 . . . . . . . . . . . . . . 15 𝑦(𝐻𝑗)
346343, 345nfel 2921 . . . . . . . . . . . . . 14 𝑦((𝐻)‘𝑗) ∈ (𝐻𝑗)
347344, 346nfan 1902 . . . . . . . . . . . . 13 𝑦((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗))
348343, 3nfel 2921 . . . . . . . . . . . . 13 𝑦((𝐻)‘𝑗) ∈ 𝑌
349347, 348nfim 1899 . . . . . . . . . . . 12 𝑦(((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ((𝐻)‘𝑗) ∈ 𝑌)
350 eleq1 2825 . . . . . . . . . . . . . 14 (𝑦 = ((𝐻)‘𝑗) → (𝑦 ∈ (𝐻𝑗) ↔ ((𝐻)‘𝑗) ∈ (𝐻𝑗)))
351350anbi2d 629 . . . . . . . . . . . . 13 (𝑦 = ((𝐻)‘𝑗) → (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) ↔ ((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗))))
352 eleq1 2825 . . . . . . . . . . . . 13 (𝑦 = ((𝐻)‘𝑗) → (𝑦𝑌 ↔ ((𝐻)‘𝑗) ∈ 𝑌))
353351, 352imbi12d 344 . . . . . . . . . . . 12 (𝑦 = ((𝐻)‘𝑗) → ((((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑦𝑌) ↔ (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ((𝐻)‘𝑗) ∈ 𝑌)))
354343, 349, 353, 292vtoclgf 3523 . . . . . . . . . . 11 (((𝐻)‘𝑗) ∈ (𝐻𝑗) → (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ((𝐻)‘𝑗) ∈ 𝑌))
355354anabsi7 669 . . . . . . . . . 10 (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ((𝐻)‘𝑗) ∈ 𝑌)
356316, 317, 340, 355syl21anc 836 . . . . . . . . 9 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝐻)‘𝑗) ∈ 𝑌)
3571eleq2i 2829 . . . . . . . . . 10 (((𝐻)‘𝑗) ∈ 𝑌 ↔ ((𝐻)‘𝑗) ∈ {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)})
358 nfcv 2907 . . . . . . . . . . 11 𝑦𝐴
359 nfcv 2907 . . . . . . . . . . . 12 𝑦𝑇
360 nfcv 2907 . . . . . . . . . . . . . 14 𝑦0
361 nfcv 2907 . . . . . . . . . . . . . 14 𝑦
362 nfcv 2907 . . . . . . . . . . . . . . 15 𝑦𝑡
363343, 362nffv 6852 . . . . . . . . . . . . . 14 𝑦(((𝐻)‘𝑗)‘𝑡)
364360, 361, 363nfbr 5152 . . . . . . . . . . . . 13 𝑦0 ≤ (((𝐻)‘𝑗)‘𝑡)
365 nfcv 2907 . . . . . . . . . . . . . 14 𝑦1
366363, 361, 365nfbr 5152 . . . . . . . . . . . . 13 𝑦(((𝐻)‘𝑗)‘𝑡) ≤ 1
367364, 366nfan 1902 . . . . . . . . . . . 12 𝑦(0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)
368359, 367nfralw 3294 . . . . . . . . . . 11 𝑦𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)
369 nfcv 2907 . . . . . . . . . . . . 13 𝑡𝑦
370 nfcv 2907 . . . . . . . . . . . . . . 15 𝑡
371 nfra1 3267 . . . . . . . . . . . . . . . . . . 19 𝑡𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁)
372 nfra1 3267 . . . . . . . . . . . . . . . . . . 19 𝑡𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡)
373371, 372nfan 1902 . . . . . . . . . . . . . . . . . 18 𝑡(∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))
374 nfra1 3267 . . . . . . . . . . . . . . . . . . . 20 𝑡𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)
375 nfcv 2907 . . . . . . . . . . . . . . . . . . . 20 𝑡𝐴
376374, 375nfrabw 3440 . . . . . . . . . . . . . . . . . . 19 𝑡{𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)}
3771, 376nfcxfr 2905 . . . . . . . . . . . . . . . . . 18 𝑡𝑌
378373, 377nfrabw 3440 . . . . . . . . . . . . . . . . 17 𝑡{𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))}
37970, 378nfmpt 5212 . . . . . . . . . . . . . . . 16 𝑡(𝑗 ∈ (0...𝑁) ↦ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
38022, 379nfcxfr 2905 . . . . . . . . . . . . . . 15 𝑡𝐻
381370, 380nfco 5821 . . . . . . . . . . . . . 14 𝑡(𝐻)
382381, 74nffv 6852 . . . . . . . . . . . . 13 𝑡((𝐻)‘𝑗)
383369, 382nfeq 2920 . . . . . . . . . . . 12 𝑡 𝑦 = ((𝐻)‘𝑗)
384 fveq1 6841 . . . . . . . . . . . . . 14 (𝑦 = ((𝐻)‘𝑗) → (𝑦𝑡) = (((𝐻)‘𝑗)‘𝑡))
385384breq2d 5117 . . . . . . . . . . . . 13 (𝑦 = ((𝐻)‘𝑗) → (0 ≤ (𝑦𝑡) ↔ 0 ≤ (((𝐻)‘𝑗)‘𝑡)))
386384breq1d 5115 . . . . . . . . . . . . 13 (𝑦 = ((𝐻)‘𝑗) → ((𝑦𝑡) ≤ 1 ↔ (((𝐻)‘𝑗)‘𝑡) ≤ 1))
387385, 386anbi12d 631 . . . . . . . . . . . 12 (𝑦 = ((𝐻)‘𝑗) → ((0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
388383, 387ralbid 3256 . . . . . . . . . . 11 (𝑦 = ((𝐻)‘𝑗) → (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
389343, 358, 368, 388elrabf 3641 . . . . . . . . . 10 (((𝐻)‘𝑗) ∈ {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)} ↔ (((𝐻)‘𝑗) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
390357, 389bitri 274 . . . . . . . . 9 (((𝐻)‘𝑗) ∈ 𝑌 ↔ (((𝐻)‘𝑗) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
391356, 390sylib 217 . . . . . . . 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)))
392391simprd 496 . . . . . . 7 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1))
393 nfcv 2907 . . . . . . . . . . . 12 𝑦(𝐷𝑗)
394 nfcv 2907 . . . . . . . . . . . . 13 𝑦 <
395 nfcv 2907 . . . . . . . . . . . . 13 𝑦(𝐸 / 𝑁)
396363, 394, 395nfbr 5152 . . . . . . . . . . . 12 𝑦(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)
397393, 396nfralw 3294 . . . . . . . . . . 11 𝑦𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)
398347, 397nfim 1899 . . . . . . . . . 10 𝑦(((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁))
399384breq1d 5115 . . . . . . . . . . . 12 (𝑦 = ((𝐻)‘𝑗) → ((𝑦𝑡) < (𝐸 / 𝑁) ↔ (((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)))
400383, 399ralbid 3256 . . . . . . . . . . 11 (𝑦 = ((𝐻)‘𝑗) → (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ↔ ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)))
401351, 400imbi12d 344 . . . . . . . . . 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))))
402291simprd 496 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡)))
403402simpld 495 . . . . . . . . . 10 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁))
404343, 398, 401, 403vtoclgf 3523 . . . . . . . . 9 (((𝐻)‘𝑗) ∈ (𝐻𝑗) → (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)))
405404anabsi7 669 . . . . . . . 8 (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁))
406316, 317, 340, 405syl21anc 836 . . . . . . 7 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁))
407 nfcv 2907 . . . . . . . . . . . 12 𝑦(𝐵𝑗)
408 nfcv 2907 . . . . . . . . . . . . 13 𝑦(1 − (𝐸 / 𝑁))
409408, 394, 363nfbr 5152 . . . . . . . . . . . 12 𝑦(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)
410407, 409nfralw 3294 . . . . . . . . . . 11 𝑦𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)
411347, 410nfim 1899 . . . . . . . . . 10 𝑦(((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡))
412384breq2d 5117 . . . . . . . . . . . 12 (𝑦 = ((𝐻)‘𝑗) → ((1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ (1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))
413383, 412ralbid 3256 . . . . . . . . . . 11 (𝑦 = ((𝐻)‘𝑗) → (∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))
414351, 413imbi12d 344 . . . . . . . . . 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))))
415402simprd 496 . . . . . . . . . 10 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))
416343, 411, 414, 415vtoclgf 3523 . . . . . . . . 9 (((𝐻)‘𝑗) ∈ (𝐻𝑗) → (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))
417416anabsi7 669 . . . . . . . 8 (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡))
418316, 317, 340, 417syl21anc 836 . . . . . . 7 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡))
419392, 406, 4183jca 1128 . . . . . 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)))
420419ex 413 . . . . 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))))
421315, 420ralrimi 3240 . . . 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)))
422309, 421jca 512 . . 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))))
423 feq1 6649 . . . . 5 (𝑥 = (𝐻) → (𝑥:(0...𝑁)⟶𝐴 ↔ (𝐻):(0...𝑁)⟶𝐴))
424 nfcv 2907 . . . . . . 7 𝑗𝑥
425310, 53nfco 5821 . . . . . . 7 𝑗(𝐻)
426424, 425nfeq 2920 . . . . . 6 𝑗 𝑥 = (𝐻)
427 nfcv 2907 . . . . . . . . 9 𝑡𝑥
428427, 381nfeq 2920 . . . . . . . 8 𝑡 𝑥 = (𝐻)
429 fveq1 6841 . . . . . . . . . . 11 (𝑥 = (𝐻) → (𝑥𝑗) = ((𝐻)‘𝑗))
430429fveq1d 6844 . . . . . . . . . 10 (𝑥 = (𝐻) → ((𝑥𝑗)‘𝑡) = (((𝐻)‘𝑗)‘𝑡))
431430breq2d 5117 . . . . . . . . 9 (𝑥 = (𝐻) → (0 ≤ ((𝑥𝑗)‘𝑡) ↔ 0 ≤ (((𝐻)‘𝑗)‘𝑡)))
432430breq1d 5115 . . . . . . . . 9 (𝑥 = (𝐻) → (((𝑥𝑗)‘𝑡) ≤ 1 ↔ (((𝐻)‘𝑗)‘𝑡) ≤ 1))
433431, 432anbi12d 631 . . . . . . . 8 (𝑥 = (𝐻) → ((0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ↔ (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
434428, 433ralbid 3256 . . . . . . 7 (𝑥 = (𝐻) → (∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
435430breq1d 5115 . . . . . . . 8 (𝑥 = (𝐻) → (((𝑥𝑗)‘𝑡) < (𝐸 / 𝑁) ↔ (((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)))
436428, 435ralbid 3256 . . . . . . 7 (𝑥 = (𝐻) → (∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑁) ↔ ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)))
437430breq2d 5117 . . . . . . . 8 (𝑥 = (𝐻) → ((1 − (𝐸 / 𝑁)) < ((𝑥𝑗)‘𝑡) ↔ (1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))
438428, 437ralbid 3256 . . . . . . 7 (𝑥 = (𝐻) → (∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < ((𝑥𝑗)‘𝑡) ↔ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))
439434, 436, 4383anbi123d 1436 . . . . . 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))))
440426, 439ralbid 3256 . . . . 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))))
441423, 440anbi12d 631 . . . 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)))))
442441spcegv 3556 . . 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)))))
44340, 422, 442sylc 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))))
44431, 443exlimddv 1938 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 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wnf 1785  wcel 2106  wnfc 2887  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cdif 3907  cin 3909  wss 3910  c0 4282   cuni 4865   class class class wbr 5105  cmpt 5188  dom cdm 5633  ran crn 5634  ccom 5637  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  Fincfn 8883  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  cn 12153  3c3 12209  +crp 12915  (,)cioo 13264  ...cfz 13424  topGenctg 17319  Clsdccld 22367   Cn ccn 22575  Compccmp 22737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-cn 22578  df-cnp 22579  df-cmp 22738  df-tx 22913  df-hmeo 23106  df-xms 23673  df-ms 23674  df-tms 23675
This theorem is referenced by:  stoweidlem60  44291
  Copyright terms: Public domain W3C validator