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Theorem stoweidlem59 40793
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 3271 . . . . . . . . . 10 𝑦{𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)}
31, 2nfcxfr 2911 . . . . . . . . 9 𝑦𝑌
4 nfcv 2913 . . . . . . . . 9 𝑧𝑌
5 nfv 1995 . . . . . . . . 9 𝑧(∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))
6 nfv 1995 . . . . . . . . 9 𝑦(∀𝑡 ∈ (𝐷𝑗)(𝑧𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑧𝑡))
7 fveq1 6331 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑦𝑡) = (𝑧𝑡))
87breq1d 4796 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝑦𝑡) < (𝐸 / 𝑁) ↔ (𝑧𝑡) < (𝐸 / 𝑁)))
98ralbidv 3135 . . . . . . . . . 10 (𝑦 = 𝑧 → (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ↔ ∀𝑡 ∈ (𝐷𝑗)(𝑧𝑡) < (𝐸 / 𝑁)))
107breq2d 4798 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ (1 − (𝐸 / 𝑁)) < (𝑧𝑡)))
1110ralbidv 3135 . . . . . . . . . 10 (𝑦 = 𝑧 → (∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑧𝑡)))
129, 11anbi12d 616 . . . . . . . . 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 3348 . . . . . . . 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 6825 . . . . . . . . . 10 (𝜑 → (𝐽 Cn 𝐾) ∈ V)
15 stoweidlem59.11 . . . . . . . . . . 11 (𝜑𝐴𝐶)
16 stoweidlem59.5 . . . . . . . . . . 11 𝐶 = (𝐽 Cn 𝐾)
1715, 16syl6sseq 3800 . . . . . . . . . 10 (𝜑𝐴 ⊆ (𝐽 Cn 𝐾))
1814, 17ssexd 4939 . . . . . . . . 9 (𝜑𝐴 ∈ V)
191, 18rabexd 4947 . . . . . . . 8 (𝜑𝑌 ∈ V)
2013, 19rabexd 4947 . . . . . . 7 (𝜑 → {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} ∈ V)
2120ralrimivw 3116 . . . . . 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 6160 . . . . . 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 12979 . . . . 5 (0...𝑁) ∈ Fin
26 fnfi 8394 . . . . 5 ((𝐻 Fn (0...𝑁) ∧ (0...𝑁) ∈ Fin) → 𝐻 ∈ Fin)
2724, 25, 26sylancl 574 . . . 4 (𝜑𝐻 ∈ Fin)
28 rnfi 8405 . . . 4 (𝐻 ∈ Fin → ran 𝐻 ∈ Fin)
2927, 28syl 17 . . 3 (𝜑 → ran 𝐻 ∈ Fin)
30 fnchoice 39710 . . 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 754 . . . . 5 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → Fn ran 𝐻)
33 ovex 6823 . . . . . . . 8 (0...𝑁) ∈ V
3433mptex 6630 . . . . . . 7 (𝑗 ∈ (0...𝑁) ↦ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))}) ∈ V
3522, 34eqeltri 2846 . . . . . 6 𝐻 ∈ V
3635rnex 7247 . . . . 5 ran 𝐻 ∈ V
37 fnex 6625 . . . . 5 (( Fn ran 𝐻 ∧ ran 𝐻 ∈ V) → ∈ V)
3832, 36, 37sylancl 574 . . . 4 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ∈ V)
39 coexg 7264 . . . 4 (( ∈ V ∧ 𝐻 ∈ V) → (𝐻) ∈ V)
4038, 35, 39sylancl 574 . . 3 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → (𝐻) ∈ V)
41 dffn3 6194 . . . . . . 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 1995 . . . . . . . . . 10 𝑤𝜑
44 nfv 1995 . . . . . . . . . . 11 𝑤 Fn ran 𝐻
45 nfra1 3090 . . . . . . . . . . 11 𝑤𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)
4644, 45nfan 1980 . . . . . . . . . 10 𝑤( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))
4743, 46nfan 1980 . . . . . . . . 9 𝑤(𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)))
48 simplrr 763 . . . . . . . . . . 11 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑤 ∈ ran 𝐻) → ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))
49 simpr 471 . . . . . . . . . . 11 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑤 ∈ ran 𝐻) → 𝑤 ∈ ran 𝐻)
50 fvelrnb 6385 . . . . . . . . . . . . . . . 16 (𝐻 Fn (0...𝑁) → (𝑤 ∈ ran 𝐻 ↔ ∃𝑎 ∈ (0...𝑁)(𝐻𝑎) = 𝑤))
51 nfv 1995 . . . . . . . . . . . . . . . . 17 𝑎(𝐻𝑗) = 𝑤
52 nfmpt1 4881 . . . . . . . . . . . . . . . . . . . 20 𝑗(𝑗 ∈ (0...𝑁) ↦ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
5322, 52nfcxfr 2911 . . . . . . . . . . . . . . . . . . 19 𝑗𝐻
54 nfcv 2913 . . . . . . . . . . . . . . . . . . 19 𝑗𝑎
5553, 54nffv 6339 . . . . . . . . . . . . . . . . . 18 𝑗(𝐻𝑎)
56 nfcv 2913 . . . . . . . . . . . . . . . . . 18 𝑗𝑤
5755, 56nfeq 2925 . . . . . . . . . . . . . . . . 17 𝑗(𝐻𝑎) = 𝑤
58 fveq2 6332 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑎 → (𝐻𝑗) = (𝐻𝑎))
5958eqeq1d 2773 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑎 → ((𝐻𝑗) = 𝑤 ↔ (𝐻𝑎) = 𝑤))
6051, 57, 59cbvrex 3317 . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ran 𝐻) → ∃𝑗 ∈ (0...𝑁)(𝐻𝑗) = 𝑤)
64 simp3 1132 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0...𝑁) ∧ (𝐻𝑗) = 𝑤) → (𝐻𝑗) = 𝑤)
65 simpr 471 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑗 ∈ (0...𝑁))
6620adantr 466 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0...𝑁)) → {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))} ∈ V)
6722fvmpt2 6433 . . . . . . . . . . . . . . . . . . . 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 573 . . . . . . . . . . . . . . . . . . 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 2913 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑡(0...𝑁)
71 nfrab1 3271 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑡{𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)}
7270, 71nfmpt 4880 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑡(𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
7369, 72nfcxfr 2911 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡𝐷
74 nfcv 2913 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡𝑗
7573, 74nffv 6339 . . . . . . . . . . . . . . . . . . . . . . 23 𝑡(𝐷𝑗)
76 nfcv 2913 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡𝑇
77 stoweidlem59.7 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝐵 = (𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
78 nfrab1 3271 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑡{𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)}
7970, 78nfmpt 4880 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑡(𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
8077, 79nfcxfr 2911 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑡𝐵
8180, 74nffv 6339 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡(𝐵𝑗)
8276, 81nfdif 3882 . . . . . . . . . . . . . . . . . . . . . . 23 𝑡(𝑇 ∖ (𝐵𝑗))
83 stoweidlem59.2 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡𝜑
84 nfv 1995 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑡 𝑗 ∈ (0...𝑁)
8583, 84nfan 1980 . . . . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → 𝐽 ∈ Comp)
9015adantr 466 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → 𝐴𝐶)
91 stoweidlem59.12 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
92913adant1r 1187 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
93 stoweidlem59.13 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
94933adant1r 1187 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
95 stoweidlem59.14 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
9695adantlr 694 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
97 stoweidlem59.15 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))
9897adantlr 694 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (0...𝑁)) ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))
99 uniexg 7102 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐽 ∈ Comp → 𝐽 ∈ V)
10088, 99syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 𝐽 ∈ V)
10187, 100syl5eqel 2854 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝑇 ∈ V)
102101adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑇 ∈ V)
103 rabexg 4945 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6433 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑗 ∈ (0...𝑁) ∧ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} ∈ V) → (𝐵𝑗) = {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
10665, 104, 105syl2anc 573 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐵𝑗) = {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
107 stoweidlem59.1 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑡𝐹
108 eqid 2771 . . . . . . . . . . . . . . . . . . . . . . . . 25 {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} = {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)}
109 elfzelz 12549 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
110109zred 11684 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℝ)
111 3re 11296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 ∈ ℝ
112 3ne0 11317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 ≠ 0
113111, 112rereccli 10992 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (1 / 3) ∈ ℝ
114 readdcl 10221 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑗 ∈ ℝ ∧ (1 / 3) ∈ ℝ) → (𝑗 + (1 / 3)) ∈ ℝ)
115110, 113, 114sylancl 574 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑁) → (𝑗 + (1 / 3)) ∈ ℝ)
116115adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑗 + (1 / 3)) ∈ ℝ)
117 stoweidlem59.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐸 ∈ ℝ+)
118117rpred 12075 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐸 ∈ ℝ)
119118adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (0...𝑁)) → 𝐸 ∈ ℝ)
120116, 119remulcld 10272 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑗 + (1 / 3)) · 𝐸) ∈ ℝ)
121 stoweidlem59.16 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐹𝐶)
122121, 16syl6eleq 2860 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
123122adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑁)) → 𝐹 ∈ (𝐽 Cn 𝐾))
124107, 86, 87, 108, 120, 123rfcnpre3 39714 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (0...𝑁)) → {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} ∈ (Clsd‘𝐽))
125106, 124eqeltrd 2850 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐵𝑗) ∈ (Clsd‘𝐽))
126 rabexg 4945 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6433 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑗 ∈ (0...𝑁) ∧ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)} ∈ V) → (𝐷𝑗) = {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
12965, 127, 128syl2anc 573 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐷𝑗) = {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
130 eqid 2771 . . . . . . . . . . . . . . . . . . . . . . . . 25 {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)} = {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)}
131 resubcl 10547 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑗 ∈ ℝ ∧ (1 / 3) ∈ ℝ) → (𝑗 − (1 / 3)) ∈ ℝ)
132110, 113, 131sylancl 574 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑁) → (𝑗 − (1 / 3)) ∈ ℝ)
133132adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑗 − (1 / 3)) ∈ ℝ)
134133, 119remulcld 10272 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑗 − (1 / 3)) · 𝐸) ∈ ℝ)
135107, 86, 87, 130, 134, 123rfcnpre4 39715 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (0...𝑁)) → {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)} ∈ (Clsd‘𝐽))
136129, 135eqeltrd 2850 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐷𝑗) ∈ (Clsd‘𝐽))
137134adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ((𝑗 − (1 / 3)) · 𝐸) ∈ ℝ)
138120adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ((𝑗 + (1 / 3)) · 𝐸) ∈ ℝ)
13986, 87, 16, 121fcnre 39706 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝐹:𝑇⟶ℝ)
140139ad2antrr 705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → 𝐹:𝑇⟶ℝ)
141 ssrab2 3836 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)} ⊆ 𝑇
142106, 141syl6eqss 3804 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐵𝑗) ⊆ 𝑇)
143142sselda 3752 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → 𝑡𝑇)
144140, 143ffvelrnd 6503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → (𝐹𝑡) ∈ ℝ)
145113, 131mpan2 671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ ℝ → (𝑗 − (1 / 3)) ∈ ℝ)
146 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ ℝ → 𝑗 ∈ ℝ)
147113, 114mpan2 671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ ℝ → (𝑗 + (1 / 3)) ∈ ℝ)
148 3pos 11316 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 0 < 3
149111, 148recgt0ii 11131 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 0 < (1 / 3)
150113, 149elrpii 12038 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (1 / 3) ∈ ℝ+
151 ltsubrp 12069 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑗 ∈ ℝ ∧ (1 / 3) ∈ ℝ+) → (𝑗 − (1 / 3)) < 𝑗)
152150, 151mpan2 671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ ℝ → (𝑗 − (1 / 3)) < 𝑗)
153 ltaddrp 12070 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑗 ∈ ℝ ∧ (1 / 3) ∈ ℝ+) → 𝑗 < (𝑗 + (1 / 3)))
154150, 153mpan2 671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ ℝ → 𝑗 < (𝑗 + (1 / 3)))
155145, 146, 147, 152, 154lttrd 10400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑗 ∈ ℝ → (𝑗 − (1 / 3)) < (𝑗 + (1 / 3)))
156110, 155syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑗 ∈ (0...𝑁) → (𝑗 − (1 / 3)) < (𝑗 + (1 / 3)))
157156adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑗 − (1 / 3)) < (𝑗 + (1 / 3)))
158117rpregt0d 12081 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
159158adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
160 ltmul1 11075 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ((𝑗 − (1 / 3)) · 𝐸) < ((𝑗 + (1 / 3)) · 𝐸))
164106eleq2d 2836 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑡 ∈ (𝐵𝑗) ↔ 𝑡 ∈ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)}))
165164biimpa 462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → 𝑡 ∈ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
166 rabid 3264 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡))
169137, 138, 144, 163, 168ltletrd 10399 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ((𝑗 − (1 / 3)) · 𝐸) < (𝐹𝑡))
170137, 144ltnled 10386 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ¬ (𝑡𝑇 ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)))
173 rabid 3264 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → (𝐷𝑗) = {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
176174, 175neleqtrrd 2872 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ (𝐵𝑗)) → ¬ 𝑡 ∈ (𝐷𝑗))
177176ex 397 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑡 ∈ (𝐵𝑗) → ¬ 𝑡 ∈ (𝐷𝑗)))
17885, 177ralrimi 3106 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (0...𝑁)) → ∀𝑡 ∈ (𝐵𝑗) ¬ 𝑡 ∈ (𝐷𝑗))
179 disj 4160 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐵𝑗) ∩ (𝐷𝑗)) = ∅ ↔ ∀𝑎 ∈ (𝐵𝑗) ¬ 𝑎 ∈ (𝐷𝑗))
180 nfcv 2913 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑎(𝐵𝑗)
18175nfcri 2907 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑡 𝑎 ∈ (𝐷𝑗)
182181nfn 1935 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑡 ¬ 𝑎 ∈ (𝐷𝑗)
183 nfv 1995 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑎 ¬ 𝑡 ∈ (𝐷𝑗)
184 eleq1 2838 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎 = 𝑡 → (𝑎 ∈ (𝐷𝑗) ↔ 𝑡 ∈ (𝐷𝑗)))
185184notbid 307 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = 𝑡 → (¬ 𝑎 ∈ (𝐷𝑗) ↔ ¬ 𝑡 ∈ (𝐷𝑗)))
186180, 81, 182, 183, 185cbvralf 3314 . . . . . . . . . . . . . . . . . . . . . . . . 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 2771 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑇 ∖ (𝐵𝑗)) = (𝑇 ∖ (𝐵𝑗))
190 stoweidlem59.19 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑁 ∈ ℕ)
191190nnrpd 12073 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑁 ∈ ℝ+)
192117, 191rpdivcld 12092 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐸 / 𝑁) ∈ ℝ+)
193192adantr 466 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐸 / 𝑁) ∈ ℝ+)
194118, 190nndivred 11271 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐸 / 𝑁) ∈ ℝ)
195113a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (1 / 3) ∈ ℝ)
196190nnge1d 11265 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → 1 ≤ 𝑁)
197 1re 10241 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1 ∈ ℝ
198 0lt1 10752 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 0 < 1
199197, 198pm3.2i 447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (1 ∈ ℝ ∧ 0 < 1)
200199a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (1 ∈ ℝ ∧ 0 < 1))
201190nnred 11237 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑁 ∈ ℝ)
202190nngt0d 11266 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → 0 < 𝑁)
203 lediv2 11115 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁) ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → (1 ≤ 𝑁 ↔ (𝐸 / 𝑁) ≤ (𝐸 / 1)))
204200, 201, 202, 158, 203syl121anc 1481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (1 ≤ 𝑁 ↔ (𝐸 / 𝑁) ≤ (𝐸 / 1)))
205196, 204mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐸 / 𝑁) ≤ (𝐸 / 1))
206117rpcnd 12077 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐸 ∈ ℂ)
207206div1d 10995 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐸 / 1) = 𝐸)
208205, 207breqtrd 4812 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐸 / 𝑁) ≤ 𝐸)
209 stoweidlem59.18 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐸 < (1 / 3))
210194, 118, 195, 208, 209lelttrd 10397 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐸 / 𝑁) < (1 / 3))
211210adantr 466 . . . . . . . . . . . . . . . . . . . . . . 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 40792 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (0...𝑁)) → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))
213 df-rex 3067 . . . . . . . . . . . . . . . . . . . . . 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 754 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ (0...𝑁)) ∧ (𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))) → 𝑥𝐴)
216 simprr1 1272 . . . . . . . . . . . . . . . . . . . . . . . . 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 6331 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = 𝑥 → (𝑦𝑡) = (𝑥𝑡))
218217breq2d 4798 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = 𝑥 → (0 ≤ (𝑦𝑡) ↔ 0 ≤ (𝑥𝑡)))
219217breq1d 4796 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = 𝑥 → ((𝑦𝑡) ≤ 1 ↔ (𝑥𝑡) ≤ 1))
220218, 219anbi12d 616 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑥 → ((0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1)))
221220ralbidv 3135 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑥 → (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1)))
222221, 1elrab2 3518 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝑌 ↔ (𝑥𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1)))
223215, 216, 222sylanbrc 572 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ (0...𝑁)) ∧ (𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))) → 𝑥𝑌)
224 simprr2 1274 . . . . . . . . . . . . . . . . . . . . . . . . 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 1276 . . . . . . . . . . . . . . . . . . . . . . . . 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 501 . . . . . . . . . . . . . . . . . . . . . . . 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 2913 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦𝑥
228 nfv 1995 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦(∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡))
229217breq1d 4796 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑥 → ((𝑦𝑡) < (𝐸 / 𝑁) ↔ (𝑥𝑡) < (𝐸 / 𝑁)))
230229ralbidv 3135 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑥 → (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ↔ ∀𝑡 ∈ (𝐷𝑗)(𝑥𝑡) < (𝐸 / 𝑁)))
231217breq2d 4798 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑥 → ((1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ (1 − (𝐸 / 𝑁)) < (𝑥𝑡)))
232231ralbidv 3135 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑥 → (∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑥𝑡)))
233230, 232anbi12d 616 . . . . . . . . . . . . . . . . . . . . . . . . 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 3511 . . . . . . . . . . . . . . . . . . . . . . . 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 572 . . . . . . . . . . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . . . . . . . . . . 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 1998 . . . . . . . . . . . . . . . . . . . . 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 4069 . . . . . . . . . . . . . . . . . . . . 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 2010 . . . . . . . . . . . . . . . . . . . 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 3010 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐻𝑗) ≠ ∅)
2432423adant3 1126 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0...𝑁) ∧ (𝐻𝑗) = 𝑤) → (𝐻𝑗) ≠ ∅)
24464, 243eqnetrrd 3011 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0...𝑁) ∧ (𝐻𝑗) = 𝑤) → 𝑤 ≠ ∅)
2452443exp 1112 . . . . . . . . . . . . . . 15 (𝜑 → (𝑗 ∈ (0...𝑁) → ((𝐻𝑗) = 𝑤𝑤 ≠ ∅)))
246245rexlimdv 3178 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑗 ∈ (0...𝑁)(𝐻𝑗) = 𝑤𝑤 ≠ ∅))
247246adantr 466 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ran 𝐻) → (∃𝑗 ∈ (0...𝑁)(𝐻𝑗) = 𝑤𝑤 ≠ ∅))
24863, 247mpd 15 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ran 𝐻) → 𝑤 ≠ ∅)
249248adantlr 694 . . . . . . . . . . 11 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑤 ∈ ran 𝐻) → 𝑤 ≠ ∅)
250 rsp 3078 . . . . . . . . . . 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 397 . . . . . . . . 9 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → (𝑤 ∈ ran 𝐻 → (𝑤) ∈ 𝑤))
25347, 252ralrimi 3106 . . . . . . . 8 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ∀𝑤 ∈ ran 𝐻(𝑤) ∈ 𝑤)
254 chfnrn 6471 . . . . . . . 8 (( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤) ∈ 𝑤) → ran ran 𝐻)
25532, 253, 254syl2anc 573 . . . . . . 7 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ran ran 𝐻)
256 nfv 1995 . . . . . . . . . 10 𝑦𝜑
257 nfcv 2913 . . . . . . . . . . . 12 𝑦
258 nfcv 2913 . . . . . . . . . . . . . . 15 𝑦(0...𝑁)
259 nfrab1 3271 . . . . . . . . . . . . . . 15 𝑦{𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))}
260258, 259nfmpt 4880 . . . . . . . . . . . . . 14 𝑦(𝑗 ∈ (0...𝑁) ↦ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
26122, 260nfcxfr 2911 . . . . . . . . . . . . 13 𝑦𝐻
262261nfrn 5506 . . . . . . . . . . . 12 𝑦ran 𝐻
263257, 262nffn 6127 . . . . . . . . . . 11 𝑦 Fn ran 𝐻
264 nfv 1995 . . . . . . . . . . . 12 𝑦(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)
265262, 264nfral 3094 . . . . . . . . . . 11 𝑦𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)
266263, 265nfan 1980 . . . . . . . . . 10 𝑦( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))
267256, 266nfan 1980 . . . . . . . . 9 𝑦(𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)))
268262nfuni 4580 . . . . . . . . 9 𝑦 ran 𝐻
269 fnunirn 6654 . . . . . . . . . . . . . . 15 (𝐻 Fn (0...𝑁) → (𝑦 ran 𝐻 ↔ ∃𝑧 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑧)))
270 nfcv 2913 . . . . . . . . . . . . . . . . . 18 𝑗𝑧
27153, 270nffv 6339 . . . . . . . . . . . . . . . . 17 𝑗(𝐻𝑧)
272271nfcri 2907 . . . . . . . . . . . . . . . 16 𝑗 𝑦 ∈ (𝐻𝑧)
273 nfv 1995 . . . . . . . . . . . . . . . 16 𝑧 𝑦 ∈ (𝐻𝑗)
274 fveq2 6332 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑗 → (𝐻𝑧) = (𝐻𝑗))
275274eleq2d 2836 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑗 → (𝑦 ∈ (𝐻𝑧) ↔ 𝑦 ∈ (𝐻𝑗)))
276272, 273, 275cbvrex 3317 . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . 12 ((𝜑𝑦 ran 𝐻) → ∃𝑗 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑗))
280 nfv 1995 . . . . . . . . . . . . . 14 𝑗𝜑
28153nfrn 5506 . . . . . . . . . . . . . . . 16 𝑗ran 𝐻
282281nfuni 4580 . . . . . . . . . . . . . . 15 𝑗 ran 𝐻
283282nfcri 2907 . . . . . . . . . . . . . 14 𝑗 𝑦 ran 𝐻
284280, 283nfan 1980 . . . . . . . . . . . . 13 𝑗(𝜑𝑦 ran 𝐻)
285 nfv 1995 . . . . . . . . . . . . 13 𝑗 𝑦𝑌
286 simp1l 1239 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ran 𝐻) ∧ 𝑗 ∈ (0...𝑁) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝜑)
287 simp2 1131 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ran 𝐻) ∧ 𝑗 ∈ (0...𝑁) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑗 ∈ (0...𝑁))
288 simp3 1132 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ran 𝐻) ∧ 𝑗 ∈ (0...𝑁) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑦 ∈ (𝐻𝑗))
28968eleq2d 2836 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑦 ∈ (𝐻𝑗) ↔ 𝑦 ∈ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))}))
290289biimpa 462 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑦 ∈ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
291 rabid 3264 . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑦𝑌)
294286, 287, 288, 293syl21anc 1475 . . . . . . . . . . . . . 14 (((𝜑𝑦 ran 𝐻) ∧ 𝑗 ∈ (0...𝑁) ∧ 𝑦 ∈ (𝐻𝑗)) → 𝑦𝑌)
2952943exp 1112 . . . . . . . . . . . . 13 ((𝜑𝑦 ran 𝐻) → (𝑗 ∈ (0...𝑁) → (𝑦 ∈ (𝐻𝑗) → 𝑦𝑌)))
296284, 285, 295rexlimd 3174 . . . . . . . . . . . 12 ((𝜑𝑦 ran 𝐻) → (∃𝑗 ∈ (0...𝑁)𝑦 ∈ (𝐻𝑗) → 𝑦𝑌))
297279, 296mpd 15 . . . . . . . . . . 11 ((𝜑𝑦 ran 𝐻) → 𝑦𝑌)
298297adantlr 694 . . . . . . . . . 10 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑦 ran 𝐻) → 𝑦𝑌)
299298ex 397 . . . . . . . . 9 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → (𝑦 ran 𝐻𝑦𝑌))
300267, 268, 3, 299ssrd 3757 . . . . . . . 8 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ran 𝐻𝑌)
301 ssrab2 3836 . . . . . . . . 9 {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)} ⊆ 𝐴
3021, 301eqsstri 3784 . . . . . . . 8 𝑌𝐴
303300, 302syl6ss 3764 . . . . . . 7 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ran 𝐻𝐴)
304255, 303sstrd 3762 . . . . . 6 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → ran 𝐴)
30542, 304fssd 6197 . . . . 5 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → :ran 𝐻𝐴)
306 dffn3 6194 . . . . . . 7 (𝐻 Fn (0...𝑁) ↔ 𝐻:(0...𝑁)⟶ran 𝐻)
30724, 306sylib 208 . . . . . 6 (𝜑𝐻:(0...𝑁)⟶ran 𝐻)
308307adantr 466 . . . . 5 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → 𝐻:(0...𝑁)⟶ran 𝐻)
309 fco 6198 . . . . 5 ((:ran 𝐻𝐴𝐻:(0...𝑁)⟶ran 𝐻) → (𝐻):(0...𝑁)⟶𝐴)
310305, 308, 309syl2anc 573 . . . 4 ((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) → (𝐻):(0...𝑁)⟶𝐴)
311 nfcv 2913 . . . . . . . 8 𝑗
312311, 281nffn 6127 . . . . . . 7 𝑗 Fn ran 𝐻
313 nfv 1995 . . . . . . . 8 𝑗(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)
314281, 313nfral 3094 . . . . . . 7 𝑗𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)
315312, 314nfan 1980 . . . . . 6 𝑗( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))
316280, 315nfan 1980 . . . . 5 𝑗(𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤)))
317 simpll 750 . . . . . . . . . 10 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → 𝜑)
318 simpr 471 . . . . . . . . . 10 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑗 ∈ (0...𝑁))
31924ad2antrr 705 . . . . . . . . . . . 12 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → 𝐻 Fn (0...𝑁))
320 fvco2 6415 . . . . . . . . . . . 12 ((𝐻 Fn (0...𝑁) ∧ 𝑗 ∈ (0...𝑁)) → ((𝐻)‘𝑗) = (‘(𝐻𝑗)))
321319, 320sylancom 576 . . . . . . . . . . 11 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝐻)‘𝑗) = (‘(𝐻𝑗)))
322 simplrr 763 . . . . . . . . . . . . 13 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))
323 fnfun 6128 . . . . . . . . . . . . . . . 16 (𝐻 Fn (0...𝑁) → Fun 𝐻)
32424, 323syl 17 . . . . . . . . . . . . . . 15 (𝜑 → Fun 𝐻)
325324ad2antrr 705 . . . . . . . . . . . . . 14 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → Fun 𝐻)
326 fndm 6130 . . . . . . . . . . . . . . . . . 18 (𝐻 Fn (0...𝑁) → dom 𝐻 = (0...𝑁))
32724, 326syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → dom 𝐻 = (0...𝑁))
328327adantr 466 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0...𝑁)) → dom 𝐻 = (0...𝑁))
32965, 328eleqtrrd 2853 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑗 ∈ dom 𝐻)
330329adantlr 694 . . . . . . . . . . . . . 14 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑗 ∈ dom 𝐻)
331 fvelrn 6495 . . . . . . . . . . . . . 14 ((Fun 𝐻𝑗 ∈ dom 𝐻) → (𝐻𝑗) ∈ ran 𝐻)
332325, 330, 331syl2anc 573 . . . . . . . . . . . . 13 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → (𝐻𝑗) ∈ ran 𝐻)
333322, 332jca 501 . . . . . . . . . . . 12 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → (∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤) ∧ (𝐻𝑗) ∈ ran 𝐻))
334242adantlr 694 . . . . . . . . . . . 12 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → (𝐻𝑗) ≠ ∅)
335 neeq1 3005 . . . . . . . . . . . . . 14 (𝑤 = (𝐻𝑗) → (𝑤 ≠ ∅ ↔ (𝐻𝑗) ≠ ∅))
336 fveq2 6332 . . . . . . . . . . . . . . 15 (𝑤 = (𝐻𝑗) → (𝑤) = (‘(𝐻𝑗)))
337 id 22 . . . . . . . . . . . . . . 15 (𝑤 = (𝐻𝑗) → 𝑤 = (𝐻𝑗))
338336, 337eleq12d 2844 . . . . . . . . . . . . . 14 (𝑤 = (𝐻𝑗) → ((𝑤) ∈ 𝑤 ↔ (‘(𝐻𝑗)) ∈ (𝐻𝑗)))
339335, 338imbi12d 333 . . . . . . . . . . . . 13 (𝑤 = (𝐻𝑗) → ((𝑤 ≠ ∅ → (𝑤) ∈ 𝑤) ↔ ((𝐻𝑗) ≠ ∅ → (‘(𝐻𝑗)) ∈ (𝐻𝑗))))
340339rspccva 3459 . . . . . . . . . . . 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 2850 . . . . . . . . . 10 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝐻)‘𝑗) ∈ (𝐻𝑗))
343257, 261nfco 5426 . . . . . . . . . . . . 13 𝑦(𝐻)
344 nfcv 2913 . . . . . . . . . . . . 13 𝑦𝑗
345343, 344nffv 6339 . . . . . . . . . . . 12 𝑦((𝐻)‘𝑗)
346 nfv 1995 . . . . . . . . . . . . . 14 𝑦(𝜑𝑗 ∈ (0...𝑁))
347261, 344nffv 6339 . . . . . . . . . . . . . . 15 𝑦(𝐻𝑗)
348345, 347nfel 2926 . . . . . . . . . . . . . 14 𝑦((𝐻)‘𝑗) ∈ (𝐻𝑗)
349346, 348nfan 1980 . . . . . . . . . . . . 13 𝑦((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗))
350345, 3nfel 2926 . . . . . . . . . . . . 13 𝑦((𝐻)‘𝑗) ∈ 𝑌
351349, 350nfim 1977 . . . . . . . . . . . 12 𝑦(((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ((𝐻)‘𝑗) ∈ 𝑌)
352 eleq1 2838 . . . . . . . . . . . . . 14 (𝑦 = ((𝐻)‘𝑗) → (𝑦 ∈ (𝐻𝑗) ↔ ((𝐻)‘𝑗) ∈ (𝐻𝑗)))
353352anbi2d 614 . . . . . . . . . . . . 13 (𝑦 = ((𝐻)‘𝑗) → (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) ↔ ((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗))))
354 eleq1 2838 . . . . . . . . . . . . 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 3415 . . . . . . . . . . 11 (((𝐻)‘𝑗) ∈ (𝐻𝑗) → (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ((𝐻)‘𝑗) ∈ 𝑌))
357356anabsi7 650 . . . . . . . . . 10 (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ((𝐻)‘𝑗) ∈ 𝑌)
358317, 318, 342, 357syl21anc 1475 . . . . . . . . 9 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝐻)‘𝑗) ∈ 𝑌)
3591eleq2i 2842 . . . . . . . . . 10 (((𝐻)‘𝑗) ∈ 𝑌 ↔ ((𝐻)‘𝑗) ∈ {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)})
360 nfcv 2913 . . . . . . . . . . 11 𝑦𝐴
361 nfcv 2913 . . . . . . . . . . . 12 𝑦𝑇
362 nfcv 2913 . . . . . . . . . . . . . 14 𝑦0
363 nfcv 2913 . . . . . . . . . . . . . 14 𝑦
364 nfcv 2913 . . . . . . . . . . . . . . 15 𝑦𝑡
365345, 364nffv 6339 . . . . . . . . . . . . . 14 𝑦(((𝐻)‘𝑗)‘𝑡)
366362, 363, 365nfbr 4833 . . . . . . . . . . . . 13 𝑦0 ≤ (((𝐻)‘𝑗)‘𝑡)
367 nfcv 2913 . . . . . . . . . . . . . 14 𝑦1
368365, 363, 367nfbr 4833 . . . . . . . . . . . . 13 𝑦(((𝐻)‘𝑗)‘𝑡) ≤ 1
369366, 368nfan 1980 . . . . . . . . . . . 12 𝑦(0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)
370361, 369nfral 3094 . . . . . . . . . . 11 𝑦𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)
371 nfcv 2913 . . . . . . . . . . . . 13 𝑡𝑦
372 nfcv 2913 . . . . . . . . . . . . . . 15 𝑡
373 nfra1 3090 . . . . . . . . . . . . . . . . . . 19 𝑡𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁)
374 nfra1 3090 . . . . . . . . . . . . . . . . . . 19 𝑡𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡)
375373, 374nfan 1980 . . . . . . . . . . . . . . . . . 18 𝑡(∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))
376 nfra1 3090 . . . . . . . . . . . . . . . . . . . 20 𝑡𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)
377 nfcv 2913 . . . . . . . . . . . . . . . . . . . 20 𝑡𝐴
378376, 377nfrab 3272 . . . . . . . . . . . . . . . . . . 19 𝑡{𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)}
3791, 378nfcxfr 2911 . . . . . . . . . . . . . . . . . 18 𝑡𝑌
380375, 379nfrab 3272 . . . . . . . . . . . . . . . . 17 𝑡{𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))}
38170, 380nfmpt 4880 . . . . . . . . . . . . . . . 16 𝑡(𝑗 ∈ (0...𝑁) ↦ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})
38222, 381nfcxfr 2911 . . . . . . . . . . . . . . 15 𝑡𝐻
383372, 382nfco 5426 . . . . . . . . . . . . . 14 𝑡(𝐻)
384383, 74nffv 6339 . . . . . . . . . . . . 13 𝑡((𝐻)‘𝑗)
385371, 384nfeq 2925 . . . . . . . . . . . 12 𝑡 𝑦 = ((𝐻)‘𝑗)
386 fveq1 6331 . . . . . . . . . . . . . 14 (𝑦 = ((𝐻)‘𝑗) → (𝑦𝑡) = (((𝐻)‘𝑗)‘𝑡))
387386breq2d 4798 . . . . . . . . . . . . 13 (𝑦 = ((𝐻)‘𝑗) → (0 ≤ (𝑦𝑡) ↔ 0 ≤ (((𝐻)‘𝑗)‘𝑡)))
388386breq1d 4796 . . . . . . . . . . . . 13 (𝑦 = ((𝐻)‘𝑗) → ((𝑦𝑡) ≤ 1 ↔ (((𝐻)‘𝑗)‘𝑡) ≤ 1))
389387, 388anbi12d 616 . . . . . . . . . . . 12 (𝑦 = ((𝐻)‘𝑗) → ((0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
390385, 389ralbid 3132 . . . . . . . . . . 11 (𝑦 = ((𝐻)‘𝑗) → (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
391345, 360, 370, 390elrabf 3511 . . . . . . . . . 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 483 . . . . . . 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 2913 . . . . . . . . . . . 12 𝑦(𝐷𝑗)
396 nfcv 2913 . . . . . . . . . . . . 13 𝑦 <
397 nfcv 2913 . . . . . . . . . . . . 13 𝑦(𝐸 / 𝑁)
398365, 396, 397nfbr 4833 . . . . . . . . . . . 12 𝑦(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)
399395, 398nfral 3094 . . . . . . . . . . 11 𝑦𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)
400349, 399nfim 1977 . . . . . . . . . 10 𝑦(((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁))
401386breq1d 4796 . . . . . . . . . . . 12 (𝑦 = ((𝐻)‘𝑗) → ((𝑦𝑡) < (𝐸 / 𝑁) ↔ (((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)))
402385, 401ralbid 3132 . . . . . . . . . . 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 483 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡)))
405404simpld 482 . . . . . . . . . 10 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁))
406345, 400, 403, 405vtoclgf 3415 . . . . . . . . 9 (((𝐻)‘𝑗) ∈ (𝐻𝑗) → (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)))
407406anabsi7 650 . . . . . . . 8 (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁))
408317, 318, 342, 407syl21anc 1475 . . . . . . 7 (((𝜑 ∧ ( Fn ran 𝐻 ∧ ∀𝑤 ∈ ran 𝐻(𝑤 ≠ ∅ → (𝑤) ∈ 𝑤))) ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁))
409 nfcv 2913 . . . . . . . . . . . 12 𝑦(𝐵𝑗)
410 nfcv 2913 . . . . . . . . . . . . 13 𝑦(1 − (𝐸 / 𝑁))
411410, 396, 365nfbr 4833 . . . . . . . . . . . 12 𝑦(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)
412409, 411nfral 3094 . . . . . . . . . . 11 𝑦𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)
413349, 412nfim 1977 . . . . . . . . . 10 𝑦(((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡))
414386breq2d 4798 . . . . . . . . . . . 12 (𝑦 = ((𝐻)‘𝑗) → ((1 − (𝐸 / 𝑁)) < (𝑦𝑡) ↔ (1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))
415385, 414ralbid 3132 . . . . . . . . . . 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 483 . . . . . . . . . 10 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑦 ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))
418345, 413, 416, 417vtoclgf 3415 . . . . . . . . 9 (((𝐻)‘𝑗) ∈ (𝐻𝑗) → (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))
419418anabsi7 650 . . . . . . . 8 (((𝜑𝑗 ∈ (0...𝑁)) ∧ ((𝐻)‘𝑗) ∈ (𝐻𝑗)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡))
420317, 318, 342, 419syl21anc 1475 . . . . . . 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 1122 . . . . . 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 397 . . . . 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 3106 . . . 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 501 . . 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 6166 . . . . 5 (𝑥 = (𝐻) → (𝑥:(0...𝑁)⟶𝐴 ↔ (𝐻):(0...𝑁)⟶𝐴))
426 nfcv 2913 . . . . . . 7 𝑗𝑥
427311, 53nfco 5426 . . . . . . 7 𝑗(𝐻)
428426, 427nfeq 2925 . . . . . 6 𝑗 𝑥 = (𝐻)
429 nfcv 2913 . . . . . . . . 9 𝑡𝑥
430429, 383nfeq 2925 . . . . . . . 8 𝑡 𝑥 = (𝐻)
431 fveq1 6331 . . . . . . . . . . 11 (𝑥 = (𝐻) → (𝑥𝑗) = ((𝐻)‘𝑗))
432431fveq1d 6334 . . . . . . . . . 10 (𝑥 = (𝐻) → ((𝑥𝑗)‘𝑡) = (((𝐻)‘𝑗)‘𝑡))
433432breq2d 4798 . . . . . . . . 9 (𝑥 = (𝐻) → (0 ≤ ((𝑥𝑗)‘𝑡) ↔ 0 ≤ (((𝐻)‘𝑗)‘𝑡)))
434432breq1d 4796 . . . . . . . . 9 (𝑥 = (𝐻) → (((𝑥𝑗)‘𝑡) ≤ 1 ↔ (((𝐻)‘𝑗)‘𝑡) ≤ 1))
435433, 434anbi12d 616 . . . . . . . 8 (𝑥 = (𝐻) → ((0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ↔ (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
436430, 435ralbid 3132 . . . . . . 7 (𝑥 = (𝐻) → (∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (((𝐻)‘𝑗)‘𝑡) ∧ (((𝐻)‘𝑗)‘𝑡) ≤ 1)))
437432breq1d 4796 . . . . . . . 8 (𝑥 = (𝐻) → (((𝑥𝑗)‘𝑡) < (𝐸 / 𝑁) ↔ (((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)))
438430, 437ralbid 3132 . . . . . . 7 (𝑥 = (𝐻) → (∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑁) ↔ ∀𝑡 ∈ (𝐷𝑗)(((𝐻)‘𝑗)‘𝑡) < (𝐸 / 𝑁)))
439432breq2d 4798 . . . . . . . 8 (𝑥 = (𝐻) → ((1 − (𝐸 / 𝑁)) < ((𝑥𝑗)‘𝑡) ↔ (1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))
440430, 439ralbid 3132 . . . . . . 7 (𝑥 = (𝐻) → (∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < ((𝑥𝑗)‘𝑡) ↔ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (((𝐻)‘𝑗)‘𝑡)))
441436, 438, 4403anbi123d 1547 . . . . . 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 3132 . . . . 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 616 . . . 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 3445 . . 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 2015 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 382  w3a 1071   = wceq 1631  wex 1852  wnf 1856  wcel 2145  wnfc 2900  wne 2943  wral 3061  wrex 3062  {crab 3065  Vcvv 3351  cdif 3720  cin 3722  wss 3723  c0 4063   cuni 4574   class class class wbr 4786  cmpt 4863  dom cdm 5249  ran crn 5250  ccom 5253  Fun wfun 6025   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  Fincfn 8109  cr 10137  0cc0 10138  1c1 10139   + caddc 10141   · cmul 10143   < clt 10276  cle 10277  cmin 10468   / cdiv 10886  cn 11222  3c3 11273  +crp 12035  (,)cioo 12380  ...cfz 12533  topGenctg 16306  Clsdccld 21041   Cn ccn 21249  Compccmp 21410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216  ax-mulf 10218
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-fi 8473  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-uz 11889  df-q 11992  df-rp 12036  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ioo 12384  df-ioc 12385  df-ico 12386  df-icc 12387  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-rlim 14428  df-sum 14625  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-mulr 16163  df-starv 16164  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ds 16172  df-unif 16173  df-hom 16174  df-cco 16175  df-rest 16291  df-topn 16292  df-0g 16310  df-gsum 16311  df-topgen 16312  df-pt 16313  df-prds 16316  df-xrs 16370  df-qtop 16375  df-imas 16376  df-xps 16378  df-mre 16454  df-mrc 16455  df-acs 16457  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-submnd 17544  df-mulg 17749  df-cntz 17957  df-cmn 18402  df-psmet 19953  df-xmet 19954  df-met 19955  df-bl 19956  df-mopn 19957  df-cnfld 19962  df-top 20919  df-topon 20936  df-topsp 20958  df-bases 20971  df-cld 21044  df-cn 21252  df-cnp 21253  df-cmp 21411  df-tx 21586  df-hmeo 21779  df-xms 22345  df-ms 22346  df-tms 22347
This theorem is referenced by:  stoweidlem60  40794
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