Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  poimirlem31 Structured version   Visualization version   GIF version

Theorem poimirlem31 33570
Description: Lemma for poimir 33572, assigning values to the vertices of the tessellation that meet the hypotheses of both poimirlem30 33569 and poimirlem28 33567. 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 4230 . . . 4 (𝑟 ∈ { ≤ , ≤ } → (𝑟 = ≤ ∨ 𝑟 = ≤ ))
2 simprr 811 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → 𝑛 ∈ (1...𝑁))
3 fz1ssfz0 12474 . . . . . . . . . 10 (1...𝑁) ⊆ (0...𝑁)
43sseli 3632 . . . . . . . . 9 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (0...𝑁))
54anim2i 592 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁)))
6 eleq1 2718 . . . . . . . . . . . . 13 (𝑖 = 𝑛 → (𝑖 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...𝑁)))
76anbi2d 740 . . . . . . . . . . . 12 (𝑖 = 𝑛 → ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁)) ↔ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁))))
87anbi2d 740 . . . . . . . . . . 11 (𝑖 = 𝑛 → ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) ↔ (𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁)))))
9 eqeq1 2655 . . . . . . . . . . . 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 3081 . . . . . . . . . . 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 333 . . . . . . . . . 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, 12chvarv 2299 . . . . . . . . 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 12382 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → 1 ≤ 𝑛)
15 1re 10077 . . . . . . . . . . . . . 14 1 ∈ ℝ
16 elfzelz 12380 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
1716zred 11520 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
18 lenlt 10154 . . . . . . . . . . . . . 14 ((1 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (1 ≤ 𝑛 ↔ ¬ 𝑛 < 1))
1915, 17, 18sylancr 696 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → (1 ≤ 𝑛 ↔ ¬ 𝑛 < 1))
2014, 19mpbid 222 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑁) → ¬ 𝑛 < 1)
21 elsni 4227 . . . . . . . . . . . . 13 (𝑛 ∈ {0} → 𝑛 = 0)
22 0lt1 10588 . . . . . . . . . . . . 13 0 < 1
2321, 22syl6eqbr 4724 . . . . . . . . . . . 12 (𝑛 ∈ {0} → 𝑛 < 1)
2420, 23nsyl 135 . . . . . . . . . . 11 (𝑛 ∈ (1...𝑁) → ¬ 𝑛 ∈ {0})
25 ltso 10156 . . . . . . . . . . . . . . 15 < Or ℝ
26 snfi 8079 . . . . . . . . . . . . . . . . 17 {0} ∈ Fin
27 fzfi 12811 . . . . . . . . . . . . . . . . . 18 (1...𝑁) ∈ Fin
28 rabfi 8226 . . . . . . . . . . . . . . . . . 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 8268 . . . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . 16 ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ∈ Fin
32 c0ex 10072 . . . . . . . . . . . . . . . . . 18 0 ∈ V
3332snid 4241 . . . . . . . . . . . . . . . . 17 0 ∈ {0}
34 elun1 3813 . . . . . . . . . . . . . . . . 17 (0 ∈ {0} → 0 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}))
35 ne0i 3954 . . . . . . . . . . . . . . . . 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 10078 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
38 snssi 4371 . . . . . . . . . . . . . . . . . 18 (0 ∈ ℝ → {0} ⊆ ℝ)
3937, 38ax-mp 5 . . . . . . . . . . . . . . . . 17 {0} ⊆ ℝ
40 ssrab2 3720 . . . . . . . . . . . . . . . . . 18 {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} ⊆ (1...𝑁)
4116ssriv 3640 . . . . . . . . . . . . . . . . . . 19 (1...𝑁) ⊆ ℤ
42 zssre 11422 . . . . . . . . . . . . . . . . . . 19 ℤ ⊆ ℝ
4341, 42sstri 3645 . . . . . . . . . . . . . . . . . 18 (1...𝑁) ⊆ ℝ
4440, 43sstri 3645 . . . . . . . . . . . . . . . . 17 {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} ⊆ ℝ
4539, 44unssi 3821 . . . . . . . . . . . . . . . 16 ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ⊆ ℝ
4631, 36, 453pm3.2i 1259 . . . . . . . . . . . . . . 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 8416 . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . 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 2718 . . . . . . . . . . . . . 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 248 . . . . . . . . . . . . 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 3786 . . . . . . . . . . . . 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 208 . . . . . . . . . . . 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 6698 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑛 → (1...𝑎) = (1...𝑛))
5453raleqdv 3174 . . . . . . . . . . . . . . 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 3396 . . . . . . . . . . . . . 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 12376 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ‘1))
57 eluzfz2 12387 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (ℤ‘1) → 𝑛 ∈ (1...𝑛))
5856, 57syl 17 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (1...𝑛))
59 simpl 472 . . . . . . . . . . . . . . . 16 ((0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) → 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏))
6059ralimi 2981 . . . . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑛 → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛))
6261breq2d 4697 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑛 → (0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
6362rspcva 3338 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...𝑛) ∧ ∀𝑏 ∈ (1...𝑛)0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏)) → 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛))
6458, 60, 63syl2an 493 . . . . . . . . . . . . . 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 207 . . . . . . . . . . . . 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 539 . . . . . . . . . . . 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 396 . . . . . . . . . . 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 3045 . . . . . . . . 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 688 . . . . . . 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 4687 . . . . . . 7 (𝑟 = ≤ → (0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ↔ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
7574rexbidv 3081 . . . . . 6 (𝑟 = ≤ → (∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ↔ ∃𝑗 ∈ (0...𝑁)0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
7673, 75syl5ibrcom 237 . . . . 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 11519 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℤ)
79 elfzm1b 12456 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
8016, 78, 79syl2anr 494 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
8180biimpd 219 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ (0...(𝑁 − 1))))
8281ex 449 . . . . . . . . . . . 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 11074 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℂ)
85 npcan1 10493 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
8684, 85syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
87 nnm1nn0 11372 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
8877, 87syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℕ0)
8988nn0zd 11518 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 − 1) ∈ ℤ)
90 uzid 11740 . . . . . . . . . . . . . . 15 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
91 peano2uz 11779 . . . . . . . . . . . . . . 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 2731 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
94 fzss2 12419 . . . . . . . . . . . . 13 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
9593, 94syl 17 . . . . . . . . . . . 12 (𝜑 → (0...(𝑁 − 1)) ⊆ (0...𝑁))
9695sseld 3635 . . . . . . . . . . 11 (𝜑 → ((𝑛 − 1) ∈ (0...(𝑁 − 1)) → (𝑛 − 1) ∈ (0...𝑁)))
9783, 96syld 47 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ (0...𝑁)))
9897anim2d 588 . . . . . . . . 9 (𝜑 → ((𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁))))
9998imp 444 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁)))
100 ovex 6718 . . . . . . . . 9 (𝑛 − 1) ∈ V
101 eleq1 2718 . . . . . . . . . . . 12 (𝑖 = (𝑛 − 1) → (𝑖 ∈ (0...𝑁) ↔ (𝑛 − 1) ∈ (0...𝑁)))
102101anbi2d 740 . . . . . . . . . . 11 (𝑖 = (𝑛 − 1) → ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁)) ↔ (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁))))
103102anbi2d 740 . . . . . . . . . 10 (𝑖 = (𝑛 − 1) → ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) ↔ (𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁)))))
104 eqeq1 2655 . . . . . . . . . . 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 3081 . . . . . . . . . 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 333 . . . . . . . . 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 3290 . . . . . . . 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 486 . . . . . . 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 2718 . . . . . . . . . . . . . . . 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 248 . . . . . . . . . . . . . . 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 3786 . . . . . . . . . . . . . . . 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 4225 . . . . . . . . . . . . . . . . 17 ((𝑛 − 1) ∈ {0} ↔ (𝑛 − 1) = 0)
113 oveq2 6698 . . . . . . . . . . . . . . . . . . 19 (𝑎 = (𝑛 − 1) → (1...𝑎) = (1...(𝑛 − 1)))
114113raleqdv 3174 . . . . . . . . . . . . . . . . . 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 3396 . . . . . . . . . . . . . . . . 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 542 . . . . . . . . . . . . . . . 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 264 . . . . . . . . . . . . . . 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 208 . . . . . . . . . . . . . 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 10901 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
121 peano2rem 10386 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
122 ltnle 10155 . . . . . . . . . . . . . . . . . . 19 (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
123121, 122mpancom 704 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
124120, 123mpbid 222 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℝ → ¬ 𝑛 ≤ (𝑛 − 1))
12517, 124syl 17 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑁) → ¬ 𝑛 ≤ (𝑛 − 1))
126 breq2 4689 . . . . . . . . . . . . . . . . 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 307 . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . 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 3814 . . . . . . . . . . . . . . . . . 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 11007 . . . . . . . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . . . . . 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 1259 . . . . . . . . . . . . . . . . . . 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 11036 . . . . . . . . . . . . . . . . . 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 150 . . . . . . . . . . . . . . . 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 508 . . . . . . . . . . . . . . . . 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 322 . . . . . . . . . . . . . . . 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 208 . . . . . . . . . . . . . . 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 pm2.63 846 . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . 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 554 . . . . . . . . . . . 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 930 . . . . . . . . . . . . . 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 11171 . . . . . . . . . . . . . . . . . 18 (1 + 0) = 1
14616zcnd 11521 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
147 ax-1cn 10032 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℂ
148 0cn 10070 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℂ
149 subadd 10322 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 0 ∈ ℂ) → ((𝑛 − 1) = 0 ↔ (1 + 0) = 𝑛))
150147, 148, 149mp3an23 1456 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℂ → ((𝑛 − 1) = 0 ↔ (1 + 0) = 𝑛))
151146, 150syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = 0 ↔ (1 + 0) = 𝑛))
152151biimpa 500 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ (1...𝑁) ∧ (𝑛 − 1) = 0) → (1 + 0) = 𝑛)
153145, 152syl5reqr 2700 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (1...𝑁) ∧ (𝑛 − 1) = 0) → 𝑛 = 1)
154 1z 11445 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℤ
155 fzsn 12421 . . . . . . . . . . . . . . . . . . . . . 22 (1 ∈ ℤ → (1...1) = {1})
156154, 155ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (1...1) = {1}
157 oveq2 6698 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1 → (1...𝑛) = (1...1))
158 sneq 4220 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1 → {𝑛} = {1})
159156, 157, 1583eqtr4a 2711 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → (1...𝑛) = {𝑛})
160159raleqdv 3174 . . . . . . . . . . . . . . . . . . 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 307 . . . . . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . . . . . 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 628 . . . . . . . . . . . . . . 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 3828 . . . . . . . . . . . . . . . . . . . 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 10493 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
167146, 166syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) = 𝑛)
168167, 56eqeltrd 2730 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) ∈ (ℤ‘1))
169 peano2zm 11458 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℤ → (𝑛 − 1) ∈ ℤ)
170 uzid 11740 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)))
171 peano2uz 11779 . . . . . . . . . . . . . . . . . . . . . . . . 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 2731 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ‘(𝑛 − 1)))
174 fzsplit2 12404 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑛 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑛 ∈ (ℤ‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
175168, 173, 174syl2anc 694 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (1...𝑁) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
176167oveq1d 6705 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
177 fzsn 12421 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
17816, 177syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) → (𝑛...𝑛) = {𝑛})
179176, 178eqtrd 2685 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
180179uneq2d 3800 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (1...𝑁) → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
181175, 180eqtrd 2685 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (1...𝑁) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
182181raleqdv 3174 . . . . . . . . . . . . . . . . . . . 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 236 . . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . . 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 148 . . . . . . . . . . . . . . . . 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 752 . . . . . . . . . . . . . . . 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 628 . . . . . . . . . . . . . . 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 394 . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑛 → (𝑃𝑏) = (𝑃𝑛))
191190neeq1d 2882 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑛 → ((𝑃𝑏) ≠ 0 ↔ (𝑃𝑛) ≠ 0))
19262, 191anbi12d 747 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑛 → ((0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃𝑛) ≠ 0)))
193192ralsng 4250 . . . . . . . . . . . . . . 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 307 . . . . . . . . . . . . . 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 508 . . . . . . . . . . . . . . 15 (¬ (0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃𝑛) ≠ 0) ↔ (¬ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∨ ¬ (𝑃𝑛) ≠ 0))
196 nne 2827 . . . . . . . . . . . . . . . 16 (¬ (𝑃𝑛) ≠ 0 ↔ (𝑃𝑛) = 0)
197196orbi2i 540 . . . . . . . . . . . . . . 15 ((¬ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∨ ¬ (𝑃𝑛) ≠ 0) ↔ (¬ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃𝑛) = 0))
198195, 197bitri 264 . . . . . . . . . . . . . 14 (¬ (0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃𝑛) ≠ 0) ↔ (¬ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃𝑛) = 0))
199194, 198syl6bb 276 . . . . . . . . . . . . 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 229 . . . . . . . . . . . 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 763 . . . . . . . . . 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 22612 . . . . . . . . . . . . . . . . . . . . . . . 24 (topGen‘ran (,)) ∈ Top
206205fconst6 6133 . . . . . . . . . . . . . . . . . . . . . . 23 ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top
207 pttop 21433 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑁) ∈ Fin ∧ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top) → (∏t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Top)
20827, 206, 207mp2an 708 . . . . . . . . . . . . . . . . . . . . . 22 (∏t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Top
209204, 208eqeltri 2726 . . . . . . . . . . . . . . . . . . . . 21 𝑅 ∈ Top
210 poimir.i . . . . . . . . . . . . . . . . . . . . . 22 𝐼 = ((0[,]1) ↑𝑚 (1...𝑁))
211 reex 10065 . . . . . . . . . . . . . . . . . . . . . . 23 ℝ ∈ V
212 unitssre 12357 . . . . . . . . . . . . . . . . . . . . . . 23 (0[,]1) ⊆ ℝ
213 mapss 7942 . . . . . . . . . . . . . . . . . . . . . . 23 ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑𝑚 (1...𝑁)) ⊆ (ℝ ↑𝑚 (1...𝑁)))
214211, 212, 213mp2an 708 . . . . . . . . . . . . . . . . . . . . . 22 ((0[,]1) ↑𝑚 (1...𝑁)) ⊆ (ℝ ↑𝑚 (1...𝑁))
215210, 214eqsstri 3668 . . . . . . . . . . . . . . . . . . . . 21 𝐼 ⊆ (ℝ ↑𝑚 (1...𝑁))
216 uniretop 22613 . . . . . . . . . . . . . . . . . . . . . . . 24 ℝ = (topGen‘ran (,))
217204, 216ptuniconst 21449 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑁) ∈ Fin ∧ (topGen‘ran (,)) ∈ Top) → (ℝ ↑𝑚 (1...𝑁)) = 𝑅)
21827, 205, 217mp2an 708 . . . . . . . . . . . . . . . . . . . . . 22 (ℝ ↑𝑚 (1...𝑁)) = 𝑅
219218restuni 21014 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Top ∧ 𝐼 ⊆ (ℝ ↑𝑚 (1...𝑁))) → 𝐼 = (𝑅t 𝐼))
220209, 215, 219mp2an 708 . . . . . . . . . . . . . . . . . . . 20 𝐼 = (𝑅t 𝐼)
221220, 218cnf 21098 . . . . . . . . . . . . . . . . . . 19 (𝐹 ∈ ((𝑅t 𝐼) Cn 𝑅) → 𝐹:𝐼⟶(ℝ ↑𝑚 (1...𝑁)))
222203, 221syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹:𝐼⟶(ℝ ↑𝑚 (1...𝑁)))
223222ad2antrr 762 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝐹:𝐼⟶(ℝ ↑𝑚 (1...𝑁)))
224 simplr 807 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝑘 ∈ ℕ)
225 elfzelz 12380 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℤ)
226225zred 11520 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℝ)
227226adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
228 nnre 11065 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
229228adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
230 nnne0 11091 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ → 𝑘 ≠ 0)
231230adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ≠ 0)
232227, 229, 231redivcld 10891 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑥 / 𝑘) ∈ ℝ)
233 elfzle1 12382 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (0...𝑘) → 0 ≤ 𝑥)
234226, 233jca 553 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ (0...𝑘) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
235 nnrp 11880 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
236235rpregt0d 11916 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
237 divge0 10930 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝑥 / 𝑘))
238234, 236, 237syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑥 / 𝑘))
239 elfzle2 12383 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (0...𝑘) → 𝑥𝑘)
240239adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑥𝑘)
241 1red 10093 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ)
242235adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ+)
243227, 241, 242ledivmuld 11963 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑥 / 𝑘) ≤ 1 ↔ 𝑥 ≤ (𝑘 · 1)))
244 nncn 11066 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
245244mulid1d 10095 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 ∈ ℕ → (𝑘 · 1) = 𝑘)
246245breq2d 4697 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ ℕ → (𝑥 ≤ (𝑘 · 1) ↔ 𝑥𝑘))
247246adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑥 ≤ (𝑘 · 1) ↔ 𝑥𝑘))
248243, 247bitrd 268 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑥 / 𝑘) ≤ 1 ↔ 𝑥𝑘))
249240, 248mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑥 / 𝑘) ≤ 1)
25037, 15elicc2i 12277 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 / 𝑘) ∈ (0[,]1) ↔ ((𝑥 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝑥 / 𝑘) ∧ (𝑥 / 𝑘) ≤ 1))
251232, 238, 249, 250syl3anbrc 1265 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑥 / 𝑘) ∈ (0[,]1))
252251ancoms 468 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ (0...𝑘)) → (𝑥 / 𝑘) ∈ (0[,]1))
253 elsni 4227 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ {𝑘} → 𝑦 = 𝑘)
254253oveq2d 6706 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ {𝑘} → (𝑥 / 𝑦) = (𝑥 / 𝑘))
255254eleq1d 2715 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ {𝑘} → ((𝑥 / 𝑦) ∈ (0[,]1) ↔ (𝑥 / 𝑘) ∈ (0[,]1)))
256252, 255syl5ibrcom 237 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ (0...𝑘)) → (𝑦 ∈ {𝑘} → (𝑥 / 𝑦) ∈ (0[,]1)))
257256impr 648 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ ℕ ∧ (𝑥 ∈ (0...𝑘) ∧ 𝑦 ∈ {𝑘})) → (𝑥 / 𝑦) ∈ (0[,]1))
258224, 257sylan 487 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑥 ∈ (0...𝑘) ∧ 𝑦 ∈ {𝑘})) → (𝑥 / 𝑦) ∈ (0[,]1))
259 elun 3786 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ({1} ∪ {0}) ↔ (𝑦 ∈ {1} ∨ 𝑦 ∈ {0}))
260 fzofzp1 12605 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (0..^𝑘) → (𝑥 + 1) ∈ (0...𝑘))
261 elsni 4227 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ {1} → 𝑦 = 1)
262261oveq2d 6706 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ {1} → (𝑥 + 𝑦) = (𝑥 + 1))
263262eleq1d 2715 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ {1} → ((𝑥 + 𝑦) ∈ (0...𝑘) ↔ (𝑥 + 1) ∈ (0...𝑘)))
264260, 263syl5ibrcom 237 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ (0..^𝑘) → (𝑦 ∈ {1} → (𝑥 + 𝑦) ∈ (0...𝑘)))
265 elfzonn0 12552 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 ∈ (0..^𝑘) → 𝑥 ∈ ℕ0)
266265nn0cnd 11391 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 ∈ (0..^𝑘) → 𝑥 ∈ ℂ)
267266addid1d 10274 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ (0..^𝑘) → (𝑥 + 0) = 𝑥)
268 elfzofz 12524 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ (0..^𝑘) → 𝑥 ∈ (0...𝑘))
269267, 268eqeltrd 2730 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (0..^𝑘) → (𝑥 + 0) ∈ (0...𝑘))
270 elsni 4227 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ {0} → 𝑦 = 0)
271270oveq2d 6706 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ {0} → (𝑥 + 𝑦) = (𝑥 + 0))
272271eleq1d 2715 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ {0} → ((𝑥 + 𝑦) ∈ (0...𝑘) ↔ (𝑥 + 0) ∈ (0...𝑘)))
273269, 272syl5ibrcom 237 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ (0..^𝑘) → (𝑦 ∈ {0} → (𝑥 + 𝑦) ∈ (0...𝑘)))
274264, 273jaod 394 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ (0..^𝑘) → ((𝑦 ∈ {1} ∨ 𝑦 ∈ {0}) → (𝑥 + 𝑦) ∈ (0...𝑘)))
275259, 274syl5bi 232 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (0..^𝑘) → (𝑦 ∈ ({1} ∪ {0}) → (𝑥 + 𝑦) ∈ (0...𝑘)))
276275imp 444 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ (0..^𝑘) ∧ 𝑦 ∈ ({1} ∪ {0})) → (𝑥 + 𝑦) ∈ (0...𝑘))
277276adantl 481 . . . . . . . . . . . . . . . . . . . . 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 6399 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
280 xp1st 7242 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺𝑘) ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(𝐺𝑘)) ∈ (ℕ0𝑚 (1...𝑁)))
281 elmapfn 7922 . . . . . . . . . . . . . . . . . . . . . . . 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 5930 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st ‘(𝐺𝑘)):(1...𝑁)⟶(0..^𝑘) ↔ ((1st ‘(𝐺𝑘)) Fn (1...𝑁) ∧ ran (1st ‘(𝐺𝑘)) ⊆ (0..^𝑘)))
285282, 283, 284sylanbrc 699 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐺𝑘)):(1...𝑁)⟶(0..^𝑘))
286285adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (1st ‘(𝐺𝑘)):(1...𝑁)⟶(0..^𝑘))
287 1ex 10073 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ V
288287fconst 6129 . . . . . . . . . . . . . . . . . . . . . . . 24 (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}):((2nd ‘(𝐺𝑘)) “ (1...𝑗))⟶{1}
28932fconst 6129 . . . . . . . . . . . . . . . . . . . . . . . 24 (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}):((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))⟶{0}
290288, 289pm3.2i 470 . . . . . . . . . . . . . . . . . . . . . . 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 7243 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6239 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (2nd ‘(𝐺𝑘)) ∈ V
294 f1oeq1 6165 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 = (2nd ‘(𝐺𝑘)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)))
295293, 294elab 3382 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((2nd ‘(𝐺𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
296292, 295sylib 208 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑘 ∈ ℕ) → (2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
297 dff1o3 6181 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(𝐺𝑘)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(𝐺𝑘))))
298297simprbi 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(𝐺𝑘)))
299 imain 6012 . . . . . . . . . . . . . . . . . . . . . . . . 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 12471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℕ0)
302301nn0red 11390 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℝ)
303302ltp1d 10992 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑁) → 𝑗 < (𝑗 + 1))
304 fzdisj 12406 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
305303, 304syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (0...𝑁) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
306305imaeq2d 5501 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑁) → ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((2nd ‘(𝐺𝑘)) “ ∅))
307 ima0 5516 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((2nd ‘(𝐺𝑘)) “ ∅) = ∅
308306, 307syl6eq 2701 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0...𝑁) → ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ∅)
309300, 308sylan9req 2706 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∩ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))) = ∅)
310 fun 6104 . . . . . . . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . . . . . . . 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 5580 . . . . . . . . . . . . . . . . . . . . . . . 24 ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∪ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)))
313 nn0p1nn 11370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ ℕ0 → (𝑗 + 1) ∈ ℕ)
314301, 313syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ ℕ)
315 nnuz 11761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ℕ = (ℤ‘1)
316314, 315syl6eleq 2740 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ (ℤ‘1))
317 elfzuz3 12377 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑁) → 𝑁 ∈ (ℤ𝑗))
318 fzsplit2 12404 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑗 + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ𝑗)) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
319316, 317, 318syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑁) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
320319imaeq2d 5501 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (0...𝑁) → ((2nd ‘(𝐺𝑘)) “ (1...𝑁)) = ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))))
321 f1ofo 6182 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(𝐺𝑘)):(1...𝑁)–onto→(1...𝑁))
322 foima 6158 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2706 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑗 ∈ (0...𝑁) ∧ (𝜑𝑘 ∈ ℕ)) → ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (1...𝑁))
325324ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (1...𝑁))
326312, 325syl5eqr 2699 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∪ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))) = (1...𝑁))
327326feq2d 6069 . . . . . . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))
329 fzfid 12812 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
330 inidm 3855 . . . . . . . . . . . . . . . . . . . . 21 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
331277, 286, 328, 329, 329, 330off 6954 . . . . . . . . . . . . . . . . . . . 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 6074 . . . . . . . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝑃:(1...𝑁)⟶(0...𝑘))
335 vex 3234 . . . . . . . . . . . . . . . . . . . . 21 𝑘 ∈ V
336335fconst 6129 . . . . . . . . . . . . . . . . . . . 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 6954 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (𝑃𝑓 / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
339210eleq2i 2722 . . . . . . . . . . . . . . . . . . 19 ((𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑𝑚 (1...𝑁)))
340 ovex 6718 . . . . . . . . . . . . . . . . . . . 20 (0[,]1) ∈ V
341 ovex 6718 . . . . . . . . . . . . . . . . . . . 20 (1...𝑁) ∈ V
342340, 341elmap 7928 . . . . . . . . . . . . . . . . . . 19 ((𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑𝑚 (1...𝑁)) ↔ (𝑃𝑓 / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
343339, 342bitri 264 . . . . . . . . . . . . . . . . . 18 ((𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (𝑃𝑓 / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
344338, 343sylibr 224 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼)
345223, 344ffvelrnd 6400 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘}))) ∈ (ℝ ↑𝑚 (1...𝑁)))
346 elmapi 7921 . . . . . . . . . . . . . . . 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 6399 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∈ ℝ)
349348an32s 863 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∈ ℝ)
350 0red 10079 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → 0 ∈ ℝ)
351349, 350ltnled 10222 . . . . . . . . . . . 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 10164 . . . . . . . . . . . . 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 695 . . . . . . . . . . . 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 250 . . . . . . . . . . 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 10838 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (0 / 𝑘) = 0)
356 oveq1 6697 . . . . . . . . . . . . . . . 16 ((𝑃𝑛) = 0 → ((𝑃𝑛) / 𝑘) = (0 / 𝑘))
357356eqeq1d 2653 . . . . . . . . . . . . . . 15 ((𝑃𝑛) = 0 → (((𝑃𝑛) / 𝑘) = 0 ↔ (0 / 𝑘) = 0))
358355, 357syl5ibrcom 237 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → ((𝑃𝑛) = 0 → ((𝑃𝑛) / 𝑘) = 0))
359358ad3antlr 767 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑃𝑛) = 0 → ((𝑃𝑛) / 𝑘) = 0))
360 ffn 6083 . . . . . . . . . . . . . . . . 17 (𝑃:(1...𝑁)⟶(0...𝑘) → 𝑃 Fn (1...𝑁))
361334, 360syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝑃 Fn (1...𝑁))
362 fnconstg 6131 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ V → ((1...𝑁) × {𝑘}) Fn (1...𝑁))
363335, 362mp1i 13 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((1...𝑁) × {𝑘}) Fn (1...𝑁))
364 eqidd 2652 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑃𝑛) = (𝑃𝑛))
365335fvconst2 6510 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {𝑘})‘𝑛) = 𝑘)
366365adantl 481 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {𝑘})‘𝑛) = 𝑘)
367361, 363, 329, 329, 330, 364, 366ofval 6948 . . . . . . . . . . . . . . 15 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = ((𝑃𝑛) / 𝑘))
368367an32s 863 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = ((𝑃𝑛) / 𝑘))
369368eqeq1d 2653 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 0 ↔ ((𝑃𝑛) / 𝑘) = 0))
370359, 369sylibrd 249 . . . . . . . . . . . 12 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑃𝑛) = 0 → ((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 0))
371 simplll 813 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → 𝜑)
372 simplr 807 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → 𝑛 ∈ (1...𝑁))
373344adantlr 751 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼)
374 ovex 6718 . . . . . . . . . . . . . 14 (𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ V
375 eleq1 2718 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑃𝑓 / ((1...𝑁) × {𝑘})) → (𝑧𝐼 ↔ (𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼))
376 fveq1 6228 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑃𝑓 / ((1...𝑁) × {𝑘})) → (𝑧𝑛) = ((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛))
377376eqeq1d 2653 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑃𝑓 / ((1...𝑁) × {𝑘})) → ((𝑧𝑛) = 0 ↔ ((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 0))
378 fveq2 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑃𝑓 / ((1...𝑁) × {𝑘})) → (𝐹𝑧) = (𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘}))))
379378fveq1d 6231 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑃𝑓 / ((1...𝑁) × {𝑘})) → ((𝐹𝑧)‘𝑛) = ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛))
380379breq1d 4695 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑃𝑓 / ((1...𝑁) × {𝑘})) → (((𝐹𝑧)‘𝑛) ≤ 0 ↔ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
381377, 380imbi12d 333 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑃𝑓 / ((1...𝑁) × {𝑘})) → (((𝑧𝑛) = 0 → ((𝐹𝑧)‘𝑛) ≤ 0) ↔ (((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0)))
382375, 381imbi12d 333 . . . . . . . . . . . . . . . 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))))
383382imbi2d 329 . . . . . . . . . . . . . . 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)))))
384383imbi2d 329 . . . . . . . . . . . . . 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))))))
385 poimir.2 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧𝐼 ∧ (𝑧𝑛) = 0)) → ((𝐹𝑧)‘𝑛) ≤ 0)
3863853exp2 1307 . . . . . . . . . . . . . 14 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑧𝐼 → ((𝑧𝑛) = 0 → ((𝐹𝑧)‘𝑛) ≤ 0))))
387374, 384, 386vtocl 3290 . . . . . . . . . . . . 13 (𝜑 → (𝑛 ∈ (1...𝑁) → ((𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 → (((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))))
388371, 372, 373, 387syl3c 66 . . . . . . . . . . . 12 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
389370, 388syld 47 . . . . . . . . . . 11 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑃𝑛) = 0 → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
390354, 389jaod 394 . . . . . . . . . 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))
391202, 390syld 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))
392391reximdva 3046 . . . . . . . 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))
393392anasss 680 . . . . . . 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))
394108, 393mpd 15 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → ∃𝑗 ∈ (0...𝑁)((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0)
395 breq 4687 . . . . . . . 8 (𝑟 = ≤ → (0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ↔ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
396 fvex 6239 . . . . . . . . 9 ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∈ V
39732, 396brcnv 5337 . . . . . . . 8 (0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ↔ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0)
398395, 397syl6bb 276 . . . . . . 7 (𝑟 = ≤ → (0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ↔ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
399398rexbidv 3081 . . . . . 6 (𝑟 = ≤ → (∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ↔ ∃𝑗 ∈ (0...𝑁)((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
400394, 399syl5ibrcom 237 . . . . 5 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (𝑟 = ≤ → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
40176, 400jaod 394 . . . 4 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → ((𝑟 = ≤ ∨ 𝑟 = ≤ ) → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
4021, 401syl5 34 . . 3 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (𝑟 ∈ { ≤ , ≤ } → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
403402exp32 630 . 2 (𝜑 → (𝑘 ∈ ℕ → (𝑛 ∈ (1...𝑁) → (𝑟 ∈ { ≤ , ≤ } → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))))
4044033imp2 1304 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 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210  {cpr 4212   cuni 4468   class class class wbr 4685   Or wor 5063   × cxp 5141  ccnv 5142  ran crn 5144  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  𝑓 cof 6937  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  Fincfn 7997  supcsup 8387  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  0cn0 11330  cz 11415  cuz 11725  +crp 11870  (,)cioo 12213  [,]cicc 12216  ...cfz 12364  ..^cfzo 12504  t crest 16128  topGenctg 16145  tcpt 16146  Topctop 20746   Cn ccn 21076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ioo 12217  df-icc 12220  df-fz 12365  df-fzo 12505  df-rest 16130  df-topgen 16151  df-pt 16152  df-top 20747  df-topon 20764  df-bases 20798  df-cn 21079
This theorem is referenced by:  poimirlem32  33571
  Copyright terms: Public domain W3C validator