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Theorem poimirlem31 35087
Description: Lemma for poimir 35089, assigning values to the vertices of the tessellation that meet the hypotheses of both poimirlem30 35086 and poimirlem28 35084. Equation (2) of [Kulpa] p. 547. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimir.i 𝐼 = ((0[,]1) ↑m (1...𝑁))
poimir.r 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))
poimir.1 (𝜑𝐹 ∈ ((𝑅t 𝐼) Cn 𝑅))
poimir.2 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧𝐼 ∧ (𝑧𝑛) = 0)) → ((𝐹𝑧)‘𝑛) ≤ 0)
poimirlem31.p 𝑃 = ((1st ‘(𝐺𝑘)) ∘f + ((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
poimirlem31.3 (𝜑𝐺:ℕ⟶((ℕ0m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
poimirlem31.4 ((𝜑𝑘 ∈ ℕ) → ran (1st ‘(𝐺𝑘)) ⊆ (0..^𝑘))
poimirlem31.5 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ))
Assertion
Ref Expression
poimirlem31 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛))
Distinct variable groups:   𝑓,𝑖,𝑗,𝑘,𝑛,𝑧   𝜑,𝑗,𝑛   𝑗,𝐹,𝑛   𝑗,𝑁,𝑛   𝜑,𝑖,𝑘   𝑓,𝑁,𝑖,𝑘   𝜑,𝑧   𝑓,𝐹,𝑘,𝑧   𝑧,𝑁   𝑖,𝑟,𝑗,𝑘,𝑛,𝑧,𝜑   𝑎,𝑏,𝑓,𝑖,𝑗,𝑘,𝑛,𝑟,𝑧,𝐹   𝐺,𝑎,𝑏,𝑓,𝑖,𝑗,𝑘,𝑛,𝑟,𝑧   𝐼,𝑎,𝑏,𝑓,𝑖,𝑗,𝑘,𝑛,𝑟,𝑧   𝑁,𝑎,𝑏,𝑟   𝑅,𝑎,𝑏,𝑓,𝑖,𝑗,𝑘,𝑛,𝑟,𝑧   𝑃,𝑎,𝑏,𝑓,𝑖,𝑛,𝑟,𝑧
Allowed substitution hints:   𝜑(𝑓,𝑎,𝑏)   𝑃(𝑗,𝑘)

Proof of Theorem poimirlem31
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpri 4550 . . . 4 (𝑟 ∈ { ≤ , ≤ } → (𝑟 = ≤ ∨ 𝑟 = ≤ ))
2 simprr 772 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → 𝑛 ∈ (1...𝑁))
3 fz1ssfz0 13002 . . . . . . . . . 10 (1...𝑁) ⊆ (0...𝑁)
43sseli 3914 . . . . . . . . 9 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (0...𝑁))
54anim2i 619 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁)))
6 eleq1 2880 . . . . . . . . . . . . 13 (𝑖 = 𝑛 → (𝑖 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...𝑁)))
76anbi2d 631 . . . . . . . . . . . 12 (𝑖 = 𝑛 → ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁)) ↔ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁))))
87anbi2d 631 . . . . . . . . . . 11 (𝑖 = 𝑛 → ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) ↔ (𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁)))))
9 eqeq1 2805 . . . . . . . . . . . 12 (𝑖 = 𝑛 → (𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) ↔ 𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < )))
109rexbidv 3259 . . . . . . . . . . 11 (𝑖 = 𝑛 → (∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) ↔ ∃𝑗 ∈ (0...𝑁)𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < )))
118, 10imbi12d 348 . . . . . . . . . 10 (𝑖 = 𝑛 → (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < )) ↔ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ))))
12 poimirlem31.5 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ))
1311, 12chvarvv 2005 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ))
14 elfzle1 12909 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → 1 ≤ 𝑛)
15 1re 10634 . . . . . . . . . . . . . 14 1 ∈ ℝ
16 elfzelz 12906 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
1716zred 12079 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
18 lenlt 10712 . . . . . . . . . . . . . 14 ((1 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (1 ≤ 𝑛 ↔ ¬ 𝑛 < 1))
1915, 17, 18sylancr 590 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → (1 ≤ 𝑛 ↔ ¬ 𝑛 < 1))
2014, 19mpbid 235 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑁) → ¬ 𝑛 < 1)
21 elsni 4545 . . . . . . . . . . . . 13 (𝑛 ∈ {0} → 𝑛 = 0)
22 0lt1 11155 . . . . . . . . . . . . 13 0 < 1
2321, 22eqbrtrdi 5072 . . . . . . . . . . . 12 (𝑛 ∈ {0} → 𝑛 < 1)
2420, 23nsyl 142 . . . . . . . . . . 11 (𝑛 ∈ (1...𝑁) → ¬ 𝑛 ∈ {0})
25 ltso 10714 . . . . . . . . . . . . . . 15 < Or ℝ
26 snfi 8581 . . . . . . . . . . . . . . . . 17 {0} ∈ Fin
27 fzfi 13339 . . . . . . . . . . . . . . . . . 18 (1...𝑁) ∈ Fin
28 rabfi 8731 . . . . . . . . . . . . . . . . . 18 ((1...𝑁) ∈ Fin → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} ∈ Fin)
2927, 28ax-mp 5 . . . . . . . . . . . . . . . . 17 {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} ∈ Fin
30 unfi 8773 . . . . . . . . . . . . . . . . 17 (({0} ∈ Fin ∧ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} ∈ Fin) → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ∈ Fin)
3126, 29, 30mp2an 691 . . . . . . . . . . . . . . . 16 ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ∈ Fin
32 c0ex 10628 . . . . . . . . . . . . . . . . . 18 0 ∈ V
3332snid 4564 . . . . . . . . . . . . . . . . 17 0 ∈ {0}
34 elun1 4106 . . . . . . . . . . . . . . . . 17 (0 ∈ {0} → 0 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}))
35 ne0i 4253 . . . . . . . . . . . . . . . . 17 (0 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ≠ ∅)
3633, 34, 35mp2b 10 . . . . . . . . . . . . . . . 16 ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ≠ ∅
37 0re 10636 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
38 snssi 4704 . . . . . . . . . . . . . . . . . 18 (0 ∈ ℝ → {0} ⊆ ℝ)
3937, 38ax-mp 5 . . . . . . . . . . . . . . . . 17 {0} ⊆ ℝ
40 ssrab2 4010 . . . . . . . . . . . . . . . . . 18 {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} ⊆ (1...𝑁)
4116ssriv 3922 . . . . . . . . . . . . . . . . . . 19 (1...𝑁) ⊆ ℤ
42 zssre 11980 . . . . . . . . . . . . . . . . . . 19 ℤ ⊆ ℝ
4341, 42sstri 3927 . . . . . . . . . . . . . . . . . 18 (1...𝑁) ⊆ ℝ
4440, 43sstri 3927 . . . . . . . . . . . . . . . . 17 {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} ⊆ ℝ
4539, 44unssi 4115 . . . . . . . . . . . . . . . 16 ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ⊆ ℝ
4631, 36, 453pm3.2i 1336 . . . . . . . . . . . . . . 15 (({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ∈ Fin ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ≠ ∅ ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ⊆ ℝ)
47 fisupcl 8921 . . . . . . . . . . . . . . 15 (( < Or ℝ ∧ (({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ∈ Fin ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ≠ ∅ ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ⊆ ℝ)) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}))
4825, 46, 47mp2an 691 . . . . . . . . . . . . . 14 sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)})
49 eleq1 2880 . . . . . . . . . . . . . 14 (𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → (𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ↔ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)})))
5048, 49mpbiri 261 . . . . . . . . . . . . 13 (𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → 𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}))
51 elun 4079 . . . . . . . . . . . . 13 (𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ↔ (𝑛 ∈ {0} ∨ 𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}))
5250, 51sylib 221 . . . . . . . . . . . 12 (𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → (𝑛 ∈ {0} ∨ 𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}))
53 oveq2 7147 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑛 → (1...𝑎) = (1...𝑛))
5453raleqdv 3367 . . . . . . . . . . . . . . 15 (𝑎 = 𝑛 → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
5554elrab 3631 . . . . . . . . . . . . . 14 (𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} ↔ (𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
56 elfzuz 12902 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ‘1))
57 eluzfz2 12914 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (ℤ‘1) → 𝑛 ∈ (1...𝑛))
5856, 57syl 17 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (1...𝑛))
59 simpl 486 . . . . . . . . . . . . . . . 16 ((0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) → 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏))
6059ralimi 3131 . . . . . . . . . . . . . . 15 (∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏))
61 fveq2 6649 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑛 → ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛))
6261breq2d 5045 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑛 → (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))
6362rspcva 3572 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...𝑛) ∧ ∀𝑏 ∈ (1...𝑛)0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏)) → 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛))
6458, 60, 63syl2an 598 . . . . . . . . . . . . . 14 ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) → 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛))
6555, 64sylbi 220 . . . . . . . . . . . . 13 (𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} → 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛))
6665orim2i 908 . . . . . . . . . . . 12 ((𝑛 ∈ {0} ∨ 𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) → (𝑛 ∈ {0} ∨ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))
6752, 66syl 17 . . . . . . . . . . 11 (𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → (𝑛 ∈ {0} ∨ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))
68 orel1 886 . . . . . . . . . . 11 𝑛 ∈ {0} → ((𝑛 ∈ {0} ∨ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)) → 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))
6924, 67, 68syl2im 40 . . . . . . . . . 10 (𝑛 ∈ (1...𝑁) → (𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))
7069reximdv 3235 . . . . . . . . 9 (𝑛 ∈ (1...𝑁) → (∃𝑗 ∈ (0...𝑁)𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → ∃𝑗 ∈ (0...𝑁)0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))
7113, 70syl5 34 . . . . . . . 8 (𝑛 ∈ (1...𝑁) → ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))
725, 71sylan2i 608 . . . . . . 7 (𝑛 ∈ (1...𝑁) → ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → ∃𝑗 ∈ (0...𝑁)0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))
732, 72mpcom 38 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → ∃𝑗 ∈ (0...𝑁)0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛))
74 breq 5035 . . . . . . 7 (𝑟 = ≤ → (0𝑟((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ↔ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))
7574rexbidv 3259 . . . . . 6 (𝑟 = ≤ → (∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ↔ ∃𝑗 ∈ (0...𝑁)0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))
7673, 75syl5ibrcom 250 . . . . 5 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (𝑟 = ≤ → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))
77 poimir.0 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℕ)
7877nnzd 12078 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℤ)
79 elfzm1b 12984 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
8016, 78, 79syl2anr 599 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
8180biimpd 232 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ (0...(𝑁 − 1))))
8281ex 416 . . . . . . . . . . . 12 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ (0...(𝑁 − 1)))))
8382pm2.43d 53 . . . . . . . . . . 11 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ (0...(𝑁 − 1))))
8477nncnd 11645 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℂ)
85 npcan1 11058 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
8684, 85syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
87 nnm1nn0 11930 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
8877, 87syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℕ0)
8988nn0zd 12077 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 − 1) ∈ ℤ)
90 uzid 12250 . . . . . . . . . . . . . . 15 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
91 peano2uz 12293 . . . . . . . . . . . . . . 15 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
9289, 90, 913syl 18 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
9386, 92eqeltrrd 2894 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
94 fzss2 12946 . . . . . . . . . . . . 13 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
9593, 94syl 17 . . . . . . . . . . . 12 (𝜑 → (0...(𝑁 − 1)) ⊆ (0...𝑁))
9695sseld 3917 . . . . . . . . . . 11 (𝜑 → ((𝑛 − 1) ∈ (0...(𝑁 − 1)) → (𝑛 − 1) ∈ (0...𝑁)))
9783, 96syld 47 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ (0...𝑁)))
9897anim2d 614 . . . . . . . . 9 (𝜑 → ((𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁))))
9998imp 410 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁)))
100 ovex 7172 . . . . . . . . 9 (𝑛 − 1) ∈ V
101 eleq1 2880 . . . . . . . . . . . 12 (𝑖 = (𝑛 − 1) → (𝑖 ∈ (0...𝑁) ↔ (𝑛 − 1) ∈ (0...𝑁)))
102101anbi2d 631 . . . . . . . . . . 11 (𝑖 = (𝑛 − 1) → ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁)) ↔ (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁))))
103102anbi2d 631 . . . . . . . . . 10 (𝑖 = (𝑛 − 1) → ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) ↔ (𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁)))))
104 eqeq1 2805 . . . . . . . . . . 11 (𝑖 = (𝑛 − 1) → (𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) ↔ (𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < )))
105104rexbidv 3259 . . . . . . . . . 10 (𝑖 = (𝑛 − 1) → (∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) ↔ ∃𝑗 ∈ (0...𝑁)(𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < )))
106103, 105imbi12d 348 . . . . . . . . 9 (𝑖 = (𝑛 − 1) → (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < )) ↔ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)(𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ))))
107100, 106, 12vtocl 3510 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)(𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ))
10899, 107syldan 594 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → ∃𝑗 ∈ (0...𝑁)(𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ))
109 eleq1 2880 . . . . . . . . . . . . . . . 16 ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ↔ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)})))
11048, 109mpbiri 261 . . . . . . . . . . . . . . 15 ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → (𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}))
111 elun 4079 . . . . . . . . . . . . . . . 16 ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ↔ ((𝑛 − 1) ∈ {0} ∨ (𝑛 − 1) ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}))
112100elsn 4543 . . . . . . . . . . . . . . . . 17 ((𝑛 − 1) ∈ {0} ↔ (𝑛 − 1) = 0)
113 oveq2 7147 . . . . . . . . . . . . . . . . . . 19 (𝑎 = (𝑛 − 1) → (1...𝑎) = (1...(𝑛 − 1)))
114113raleqdv 3367 . . . . . . . . . . . . . . . . . 18 (𝑎 = (𝑛 − 1) → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
115114elrab 3631 . . . . . . . . . . . . . . . . 17 ((𝑛 − 1) ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} ↔ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
116112, 115orbi12i 912 . . . . . . . . . . . . . . . 16 (((𝑛 − 1) ∈ {0} ∨ (𝑛 − 1) ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ↔ ((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0))))
117111, 116bitri 278 . . . . . . . . . . . . . . 15 ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ↔ ((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0))))
118110, 117sylib 221 . . . . . . . . . . . . . 14 ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → ((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0))))
119118a1i 11 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → ((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))))
120 ltm1 11475 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
121 peano2rem 10946 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
122 ltnle 10713 . . . . . . . . . . . . . . . . . . 19 (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
123121, 122mpancom 687 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
124120, 123mpbid 235 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℝ → ¬ 𝑛 ≤ (𝑛 − 1))
12517, 124syl 17 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑁) → ¬ 𝑛 ≤ (𝑛 − 1))
126 breq2 5037 . . . . . . . . . . . . . . . . 17 ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → (𝑛 ≤ (𝑛 − 1) ↔ 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < )))
127126notbid 321 . . . . . . . . . . . . . . . 16 ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → (¬ 𝑛 ≤ (𝑛 − 1) ↔ ¬ 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < )))
128125, 127syl5ibcom 248 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → ¬ 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < )))
129 elun2 4107 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} → 𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}))
130 fimaxre2 11578 . . . . . . . . . . . . . . . . . . . . 21 ((({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ⊆ ℝ ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)})𝑦𝑥)
13145, 31, 130mp2an 691 . . . . . . . . . . . . . . . . . . . 20 𝑥 ∈ ℝ ∀𝑦 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)})𝑦𝑥
13245, 36, 1313pm3.2i 1336 . . . . . . . . . . . . . . . . . . 19 (({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ⊆ ℝ ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)})𝑦𝑥)
133132suprubii 11607 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) → 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ))
134129, 133syl 17 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} → 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ))
135134con3i 157 . . . . . . . . . . . . . . . 16 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → ¬ 𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)})
136 ianor 979 . . . . . . . . . . . . . . . . 17 (¬ (𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) ↔ (¬ 𝑛 ∈ (1...𝑁) ∨ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
137136, 55xchnxbir 336 . . . . . . . . . . . . . . . 16 𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} ↔ (¬ 𝑛 ∈ (1...𝑁) ∨ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
138135, 137sylib 221 . . . . . . . . . . . . . . 15 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → (¬ 𝑛 ∈ (1...𝑁) ∨ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
139128, 138syl6 35 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → (¬ 𝑛 ∈ (1...𝑁) ∨ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0))))
140 pm2.63 938 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...𝑁) ∨ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) → ((¬ 𝑛 ∈ (1...𝑁) ∨ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) → ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
141140orcs 872 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → ((¬ 𝑛 ∈ (1...𝑁) ∨ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) → ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
142139, 141syld 47 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
143119, 142jcad 516 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → (((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0))) ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0))))
144 andir 1006 . . . . . . . . . . . . . 14 ((((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0))) ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) ↔ (((𝑛 − 1) = 0 ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) ∨ (((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0))))
145 1p0e1 11753 . . . . . . . . . . . . . . . . . 18 (1 + 0) = 1
14616zcnd 12080 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
147 ax-1cn 10588 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℂ
148 0cn 10626 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℂ
149 subadd 10882 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 0 ∈ ℂ) → ((𝑛 − 1) = 0 ↔ (1 + 0) = 𝑛))
150147, 148, 149mp3an23 1450 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℂ → ((𝑛 − 1) = 0 ↔ (1 + 0) = 𝑛))
151146, 150syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = 0 ↔ (1 + 0) = 𝑛))
152151biimpa 480 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ (1...𝑁) ∧ (𝑛 − 1) = 0) → (1 + 0) = 𝑛)
153145, 152syl5reqr 2851 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (1...𝑁) ∧ (𝑛 − 1) = 0) → 𝑛 = 1)
154 1z 12004 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℤ
155 fzsn 12948 . . . . . . . . . . . . . . . . . . . . . 22 (1 ∈ ℤ → (1...1) = {1})
156154, 155ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (1...1) = {1}
157 oveq2 7147 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1 → (1...𝑛) = (1...1))
158 sneq 4538 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1 → {𝑛} = {1})
159156, 157, 1583eqtr4a 2862 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → (1...𝑛) = {𝑛})
160159raleqdv 3367 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 1 → (∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) ↔ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
161160notbid 321 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → (¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) ↔ ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
162161biimpd 232 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → (¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) → ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
163153, 162syl 17 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (1...𝑁) ∧ (𝑛 − 1) = 0) → (¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) → ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
164163expimpd 457 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) = 0 ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) → ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
165 ralun 4122 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) → ∀𝑏 ∈ ((1...(𝑛 − 1)) ∪ {𝑛})(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0))
166 npcan1 11058 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
167146, 166syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) = 𝑛)
168167, 56eqeltrd 2893 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) ∈ (ℤ‘1))
169 peano2zm 12017 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℤ → (𝑛 − 1) ∈ ℤ)
170 uzid 12250 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)))
171 peano2uz 12293 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)) → ((𝑛 − 1) + 1) ∈ (ℤ‘(𝑛 − 1)))
17216, 169, 170, 1714syl 19 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) ∈ (ℤ‘(𝑛 − 1)))
173167, 172eqeltrrd 2894 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ‘(𝑛 − 1)))
174 fzsplit2 12931 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑛 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑛 ∈ (ℤ‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
175168, 173, 174syl2anc 587 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (1...𝑁) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
176167oveq1d 7154 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
177 fzsn 12948 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
17816, 177syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) → (𝑛...𝑛) = {𝑛})
179176, 178eqtrd 2836 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
180179uneq2d 4093 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (1...𝑁) → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
181175, 180eqtrd 2836 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (1...𝑁) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
182181raleqdv 3367 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) ↔ ∀𝑏 ∈ ((1...(𝑛 − 1)) ∪ {𝑛})(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
183165, 182syl5ibr 249 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) → ((∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
184183expdimp 456 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
185184con3d 155 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) → (¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) → ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
186185adantrl 715 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (1...𝑁) ∧ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0))) → (¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) → ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
187186expimpd 457 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...𝑁) → ((((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) → ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
188164, 187jaod 856 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → ((((𝑛 − 1) = 0 ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) ∨ (((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0))) → ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
189144, 188syl5bi 245 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → ((((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0))) ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) → ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)))
190 fveq2 6649 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑛 → (𝑃𝑏) = (𝑃𝑛))
191190neeq1d 3049 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑛 → ((𝑃𝑏) ≠ 0 ↔ (𝑃𝑛) ≠ 0))
19262, 191anbi12d 633 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑛 → ((0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃𝑛) ≠ 0)))
193192ralsng 4576 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃𝑛) ≠ 0)))
194193notbid 321 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → (¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) ↔ ¬ (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃𝑛) ≠ 0)))
195 ianor 979 . . . . . . . . . . . . . . 15 (¬ (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃𝑛) ≠ 0) ↔ (¬ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ ¬ (𝑃𝑛) ≠ 0))
196 nne 2994 . . . . . . . . . . . . . . . 16 (¬ (𝑃𝑛) ≠ 0 ↔ (𝑃𝑛) = 0)
197196orbi2i 910 . . . . . . . . . . . . . . 15 ((¬ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ ¬ (𝑃𝑛) ≠ 0) ↔ (¬ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃𝑛) = 0))
198195, 197bitri 278 . . . . . . . . . . . . . 14 (¬ (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃𝑛) ≠ 0) ↔ (¬ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃𝑛) = 0))
199194, 198syl6bb 290 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → (¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) ↔ (¬ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃𝑛) = 0)))
200189, 199sylibd 242 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑁) → ((((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0))) ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) → (¬ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃𝑛) = 0)))
201143, 200syld 47 . . . . . . . . . . 11 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → (¬ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃𝑛) = 0)))
202201ad2antlr 726 . . . . . . . . . 10 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → (¬ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃𝑛) = 0)))
203 poimir.1 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹 ∈ ((𝑅t 𝐼) Cn 𝑅))
204 poimir.r . . . . . . . . . . . . . . . . . . . . . 22 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))
205 retop 23371 . . . . . . . . . . . . . . . . . . . . . . . 24 (topGen‘ran (,)) ∈ Top
206205fconst6 6547 . . . . . . . . . . . . . . . . . . . . . . 23 ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top
207 pttop 22191 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑁) ∈ Fin ∧ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top) → (∏t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Top)
20827, 206, 207mp2an 691 . . . . . . . . . . . . . . . . . . . . . 22 (∏t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Top
209204, 208eqeltri 2889 . . . . . . . . . . . . . . . . . . . . 21 𝑅 ∈ Top
210 poimir.i . . . . . . . . . . . . . . . . . . . . . 22 𝐼 = ((0[,]1) ↑m (1...𝑁))
211 reex 10621 . . . . . . . . . . . . . . . . . . . . . . 23 ℝ ∈ V
212 unitssre 12881 . . . . . . . . . . . . . . . . . . . . . . 23 (0[,]1) ⊆ ℝ
213 mapss 8440 . . . . . . . . . . . . . . . . . . . . . . 23 ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑m (1...𝑁)) ⊆ (ℝ ↑m (1...𝑁)))
214211, 212, 213mp2an 691 . . . . . . . . . . . . . . . . . . . . . 22 ((0[,]1) ↑m (1...𝑁)) ⊆ (ℝ ↑m (1...𝑁))
215210, 214eqsstri 3952 . . . . . . . . . . . . . . . . . . . . 21 𝐼 ⊆ (ℝ ↑m (1...𝑁))
216 uniretop 23372 . . . . . . . . . . . . . . . . . . . . . . . 24 ℝ = (topGen‘ran (,))
217204, 216ptuniconst 22207 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑁) ∈ Fin ∧ (topGen‘ran (,)) ∈ Top) → (ℝ ↑m (1...𝑁)) = 𝑅)
21827, 205, 217mp2an 691 . . . . . . . . . . . . . . . . . . . . . 22 (ℝ ↑m (1...𝑁)) = 𝑅
219218restuni 21771 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Top ∧ 𝐼 ⊆ (ℝ ↑m (1...𝑁))) → 𝐼 = (𝑅t 𝐼))
220209, 215, 219mp2an 691 . . . . . . . . . . . . . . . . . . . 20 𝐼 = (𝑅t 𝐼)
221220, 218cnf 21855 . . . . . . . . . . . . . . . . . . 19 (𝐹 ∈ ((𝑅t 𝐼) Cn 𝑅) → 𝐹:𝐼⟶(ℝ ↑m (1...𝑁)))
222203, 221syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹:𝐼⟶(ℝ ↑m (1...𝑁)))
223222ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝐹:𝐼⟶(ℝ ↑m (1...𝑁)))
224 simplr 768 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝑘 ∈ ℕ)
225 elfzelz 12906 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℤ)
226225zred 12079 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℝ)
227226adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
228 nnre 11636 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
229228adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
230 nnne0 11663 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ → 𝑘 ≠ 0)
231230adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ≠ 0)
232227, 229, 231redivcld 11461 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑥 / 𝑘) ∈ ℝ)
233 elfzle1 12909 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (0...𝑘) → 0 ≤ 𝑥)
234226, 233jca 515 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ (0...𝑘) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
235 nnrp 12392 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
236235rpregt0d 12429 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
237 divge0 11502 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝑥 / 𝑘))
238234, 236, 237syl2an 598 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑥 / 𝑘))
239 elfzle2 12910 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (0...𝑘) → 𝑥𝑘)
240239adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑥𝑘)
241 1red 10635 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ)
242235adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ+)
243227, 241, 242ledivmuld 12476 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑥 / 𝑘) ≤ 1 ↔ 𝑥 ≤ (𝑘 · 1)))
244 nncn 11637 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
245244mulid1d 10651 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 ∈ ℕ → (𝑘 · 1) = 𝑘)
246245breq2d 5045 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ ℕ → (𝑥 ≤ (𝑘 · 1) ↔ 𝑥𝑘))
247246adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑥 ≤ (𝑘 · 1) ↔ 𝑥𝑘))
248243, 247bitrd 282 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑥 / 𝑘) ≤ 1 ↔ 𝑥𝑘))
249240, 248mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑥 / 𝑘) ≤ 1)
250 elicc01 12848 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 / 𝑘) ∈ (0[,]1) ↔ ((𝑥 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝑥 / 𝑘) ∧ (𝑥 / 𝑘) ≤ 1))
251232, 238, 249, 250syl3anbrc 1340 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑥 / 𝑘) ∈ (0[,]1))
252251ancoms 462 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ (0...𝑘)) → (𝑥 / 𝑘) ∈ (0[,]1))
253 elsni 4545 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ {𝑘} → 𝑦 = 𝑘)
254253oveq2d 7155 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ {𝑘} → (𝑥 / 𝑦) = (𝑥 / 𝑘))
255254eleq1d 2877 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ {𝑘} → ((𝑥 / 𝑦) ∈ (0[,]1) ↔ (𝑥 / 𝑘) ∈ (0[,]1)))
256252, 255syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ (0...𝑘)) → (𝑦 ∈ {𝑘} → (𝑥 / 𝑦) ∈ (0[,]1)))
257256impr 458 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ ℕ ∧ (𝑥 ∈ (0...𝑘) ∧ 𝑦 ∈ {𝑘})) → (𝑥 / 𝑦) ∈ (0[,]1))
258224, 257sylan 583 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑥 ∈ (0...𝑘) ∧ 𝑦 ∈ {𝑘})) → (𝑥 / 𝑦) ∈ (0[,]1))
259 elun 4079 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ({1} ∪ {0}) ↔ (𝑦 ∈ {1} ∨ 𝑦 ∈ {0}))
260 fzofzp1 13133 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (0..^𝑘) → (𝑥 + 1) ∈ (0...𝑘))
261 elsni 4545 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ {1} → 𝑦 = 1)
262261oveq2d 7155 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ {1} → (𝑥 + 𝑦) = (𝑥 + 1))
263262eleq1d 2877 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ {1} → ((𝑥 + 𝑦) ∈ (0...𝑘) ↔ (𝑥 + 1) ∈ (0...𝑘)))
264260, 263syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ (0..^𝑘) → (𝑦 ∈ {1} → (𝑥 + 𝑦) ∈ (0...𝑘)))
265 elfzonn0 13081 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 ∈ (0..^𝑘) → 𝑥 ∈ ℕ0)
266265nn0cnd 11949 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 ∈ (0..^𝑘) → 𝑥 ∈ ℂ)
267266addid1d 10833 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ (0..^𝑘) → (𝑥 + 0) = 𝑥)
268 elfzofz 13052 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ (0..^𝑘) → 𝑥 ∈ (0...𝑘))
269267, 268eqeltrd 2893 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (0..^𝑘) → (𝑥 + 0) ∈ (0...𝑘))
270 elsni 4545 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ {0} → 𝑦 = 0)
271270oveq2d 7155 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ {0} → (𝑥 + 𝑦) = (𝑥 + 0))
272271eleq1d 2877 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ {0} → ((𝑥 + 𝑦) ∈ (0...𝑘) ↔ (𝑥 + 0) ∈ (0...𝑘)))
273269, 272syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ (0..^𝑘) → (𝑦 ∈ {0} → (𝑥 + 𝑦) ∈ (0...𝑘)))
274264, 273jaod 856 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ (0..^𝑘) → ((𝑦 ∈ {1} ∨ 𝑦 ∈ {0}) → (𝑥 + 𝑦) ∈ (0...𝑘)))
275259, 274syl5bi 245 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (0..^𝑘) → (𝑦 ∈ ({1} ∪ {0}) → (𝑥 + 𝑦) ∈ (0...𝑘)))
276275imp 410 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ (0..^𝑘) ∧ 𝑦 ∈ ({1} ∪ {0})) → (𝑥 + 𝑦) ∈ (0...𝑘))
277276adantl 485 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑥 ∈ (0..^𝑘) ∧ 𝑦 ∈ ({1} ∪ {0}))) → (𝑥 + 𝑦) ∈ (0...𝑘))
278 poimirlem31.3 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐺:ℕ⟶((ℕ0m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
279278ffvelrnda 6832 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ∈ ((ℕ0m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
280 xp1st 7707 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺𝑘) ∈ ((ℕ0m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(𝐺𝑘)) ∈ (ℕ0m (1...𝑁)))
281 elmapfn 8416 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st ‘(𝐺𝑘)) ∈ (ℕ0m (1...𝑁)) → (1st ‘(𝐺𝑘)) Fn (1...𝑁))
282279, 280, 2813syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐺𝑘)) Fn (1...𝑁))
283 poimirlem31.4 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘 ∈ ℕ) → ran (1st ‘(𝐺𝑘)) ⊆ (0..^𝑘))
284 df-f 6332 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st ‘(𝐺𝑘)):(1...𝑁)⟶(0..^𝑘) ↔ ((1st ‘(𝐺𝑘)) Fn (1...𝑁) ∧ ran (1st ‘(𝐺𝑘)) ⊆ (0..^𝑘)))
285282, 283, 284sylanbrc 586 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐺𝑘)):(1...𝑁)⟶(0..^𝑘))
286285adantr 484 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (1st ‘(𝐺𝑘)):(1...𝑁)⟶(0..^𝑘))
287 1ex 10630 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ V
288287fconst 6543 . . . . . . . . . . . . . . . . . . . . . . . 24 (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}):((2nd ‘(𝐺𝑘)) “ (1...𝑗))⟶{1}
28932fconst 6543 . . . . . . . . . . . . . . . . . . . . . . . 24 (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}):((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))⟶{0}
290288, 289pm3.2i 474 . . . . . . . . . . . . . . . . . . . . . . 23 ((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}):((2nd ‘(𝐺𝑘)) “ (1...𝑗))⟶{1} ∧ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}):((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))⟶{0})
291 xp2nd 7708 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐺𝑘) ∈ ((ℕ0m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(𝐺𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
292279, 291syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑘 ∈ ℕ) → (2nd ‘(𝐺𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
293 fvex 6662 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (2nd ‘(𝐺𝑘)) ∈ V
294 f1oeq1 6583 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 = (2nd ‘(𝐺𝑘)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)))
295293, 294elab 3618 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((2nd ‘(𝐺𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
296292, 295sylib 221 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑘 ∈ ℕ) → (2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
297 dff1o3 6600 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(𝐺𝑘)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(𝐺𝑘))))
298297simprbi 500 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(𝐺𝑘)))
299 imain 6413 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Fun (2nd ‘(𝐺𝑘)) → ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∩ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))))
300296, 298, 2993syl 18 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑘 ∈ ℕ) → ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∩ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))))
301 elfznn0 12999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℕ0)
302301nn0red 11948 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℝ)
303302ltp1d 11563 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑁) → 𝑗 < (𝑗 + 1))
304 fzdisj 12933 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
305303, 304syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (0...𝑁) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
306305imaeq2d 5900 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑁) → ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((2nd ‘(𝐺𝑘)) “ ∅))
307 ima0 5916 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((2nd ‘(𝐺𝑘)) “ ∅) = ∅
308306, 307eqtrdi 2852 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0...𝑁) → ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ∅)
309300, 308sylan9req 2857 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∩ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))) = ∅)
310 fun 6518 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}):((2nd ‘(𝐺𝑘)) “ (1...𝑗))⟶{1} ∧ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}):((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))⟶{0}) ∧ (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∩ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))) = ∅) → ((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∪ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}))
311290, 309, 310sylancr 590 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∪ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}))
312 imaundi 5979 . . . . . . . . . . . . . . . . . . . . . . . 24 ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∪ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)))
313 nn0p1nn 11928 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ ℕ0 → (𝑗 + 1) ∈ ℕ)
314301, 313syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ ℕ)
315 nnuz 12273 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ℕ = (ℤ‘1)
316314, 315eleqtrdi 2903 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ (ℤ‘1))
317 elfzuz3 12903 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑁) → 𝑁 ∈ (ℤ𝑗))
318 fzsplit2 12931 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑗 + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ𝑗)) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
319316, 317, 318syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑁) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
320319imaeq2d 5900 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (0...𝑁) → ((2nd ‘(𝐺𝑘)) “ (1...𝑁)) = ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))))
321 f1ofo 6601 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(𝐺𝑘)):(1...𝑁)–onto→(1...𝑁))
322 foima 6574 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((2nd ‘(𝐺𝑘)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(𝐺𝑘)) “ (1...𝑁)) = (1...𝑁))
323296, 321, 3223syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑘 ∈ ℕ) → ((2nd ‘(𝐺𝑘)) “ (1...𝑁)) = (1...𝑁))
324320, 323sylan9req 2857 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑗 ∈ (0...𝑁) ∧ (𝜑𝑘 ∈ ℕ)) → ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (1...𝑁))
325324ancoms 462 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (1...𝑁))
326312, 325syl5eqr 2850 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∪ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))) = (1...𝑁))
327326feq2d 6477 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∪ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}) ↔ ((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0})))
328311, 327mpbid 235 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))
329 fzfid 13340 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
330 inidm 4148 . . . . . . . . . . . . . . . . . . . . 21 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
331277, 286, 328, 329, 329, 330off 7408 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((1st ‘(𝐺𝑘)) ∘f + ((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝑘))
332 poimirlem31.p . . . . . . . . . . . . . . . . . . . . 21 𝑃 = ((1st ‘(𝐺𝑘)) ∘f + ((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
333332feq1i 6482 . . . . . . . . . . . . . . . . . . . 20 (𝑃:(1...𝑁)⟶(0...𝑘) ↔ ((1st ‘(𝐺𝑘)) ∘f + ((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝑘))
334331, 333sylibr 237 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝑃:(1...𝑁)⟶(0...𝑘))
335 vex 3447 . . . . . . . . . . . . . . . . . . . . 21 𝑘 ∈ V
336335fconst 6543 . . . . . . . . . . . . . . . . . . . 20 ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘}
337336a1i 11 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘})
338258, 334, 337, 329, 329, 330off 7408 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (𝑃f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
339210eleq2i 2884 . . . . . . . . . . . . . . . . . . 19 ((𝑃f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (𝑃f / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑m (1...𝑁)))
340 ovex 7172 . . . . . . . . . . . . . . . . . . . 20 (0[,]1) ∈ V
341 ovex 7172 . . . . . . . . . . . . . . . . . . . 20 (1...𝑁) ∈ V
342340, 341elmap 8422 . . . . . . . . . . . . . . . . . . 19 ((𝑃f / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑m (1...𝑁)) ↔ (𝑃f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
343339, 342bitri 278 . . . . . . . . . . . . . . . . . 18 ((𝑃f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (𝑃f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
344338, 343sylibr 237 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (𝑃f / ((1...𝑁) × {𝑘})) ∈ 𝐼)
345223, 344ffvelrnd 6833 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (𝐹‘(𝑃f / ((1...𝑁) × {𝑘}))) ∈ (ℝ ↑m (1...𝑁)))
346 elmapi 8415 . . . . . . . . . . . . . . . 16 ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘}))) ∈ (ℝ ↑m (1...𝑁)) → (𝐹‘(𝑃f / ((1...𝑁) × {𝑘}))):(1...𝑁)⟶ℝ)
347345, 346syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (𝐹‘(𝑃f / ((1...𝑁) × {𝑘}))):(1...𝑁)⟶ℝ)
348347ffvelrnda 6832 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∈ ℝ)
349348an32s 651 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∈ ℝ)
350 0red 10637 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → 0 ∈ ℝ)
351349, 350ltnled 10780 . . . . . . . . . . . 12 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) < 0 ↔ ¬ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))
352 ltle 10722 . . . . . . . . . . . . 13 ((((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∈ ℝ ∧ 0 ∈ ℝ) → (((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) < 0 → ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
353349, 37, 352sylancl 589 . . . . . . . . . . . 12 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) < 0 → ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
354351, 353sylbird 263 . . . . . . . . . . 11 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (¬ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) → ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
355244, 230div0d 11408 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (0 / 𝑘) = 0)
356 oveq1 7146 . . . . . . . . . . . . . . . 16 ((𝑃𝑛) = 0 → ((𝑃𝑛) / 𝑘) = (0 / 𝑘))
357356eqeq1d 2803 . . . . . . . . . . . . . . 15 ((𝑃𝑛) = 0 → (((𝑃𝑛) / 𝑘) = 0 ↔ (0 / 𝑘) = 0))
358355, 357syl5ibrcom 250 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → ((𝑃𝑛) = 0 → ((𝑃𝑛) / 𝑘) = 0))
359358ad3antlr 730 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑃𝑛) = 0 → ((𝑃𝑛) / 𝑘) = 0))
360334ffnd 6492 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝑃 Fn (1...𝑁))
361 fnconstg 6545 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ V → ((1...𝑁) × {𝑘}) Fn (1...𝑁))
362335, 361mp1i 13 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((1...𝑁) × {𝑘}) Fn (1...𝑁))
363 eqidd 2802 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑃𝑛) = (𝑃𝑛))
364335fvconst2 6947 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {𝑘})‘𝑛) = 𝑘)
365364adantl 485 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {𝑘})‘𝑛) = 𝑘)
366360, 362, 329, 329, 330, 363, 365ofval 7402 . . . . . . . . . . . . . . 15 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑃f / ((1...𝑁) × {𝑘}))‘𝑛) = ((𝑃𝑛) / 𝑘))
367366an32s 651 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑃f / ((1...𝑁) × {𝑘}))‘𝑛) = ((𝑃𝑛) / 𝑘))
368367eqeq1d 2803 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (((𝑃f / ((1...𝑁) × {𝑘}))‘𝑛) = 0 ↔ ((𝑃𝑛) / 𝑘) = 0))
369359, 368sylibrd 262 . . . . . . . . . . . 12 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑃𝑛) = 0 → ((𝑃f / ((1...𝑁) × {𝑘}))‘𝑛) = 0))
370 simplll 774 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → 𝜑)
371 simplr 768 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → 𝑛 ∈ (1...𝑁))
372344adantlr 714 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (𝑃f / ((1...𝑁) × {𝑘})) ∈ 𝐼)
373 ovex 7172 . . . . . . . . . . . . . 14 (𝑃f / ((1...𝑁) × {𝑘})) ∈ V
374 eleq1 2880 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑃f / ((1...𝑁) × {𝑘})) → (𝑧𝐼 ↔ (𝑃f / ((1...𝑁) × {𝑘})) ∈ 𝐼))
375 fveq1 6648 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑃f / ((1...𝑁) × {𝑘})) → (𝑧𝑛) = ((𝑃f / ((1...𝑁) × {𝑘}))‘𝑛))
376375eqeq1d 2803 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑃f / ((1...𝑁) × {𝑘})) → ((𝑧𝑛) = 0 ↔ ((𝑃f / ((1...𝑁) × {𝑘}))‘𝑛) = 0))
377 fveq2 6649 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑃f / ((1...𝑁) × {𝑘})) → (𝐹𝑧) = (𝐹‘(𝑃f / ((1...𝑁) × {𝑘}))))
378377fveq1d 6651 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑃f / ((1...𝑁) × {𝑘})) → ((𝐹𝑧)‘𝑛) = ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛))
379378breq1d 5043 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑃f / ((1...𝑁) × {𝑘})) → (((𝐹𝑧)‘𝑛) ≤ 0 ↔ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
380376, 379imbi12d 348 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑃f / ((1...𝑁) × {𝑘})) → (((𝑧𝑛) = 0 → ((𝐹𝑧)‘𝑛) ≤ 0) ↔ (((𝑃f / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0)))
381374, 380imbi12d 348 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑃f / ((1...𝑁) × {𝑘})) → ((𝑧𝐼 → ((𝑧𝑛) = 0 → ((𝐹𝑧)‘𝑛) ≤ 0)) ↔ ((𝑃f / ((1...𝑁) × {𝑘})) ∈ 𝐼 → (((𝑃f / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))))
382381imbi2d 344 . . . . . . . . . . . . . . 15 (𝑧 = (𝑃f / ((1...𝑁) × {𝑘})) → ((𝑛 ∈ (1...𝑁) → (𝑧𝐼 → ((𝑧𝑛) = 0 → ((𝐹𝑧)‘𝑛) ≤ 0))) ↔ (𝑛 ∈ (1...𝑁) → ((𝑃f / ((1...𝑁) × {𝑘})) ∈ 𝐼 → (((𝑃f / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0)))))
383382imbi2d 344 . . . . . . . . . . . . . 14 (𝑧 = (𝑃f / ((1...𝑁) × {𝑘})) → ((𝜑 → (𝑛 ∈ (1...𝑁) → (𝑧𝐼 → ((𝑧𝑛) = 0 → ((𝐹𝑧)‘𝑛) ≤ 0)))) ↔ (𝜑 → (𝑛 ∈ (1...𝑁) → ((𝑃f / ((1...𝑁) × {𝑘})) ∈ 𝐼 → (((𝑃f / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))))))
384 poimir.2 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧𝐼 ∧ (𝑧𝑛) = 0)) → ((𝐹𝑧)‘𝑛) ≤ 0)
3853843exp2 1351 . . . . . . . . . . . . . 14 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑧𝐼 → ((𝑧𝑛) = 0 → ((𝐹𝑧)‘𝑛) ≤ 0))))
386373, 383, 385vtocl 3510 . . . . . . . . . . . . 13 (𝜑 → (𝑛 ∈ (1...𝑁) → ((𝑃f / ((1...𝑁) × {𝑘})) ∈ 𝐼 → (((𝑃f / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))))
387370, 371, 372, 386syl3c 66 . . . . . . . . . . . 12 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (((𝑃f / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
388369, 387syld 47 . . . . . . . . . . 11 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑃𝑛) = 0 → ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
389354, 388jaod 856 . . . . . . . . . 10 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((¬ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃𝑛) = 0) → ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
390202, 389syld 47 . . . . . . . . 9 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
391390reximdva 3236 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑗 ∈ (0...𝑁)(𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → ∃𝑗 ∈ (0...𝑁)((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
392391anasss 470 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (∃𝑗 ∈ (0...𝑁)(𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}), ℝ, < ) → ∃𝑗 ∈ (0...𝑁)((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
393108, 392mpd 15 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → ∃𝑗 ∈ (0...𝑁)((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0)
394 breq 5035 . . . . . . . 8 (𝑟 = ≤ → (0𝑟((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ↔ 0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))
395 fvex 6662 . . . . . . . . 9 ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ∈ V
39632, 395brcnv 5721 . . . . . . . 8 (0 ≤ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ↔ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0)
397394, 396syl6bb 290 . . . . . . 7 (𝑟 = ≤ → (0𝑟((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ↔ ((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
398397rexbidv 3259 . . . . . 6 (𝑟 = ≤ → (∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ↔ ∃𝑗 ∈ (0...𝑁)((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
399393, 398syl5ibrcom 250 . . . . 5 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (𝑟 = ≤ → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))
40076, 399jaod 856 . . . 4 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → ((𝑟 = ≤ ∨ 𝑟 = ≤ ) → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))
4011, 400syl5 34 . . 3 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (𝑟 ∈ { ≤ , ≤ } → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))
402401exp32 424 . 2 (𝜑 → (𝑘 ∈ ℕ → (𝑛 ∈ (1...𝑁) → (𝑟 ∈ { ≤ , ≤ } → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛)))))
4034023imp2 1346 1 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃f / ((1...𝑁) × {𝑘})))‘𝑛))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  {cab 2779  wne 2990  wral 3109  wrex 3110  {crab 3113  Vcvv 3444  cun 3882  cin 3883  wss 3884  c0 4246  {csn 4528  {cpr 4530   cuni 4803   class class class wbr 5033   Or wor 5441   × cxp 5521  ccnv 5522  ran crn 5524  cima 5526  Fun wfun 6322   Fn wfn 6323  wf 6324  ontowfo 6326  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7139  f cof 7391  1st c1st 7673  2nd c2nd 7674  m cmap 8393  Fincfn 8496  supcsup 8892  cc 10528  cr 10529  0cc0 10530  1c1 10531   + caddc 10533   · cmul 10535   < clt 10668  cle 10669  cmin 10863   / cdiv 11290  cn 11629  0cn0 11889  cz 11973  cuz 12235  +crp 12381  (,)cioo 12730  [,]cicc 12733  ...cfz 12889  ..^cfzo 13032  t crest 16690  topGenctg 16707  tcpt 16708  Topctop 21502   Cn ccn 21833
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fi 8863  df-sup 8894  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-ioo 12734  df-icc 12737  df-fz 12890  df-fzo 13033  df-rest 16692  df-topgen 16713  df-pt 16714  df-top 21503  df-topon 21520  df-bases 21555  df-cn 21836
This theorem is referenced by:  poimirlem32  35088
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