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Theorem poimirlem31 32393
Description: Lemma for poimir 32395, assigning values to the vertices of the tessellation that meet the hypotheses of both poimirlem30 32392 and poimirlem28 32390. 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 4144 . . . 4 (𝑟 ∈ { ≤ , ≤ } → (𝑟 = ≤ ∨ 𝑟 = ≤ ))
2 simprr 791 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → 𝑛 ∈ (1...𝑁))
3 1eluzge0 11566 . . . . . . . . . . 11 1 ∈ (ℤ‘0)
4 fzss1 12208 . . . . . . . . . . 11 (1 ∈ (ℤ‘0) → (1...𝑁) ⊆ (0...𝑁))
53, 4ax-mp 5 . . . . . . . . . 10 (1...𝑁) ⊆ (0...𝑁)
65sseli 3563 . . . . . . . . 9 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (0...𝑁))
76anim2i 590 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁)))
8 eleq1 2675 . . . . . . . . . . . . 13 (𝑖 = 𝑛 → (𝑖 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...𝑁)))
98anbi2d 735 . . . . . . . . . . . 12 (𝑖 = 𝑛 → ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁)) ↔ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁))))
109anbi2d 735 . . . . . . . . . . 11 (𝑖 = 𝑛 → ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) ↔ (𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁)))))
11 eqeq1 2613 . . . . . . . . . . . 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, < )))
1211rexbidv 3033 . . . . . . . . . . 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, < )))
1310, 12imbi12d 332 . . . . . . . . . 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, < ))))
14 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, < ))
1513, 14chvarv 2250 . . . . . . . . 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, < ))
16 elfzle1 12172 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → 1 ≤ 𝑛)
17 1re 9895 . . . . . . . . . . . . . 14 1 ∈ ℝ
18 elfzelz 12170 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
1918zred 11316 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
20 lenlt 9967 . . . . . . . . . . . . . 14 ((1 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (1 ≤ 𝑛 ↔ ¬ 𝑛 < 1))
2117, 19, 20sylancr 693 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → (1 ≤ 𝑛 ↔ ¬ 𝑛 < 1))
2216, 21mpbid 220 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑁) → ¬ 𝑛 < 1)
23 elsni 4141 . . . . . . . . . . . . 13 (𝑛 ∈ {0} → 𝑛 = 0)
24 0lt1 10401 . . . . . . . . . . . . 13 0 < 1
2523, 24syl6eqbr 4616 . . . . . . . . . . . 12 (𝑛 ∈ {0} → 𝑛 < 1)
2622, 25nsyl 133 . . . . . . . . . . 11 (𝑛 ∈ (1...𝑁) → ¬ 𝑛 ∈ {0})
27 ltso 9969 . . . . . . . . . . . . . . 15 < Or ℝ
28 snfi 7900 . . . . . . . . . . . . . . . . 17 {0} ∈ Fin
29 fzfi 12590 . . . . . . . . . . . . . . . . . 18 (1...𝑁) ∈ Fin
30 rabfi 8047 . . . . . . . . . . . . . . . . . 18 ((1...𝑁) ∈ Fin → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} ∈ Fin)
3129, 30ax-mp 5 . . . . . . . . . . . . . . . . 17 {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} ∈ Fin
32 unfi 8089 . . . . . . . . . . . . . . . . 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)
3328, 31, 32mp2an 703 . . . . . . . . . . . . . . . 16 ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ∈ Fin
34 c0ex 9890 . . . . . . . . . . . . . . . . . 18 0 ∈ V
3534snid 4154 . . . . . . . . . . . . . . . . 17 0 ∈ {0}
36 elun1 3741 . . . . . . . . . . . . . . . . 17 (0 ∈ {0} → 0 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}))
37 ne0i 3879 . . . . . . . . . . . . . . . . 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)}) ≠ ∅)
3835, 36, 37mp2b 10 . . . . . . . . . . . . . . . 16 ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ≠ ∅
39 0re 9896 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
40 snssi 4279 . . . . . . . . . . . . . . . . . 18 (0 ∈ ℝ → {0} ⊆ ℝ)
4139, 40ax-mp 5 . . . . . . . . . . . . . . . . 17 {0} ⊆ ℝ
42 ssrab2 3649 . . . . . . . . . . . . . . . . . 18 {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} ⊆ (1...𝑁)
4318ssriv 3571 . . . . . . . . . . . . . . . . . . 19 (1...𝑁) ⊆ ℤ
44 zssre 11219 . . . . . . . . . . . . . . . . . . 19 ℤ ⊆ ℝ
4543, 44sstri 3576 . . . . . . . . . . . . . . . . . 18 (1...𝑁) ⊆ ℝ
4642, 45sstri 3576 . . . . . . . . . . . . . . . . 17 {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} ⊆ ℝ
4741, 46unssi 3749 . . . . . . . . . . . . . . . 16 ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)}) ⊆ ℝ
4833, 38, 473pm3.2i 1231 . . . . . . . . . . . . . . 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)
49 fisupcl 8235 . . . . . . . . . . . . . . 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)}))
5027, 48, 49mp2an 703 . . . . . . . . . . . . . 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)})
51 eleq1 2675 . . . . . . . . . . . . . 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)})))
5250, 51mpbiri 246 . . . . . . . . . . . . 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)}))
53 elun 3714 . . . . . . . . . . . . 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)}))
5452, 53sylib 206 . . . . . . . . . . . 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)}))
55 oveq2 6534 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑛 → (1...𝑎) = (1...𝑛))
5655raleqdv 3120 . . . . . . . . . . . . . . 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)))
5756elrab 3330 . . . . . . . . . . . . . 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)))
58 elfzuz 12166 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ‘1))
59 eluzfz2 12177 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (ℤ‘1) → 𝑛 ∈ (1...𝑛))
6058, 59syl 17 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (1...𝑛))
61 simpl 471 . . . . . . . . . . . . . . . 16 ((0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) → 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏))
6261ralimi 2935 . . . . . . . . . . . . . . 15 (∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏))
63 fveq2 6087 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑛 → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛))
6463breq2d 4589 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑛 → (0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
6564rspcva 3279 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...𝑛) ∧ ∀𝑏 ∈ (1...𝑛)0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏)) → 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛))
6660, 62, 65syl2an 492 . . . . . . . . . . . . . 14 ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)) → 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛))
6757, 66sylbi 205 . . . . . . . . . . . . 13 (𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)} → 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛))
6867orim2i 538 . . . . . . . . . . . 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)))
6954, 68syl 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)))
70 orel1 395 . . . . . . . . . . 11 𝑛 ∈ {0} → ((𝑛 ∈ {0} ∨ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)) → 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
7126, 69, 70syl2im 39 . . . . . . . . . 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)))
7271reximdv 2998 . . . . . . . . 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)))
7315, 72syl5 33 . . . . . . . 8 (𝑛 ∈ (1...𝑁) → ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
747, 73sylan2i 684 . . . . . . 7 (𝑛 ∈ (1...𝑁) → ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → ∃𝑗 ∈ (0...𝑁)0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
752, 74mpcom 37 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → ∃𝑗 ∈ (0...𝑁)0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛))
76 breq 4579 . . . . . . 7 (𝑟 = ≤ → (0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ↔ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
7776rexbidv 3033 . . . . . 6 (𝑟 = ≤ → (∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ↔ ∃𝑗 ∈ (0...𝑁)0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
7875, 77syl5ibrcom 235 . . . . 5 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (𝑟 = ≤ → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
79 poimir.0 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℕ)
8079nnzd 11315 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℤ)
81 elfzm1b 12244 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
8218, 80, 81syl2anr 493 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
8382biimpd 217 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ (0...(𝑁 − 1))))
8483ex 448 . . . . . . . . . . . 12 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ (0...(𝑁 − 1)))))
8584pm2.43d 50 . . . . . . . . . . 11 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ (0...(𝑁 − 1))))
8679nncnd 10885 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℂ)
87 npcan1 10306 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
8886, 87syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
89 nnm1nn0 11183 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
9079, 89syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℕ0)
9190nn0zd 11314 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 − 1) ∈ ℤ)
92 uzid 11536 . . . . . . . . . . . . . . 15 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
93 peano2uz 11575 . . . . . . . . . . . . . . 15 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
9491, 92, 933syl 18 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
9588, 94eqeltrrd 2688 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
96 fzss2 12209 . . . . . . . . . . . . 13 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
9795, 96syl 17 . . . . . . . . . . . 12 (𝜑 → (0...(𝑁 − 1)) ⊆ (0...𝑁))
9897sseld 3566 . . . . . . . . . . 11 (𝜑 → ((𝑛 − 1) ∈ (0...(𝑁 − 1)) → (𝑛 − 1) ∈ (0...𝑁)))
9985, 98syld 45 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ (0...𝑁)))
10099anim2d 586 . . . . . . . . 9 (𝜑 → ((𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁))))
101100imp 443 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁)))
102 ovex 6554 . . . . . . . . 9 (𝑛 − 1) ∈ V
103 eleq1 2675 . . . . . . . . . . . 12 (𝑖 = (𝑛 − 1) → (𝑖 ∈ (0...𝑁) ↔ (𝑛 − 1) ∈ (0...𝑁)))
104103anbi2d 735 . . . . . . . . . . 11 (𝑖 = (𝑛 − 1) → ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁)) ↔ (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁))))
105104anbi2d 735 . . . . . . . . . 10 (𝑖 = (𝑛 − 1) → ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) ↔ (𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁)))))
106 eqeq1 2613 . . . . . . . . . . 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, < )))
107106rexbidv 3033 . . . . . . . . . 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, < )))
108105, 107imbi12d 332 . . . . . . . . 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, < ))))
109102, 108, 14vtocl 3231 . . . . . . . 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, < ))
110101, 109syldan 485 . . . . . . 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, < ))
111 eleq1 2675 . . . . . . . . . . . . . . . 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)})))
11250, 111mpbiri 246 . . . . . . . . . . . . . . 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)}))
113 elun 3714 . . . . . . . . . . . . . . . 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)}))
114102elsn 4139 . . . . . . . . . . . . . . . . 17 ((𝑛 − 1) ∈ {0} ↔ (𝑛 − 1) = 0)
115 oveq2 6534 . . . . . . . . . . . . . . . . . . 19 (𝑎 = (𝑛 − 1) → (1...𝑎) = (1...(𝑛 − 1)))
116115raleqdv 3120 . . . . . . . . . . . . . . . . . 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)))
117116elrab 3330 . . . . . . . . . . . . . . . . 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)))
118114, 117orbi12i 541 . . . . . . . . . . . . . . . 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))))
119113, 118bitri 262 . . . . . . . . . . . . . . 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))))
120112, 119sylib 206 . . . . . . . . . . . . . 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))))
121120a1i 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)))))
122 ltm1 10714 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
123 peano2rem 10199 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
124 ltnle 9968 . . . . . . . . . . . . . . . . . . 19 (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
125123, 124mpancom 699 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
126122, 125mpbid 220 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℝ → ¬ 𝑛 ≤ (𝑛 − 1))
12719, 126syl 17 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑁) → ¬ 𝑛 ≤ (𝑛 − 1))
128 breq2 4581 . . . . . . . . . . . . . . . . 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, < )))
129128notbid 306 . . . . . . . . . . . . . . . 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, < )))
130127, 129syl5ibcom 233 . . . . . . . . . . . . . . 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, < )))
131 elun2 3742 . . . . . . . . . . . . . . . . . 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)}))
132 fimaxre2 10820 . . . . . . . . . . . . . . . . . . . . 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)
13347, 33, 132mp2an 703 . . . . . . . . . . . . . . . . . . . 20 𝑥 ∈ ℝ ∀𝑦 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0)})𝑦𝑥
13447, 38, 1333pm3.2i 1231 . . . . . . . . . . . . . . . . . . 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)
135134suprubii 10847 . . . . . . . . . . . . . . . . . 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, < ))
136131, 135syl 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, < ))
137136con3i 148 . . . . . . . . . . . . . . . 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)})
138 ianor 507 . . . . . . . . . . . . . . . . 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)))
139138, 57xchnxbir 321 . . . . . . . . . . . . . . . 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)))
140137, 139sylib 206 . . . . . . . . . . . . . . 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)))
141130, 140syl6 34 . . . . . . . . . . . . . 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))))
142 pm2.63 824 . . . . . . . . . . . . . . 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)))
143142orcs 407 . . . . . . . . . . . . . 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)))
144141, 143syld 45 . . . . . . . . . . . . 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)))
145121, 144jcad 553 . . . . . . . . . . . 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))))
146 andir 907 . . . . . . . . . . . . . 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))))
147 1p0e1 10982 . . . . . . . . . . . . . . . . . 18 (1 + 0) = 1
14818zcnd 11317 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
149 ax-1cn 9850 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℂ
150 0cn 9888 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℂ
151 subadd 10135 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 0 ∈ ℂ) → ((𝑛 − 1) = 0 ↔ (1 + 0) = 𝑛))
152149, 150, 151mp3an23 1407 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℂ → ((𝑛 − 1) = 0 ↔ (1 + 0) = 𝑛))
153148, 152syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = 0 ↔ (1 + 0) = 𝑛))
154153biimpa 499 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ (1...𝑁) ∧ (𝑛 − 1) = 0) → (1 + 0) = 𝑛)
155147, 154syl5reqr 2658 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (1...𝑁) ∧ (𝑛 − 1) = 0) → 𝑛 = 1)
156 1z 11242 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℤ
157 fzsn 12211 . . . . . . . . . . . . . . . . . . . . . 22 (1 ∈ ℤ → (1...1) = {1})
158156, 157ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (1...1) = {1}
159 oveq2 6534 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1 → (1...𝑛) = (1...1))
160 sneq 4134 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1 → {𝑛} = {1})
161158, 159, 1603eqtr4a 2669 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → (1...𝑛) = {𝑛})
162161raleqdv 3120 . . . . . . . . . . . . . . . . . . 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)))
163162notbid 306 . . . . . . . . . . . . . . . . . 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)))
164163biimpd 217 . . . . . . . . . . . . . . . . 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)))
165155, 164syl 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)))
166165expimpd 626 . . . . . . . . . . . . . . 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)))
167 ralun 3756 . . . . . . . . . . . . . . . . . . . 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))
168 npcan1 10306 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
169148, 168syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) = 𝑛)
170169, 58eqeltrd 2687 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) ∈ (ℤ‘1))
171 peano2zm 11255 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℤ → (𝑛 − 1) ∈ ℤ)
172 uzid 11536 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)))
173 peano2uz 11575 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)) → ((𝑛 − 1) + 1) ∈ (ℤ‘(𝑛 − 1)))
17418, 171, 172, 1734syl 19 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) ∈ (ℤ‘(𝑛 − 1)))
175169, 174eqeltrrd 2688 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ‘(𝑛 − 1)))
176 fzsplit2 12194 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑛 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑛 ∈ (ℤ‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
177170, 175, 176syl2anc 690 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (1...𝑁) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
178169oveq1d 6541 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
179 fzsn 12211 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
18018, 179syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) → (𝑛...𝑛) = {𝑛})
181178, 180eqtrd 2643 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
182181uneq2d 3728 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (1...𝑁) → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
183177, 182eqtrd 2643 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (1...𝑁) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
184183raleqdv 3120 . . . . . . . . . . . . . . . . . . . 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)))
185167, 184syl5ibr 234 . . . . . . . . . . . . . . . . . . 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)))
186185expdimp 451 . . . . . . . . . . . . . . . . . 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)))
187186con3d 146 . . . . . . . . . . . . . . . . 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)))
188187adantrl 747 . . . . . . . . . . . . . . . 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)))
189188expimpd 626 . . . . . . . . . . . . . . 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)))
190166, 189jaod 393 . . . . . . . . . . . . . 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)))
191146, 190syl5bi 230 . . . . . . . . . . . . 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)))
192 fveq2 6087 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑛 → (𝑃𝑏) = (𝑃𝑛))
193192neeq1d 2840 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑛 → ((𝑃𝑏) ≠ 0 ↔ (𝑃𝑛) ≠ 0))
19464, 193anbi12d 742 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑛 → ((0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃𝑛) ≠ 0)))
195194ralsng 4164 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃𝑛) ≠ 0)))
196195notbid 306 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → (¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) ↔ ¬ (0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃𝑛) ≠ 0)))
197 ianor 507 . . . . . . . . . . . . . . 15 (¬ (0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃𝑛) ≠ 0) ↔ (¬ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∨ ¬ (𝑃𝑛) ≠ 0))
198 nne 2785 . . . . . . . . . . . . . . . 16 (¬ (𝑃𝑛) ≠ 0 ↔ (𝑃𝑛) = 0)
199198orbi2i 539 . . . . . . . . . . . . . . 15 ((¬ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∨ ¬ (𝑃𝑛) ≠ 0) ↔ (¬ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃𝑛) = 0))
200197, 199bitri 262 . . . . . . . . . . . . . 14 (¬ (0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃𝑛) ≠ 0) ↔ (¬ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃𝑛) = 0))
201196, 200syl6bb 274 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → (¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃𝑏) ≠ 0) ↔ (¬ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃𝑛) = 0)))
202191, 201sylibd 227 . . . . . . . . . . . 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)))
203145, 202syld 45 . . . . . . . . . . 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)))
204203ad2antlr 758 . . . . . . . . . 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)))
205 poimir.1 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹 ∈ ((𝑅t 𝐼) Cn 𝑅))
206 poimir.r . . . . . . . . . . . . . . . . . . . . . 22 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))
207 retop 22322 . . . . . . . . . . . . . . . . . . . . . . . 24 (topGen‘ran (,)) ∈ Top
208207fconst6 5992 . . . . . . . . . . . . . . . . . . . . . . 23 ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top
209 pttop 21142 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑁) ∈ Fin ∧ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top) → (∏t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Top)
21029, 208, 209mp2an 703 . . . . . . . . . . . . . . . . . . . . . 22 (∏t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Top
211206, 210eqeltri 2683 . . . . . . . . . . . . . . . . . . . . 21 𝑅 ∈ Top
212 poimir.i . . . . . . . . . . . . . . . . . . . . . 22 𝐼 = ((0[,]1) ↑𝑚 (1...𝑁))
213 reex 9883 . . . . . . . . . . . . . . . . . . . . . . 23 ℝ ∈ V
214 unitssre 12148 . . . . . . . . . . . . . . . . . . . . . . 23 (0[,]1) ⊆ ℝ
215 mapss 7763 . . . . . . . . . . . . . . . . . . . . . . 23 ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑𝑚 (1...𝑁)) ⊆ (ℝ ↑𝑚 (1...𝑁)))
216213, 214, 215mp2an 703 . . . . . . . . . . . . . . . . . . . . . 22 ((0[,]1) ↑𝑚 (1...𝑁)) ⊆ (ℝ ↑𝑚 (1...𝑁))
217212, 216eqsstri 3597 . . . . . . . . . . . . . . . . . . . . 21 𝐼 ⊆ (ℝ ↑𝑚 (1...𝑁))
218 uniretop 22323 . . . . . . . . . . . . . . . . . . . . . . . 24 ℝ = (topGen‘ran (,))
219206, 218ptuniconst 21158 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑁) ∈ Fin ∧ (topGen‘ran (,)) ∈ Top) → (ℝ ↑𝑚 (1...𝑁)) = 𝑅)
22029, 207, 219mp2an 703 . . . . . . . . . . . . . . . . . . . . . 22 (ℝ ↑𝑚 (1...𝑁)) = 𝑅
221220restuni 20723 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Top ∧ 𝐼 ⊆ (ℝ ↑𝑚 (1...𝑁))) → 𝐼 = (𝑅t 𝐼))
222211, 217, 221mp2an 703 . . . . . . . . . . . . . . . . . . . 20 𝐼 = (𝑅t 𝐼)
223222, 220cnf 20807 . . . . . . . . . . . . . . . . . . 19 (𝐹 ∈ ((𝑅t 𝐼) Cn 𝑅) → 𝐹:𝐼⟶(ℝ ↑𝑚 (1...𝑁)))
224205, 223syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹:𝐼⟶(ℝ ↑𝑚 (1...𝑁)))
225224ad2antrr 757 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝐹:𝐼⟶(ℝ ↑𝑚 (1...𝑁)))
226 simplr 787 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝑘 ∈ ℕ)
227 elfzelz 12170 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℤ)
228227zred 11316 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℝ)
229228adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
230 nnre 10876 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
231230adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
232 nnne0 10902 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ → 𝑘 ≠ 0)
233232adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ≠ 0)
234229, 231, 233redivcld 10704 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑥 / 𝑘) ∈ ℝ)
235 elfzle1 12172 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (0...𝑘) → 0 ≤ 𝑥)
236228, 235jca 552 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ (0...𝑘) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
237 nnrp 11676 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
238237rpregt0d 11712 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
239 divge0 10743 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝑥 / 𝑘))
240236, 238, 239syl2an 492 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑥 / 𝑘))
241 elfzle2 12173 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (0...𝑘) → 𝑥𝑘)
242241adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑥𝑘)
243 1red 9911 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ)
244237adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ+)
245229, 243, 244ledivmuld 11759 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑥 / 𝑘) ≤ 1 ↔ 𝑥 ≤ (𝑘 · 1)))
246 nncn 10877 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
247246mulid1d 9913 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 ∈ ℕ → (𝑘 · 1) = 𝑘)
248247breq2d 4589 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ ℕ → (𝑥 ≤ (𝑘 · 1) ↔ 𝑥𝑘))
249248adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑥 ≤ (𝑘 · 1) ↔ 𝑥𝑘))
250245, 249bitrd 266 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑥 / 𝑘) ≤ 1 ↔ 𝑥𝑘))
251242, 250mpbird 245 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑥 / 𝑘) ≤ 1)
25239, 17elicc2i 12068 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 / 𝑘) ∈ (0[,]1) ↔ ((𝑥 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝑥 / 𝑘) ∧ (𝑥 / 𝑘) ≤ 1))
253234, 240, 251, 252syl3anbrc 1238 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑥 / 𝑘) ∈ (0[,]1))
254253ancoms 467 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ (0...𝑘)) → (𝑥 / 𝑘) ∈ (0[,]1))
255 elsni 4141 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ {𝑘} → 𝑦 = 𝑘)
256255oveq2d 6542 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ {𝑘} → (𝑥 / 𝑦) = (𝑥 / 𝑘))
257256eleq1d 2671 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ {𝑘} → ((𝑥 / 𝑦) ∈ (0[,]1) ↔ (𝑥 / 𝑘) ∈ (0[,]1)))
258254, 257syl5ibrcom 235 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ ℕ ∧ 𝑥 ∈ (0...𝑘)) → (𝑦 ∈ {𝑘} → (𝑥 / 𝑦) ∈ (0[,]1)))
259258impr 646 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ ℕ ∧ (𝑥 ∈ (0...𝑘) ∧ 𝑦 ∈ {𝑘})) → (𝑥 / 𝑦) ∈ (0[,]1))
260226, 259sylan 486 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑥 ∈ (0...𝑘) ∧ 𝑦 ∈ {𝑘})) → (𝑥 / 𝑦) ∈ (0[,]1))
261 elun 3714 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ({1} ∪ {0}) ↔ (𝑦 ∈ {1} ∨ 𝑦 ∈ {0}))
262 fzofzp1 12388 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (0..^𝑘) → (𝑥 + 1) ∈ (0...𝑘))
263 elsni 4141 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ {1} → 𝑦 = 1)
264263oveq2d 6542 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ {1} → (𝑥 + 𝑦) = (𝑥 + 1))
265264eleq1d 2671 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ {1} → ((𝑥 + 𝑦) ∈ (0...𝑘) ↔ (𝑥 + 1) ∈ (0...𝑘)))
266262, 265syl5ibrcom 235 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ (0..^𝑘) → (𝑦 ∈ {1} → (𝑥 + 𝑦) ∈ (0...𝑘)))
267 elfzonn0 12337 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 ∈ (0..^𝑘) → 𝑥 ∈ ℕ0)
268267nn0cnd 11202 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 ∈ (0..^𝑘) → 𝑥 ∈ ℂ)
269268addid1d 10087 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ (0..^𝑘) → (𝑥 + 0) = 𝑥)
270 elfzofz 12311 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ (0..^𝑘) → 𝑥 ∈ (0...𝑘))
271269, 270eqeltrd 2687 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (0..^𝑘) → (𝑥 + 0) ∈ (0...𝑘))
272 elsni 4141 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ {0} → 𝑦 = 0)
273272oveq2d 6542 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ {0} → (𝑥 + 𝑦) = (𝑥 + 0))
274273eleq1d 2671 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ {0} → ((𝑥 + 𝑦) ∈ (0...𝑘) ↔ (𝑥 + 0) ∈ (0...𝑘)))
275271, 274syl5ibrcom 235 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ (0..^𝑘) → (𝑦 ∈ {0} → (𝑥 + 𝑦) ∈ (0...𝑘)))
276266, 275jaod 393 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ (0..^𝑘) → ((𝑦 ∈ {1} ∨ 𝑦 ∈ {0}) → (𝑥 + 𝑦) ∈ (0...𝑘)))
277261, 276syl5bi 230 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (0..^𝑘) → (𝑦 ∈ ({1} ∪ {0}) → (𝑥 + 𝑦) ∈ (0...𝑘)))
278277imp 443 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ (0..^𝑘) ∧ 𝑦 ∈ ({1} ∪ {0})) → (𝑥 + 𝑦) ∈ (0...𝑘))
279278adantl 480 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑥 ∈ (0..^𝑘) ∧ 𝑦 ∈ ({1} ∪ {0}))) → (𝑥 + 𝑦) ∈ (0...𝑘))
280 poimirlem31.3 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐺:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
281280ffvelrnda 6251 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
282 xp1st 7066 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺𝑘) ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(𝐺𝑘)) ∈ (ℕ0𝑚 (1...𝑁)))
283 elmapfn 7743 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st ‘(𝐺𝑘)) ∈ (ℕ0𝑚 (1...𝑁)) → (1st ‘(𝐺𝑘)) Fn (1...𝑁))
284281, 282, 2833syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐺𝑘)) Fn (1...𝑁))
285 poimirlem31.4 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘 ∈ ℕ) → ran (1st ‘(𝐺𝑘)) ⊆ (0..^𝑘))
286 df-f 5793 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st ‘(𝐺𝑘)):(1...𝑁)⟶(0..^𝑘) ↔ ((1st ‘(𝐺𝑘)) Fn (1...𝑁) ∧ ran (1st ‘(𝐺𝑘)) ⊆ (0..^𝑘)))
287284, 285, 286sylanbrc 694 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐺𝑘)):(1...𝑁)⟶(0..^𝑘))
288287adantr 479 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (1st ‘(𝐺𝑘)):(1...𝑁)⟶(0..^𝑘))
289 1ex 9891 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ V
290289fconst 5988 . . . . . . . . . . . . . . . . . . . . . . . 24 (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}):((2nd ‘(𝐺𝑘)) “ (1...𝑗))⟶{1}
29134fconst 5988 . . . . . . . . . . . . . . . . . . . . . . . 24 (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}):((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))⟶{0}
292290, 291pm3.2i 469 . . . . . . . . . . . . . . . . . . . . . . 23 ((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}):((2nd ‘(𝐺𝑘)) “ (1...𝑗))⟶{1} ∧ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}):((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))⟶{0})
293 xp2nd 7067 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐺𝑘) ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(𝐺𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
294281, 293syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑘 ∈ ℕ) → (2nd ‘(𝐺𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
295 fvex 6097 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (2nd ‘(𝐺𝑘)) ∈ V
296 f1oeq1 6024 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 = (2nd ‘(𝐺𝑘)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)))
297295, 296elab 3318 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((2nd ‘(𝐺𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
298294, 297sylib 206 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑘 ∈ ℕ) → (2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
299 dff1o3 6040 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(𝐺𝑘)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(𝐺𝑘))))
300299simprbi 478 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(𝐺𝑘)))
301 imain 5873 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Fun (2nd ‘(𝐺𝑘)) → ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∩ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))))
302298, 300, 3013syl 18 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑘 ∈ ℕ) → ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∩ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))))
303 elfznn0 12259 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℕ0)
304303nn0red 11201 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℝ)
305304ltp1d 10805 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑁) → 𝑗 < (𝑗 + 1))
306 fzdisj 12196 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
307305, 306syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (0...𝑁) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
308307imaeq2d 5371 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑁) → ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((2nd ‘(𝐺𝑘)) “ ∅))
309 ima0 5386 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((2nd ‘(𝐺𝑘)) “ ∅) = ∅
310308, 309syl6eq 2659 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0...𝑁) → ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ∅)
311302, 310sylan9req 2664 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∩ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))) = ∅)
312 fun 5964 . . . . . . . . . . . . . . . . . . . . . . 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}))
313292, 311, 312sylancr 693 . . . . . . . . . . . . . . . . . . . . . 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}))
314 imaundi 5449 . . . . . . . . . . . . . . . . . . . . . . . 24 ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∪ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)))
315 nn0p1nn 11181 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ ℕ0 → (𝑗 + 1) ∈ ℕ)
316303, 315syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ ℕ)
317 nnuz 11557 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ℕ = (ℤ‘1)
318316, 317syl6eleq 2697 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ (ℤ‘1))
319 elfzuz3 12167 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑁) → 𝑁 ∈ (ℤ𝑗))
320 fzsplit2 12194 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑗 + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ𝑗)) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
321318, 319, 320syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑁) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
322321imaeq2d 5371 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (0...𝑁) → ((2nd ‘(𝐺𝑘)) “ (1...𝑁)) = ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))))
323 f1ofo 6041 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((2nd ‘(𝐺𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(𝐺𝑘)):(1...𝑁)–onto→(1...𝑁))
324 foima 6017 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((2nd ‘(𝐺𝑘)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(𝐺𝑘)) “ (1...𝑁)) = (1...𝑁))
325298, 323, 3243syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑘 ∈ ℕ) → ((2nd ‘(𝐺𝑘)) “ (1...𝑁)) = (1...𝑁))
326322, 325sylan9req 2664 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑗 ∈ (0...𝑁) ∧ (𝜑𝑘 ∈ ℕ)) → ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (1...𝑁))
327326ancoms 467 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((2nd ‘(𝐺𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (1...𝑁))
328314, 327syl5eqr 2657 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((2nd ‘(𝐺𝑘)) “ (1...𝑗)) ∪ ((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁))) = (1...𝑁))
329328feq2d 5929 . . . . . . . . . . . . . . . . . . . . . 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})))
330313, 329mpbid 220 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))
331 fzfid 12591 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
332 inidm 3783 . . . . . . . . . . . . . . . . . . . . 21 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
333279, 288, 330, 331, 331, 332off 6787 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((1st ‘(𝐺𝑘)) ∘𝑓 + ((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝑘))
334 poimirlem31.p . . . . . . . . . . . . . . . . . . . . 21 𝑃 = ((1st ‘(𝐺𝑘)) ∘𝑓 + ((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
335334feq1i 5934 . . . . . . . . . . . . . . . . . . . 20 (𝑃:(1...𝑁)⟶(0...𝑘) ↔ ((1st ‘(𝐺𝑘)) ∘𝑓 + ((((2nd ‘(𝐺𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝑘))
336333, 335sylibr 222 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝑃:(1...𝑁)⟶(0...𝑘))
337 vex 3175 . . . . . . . . . . . . . . . . . . . . 21 𝑘 ∈ V
338337fconst 5988 . . . . . . . . . . . . . . . . . . . 20 ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘}
339338a1i 11 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘})
340260, 336, 339, 331, 331, 332off 6787 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (𝑃𝑓 / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
341212eleq2i 2679 . . . . . . . . . . . . . . . . . . 19 ((𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑𝑚 (1...𝑁)))
342 ovex 6554 . . . . . . . . . . . . . . . . . . . 20 (0[,]1) ∈ V
343 ovex 6554 . . . . . . . . . . . . . . . . . . . 20 (1...𝑁) ∈ V
344342, 343elmap 7749 . . . . . . . . . . . . . . . . . . 19 ((𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑𝑚 (1...𝑁)) ↔ (𝑃𝑓 / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
345341, 344bitri 262 . . . . . . . . . . . . . . . . . 18 ((𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (𝑃𝑓 / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
346340, 345sylibr 222 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼)
347225, 346ffvelrnd 6252 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘}))) ∈ (ℝ ↑𝑚 (1...𝑁)))
348 elmapi 7742 . . . . . . . . . . . . . . . 16 ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘}))) ∈ (ℝ ↑𝑚 (1...𝑁)) → (𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘}))):(1...𝑁)⟶ℝ)
349347, 348syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘}))):(1...𝑁)⟶ℝ)
350349ffvelrnda 6251 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∈ ℝ)
351350an32s 841 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∈ ℝ)
352 0red 9897 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → 0 ∈ ℝ)
353351, 352ltnled 10035 . . . . . . . . . . . 12 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) < 0 ↔ ¬ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
354 ltle 9977 . . . . . . . . . . . . 13 ((((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∈ ℝ ∧ 0 ∈ ℝ) → (((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) < 0 → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
355351, 39, 354sylancl 692 . . . . . . . . . . . 12 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) < 0 → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
356353, 355sylbird 248 . . . . . . . . . . 11 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (¬ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
357246, 232div0d 10651 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (0 / 𝑘) = 0)
358 oveq1 6533 . . . . . . . . . . . . . . . 16 ((𝑃𝑛) = 0 → ((𝑃𝑛) / 𝑘) = (0 / 𝑘))
359358eqeq1d 2611 . . . . . . . . . . . . . . 15 ((𝑃𝑛) = 0 → (((𝑃𝑛) / 𝑘) = 0 ↔ (0 / 𝑘) = 0))
360357, 359syl5ibrcom 235 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → ((𝑃𝑛) = 0 → ((𝑃𝑛) / 𝑘) = 0))
361360ad3antlr 762 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑃𝑛) = 0 → ((𝑃𝑛) / 𝑘) = 0))
362 ffn 5943 . . . . . . . . . . . . . . . . 17 (𝑃:(1...𝑁)⟶(0...𝑘) → 𝑃 Fn (1...𝑁))
363336, 362syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝑃 Fn (1...𝑁))
364 fnconstg 5990 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ V → ((1...𝑁) × {𝑘}) Fn (1...𝑁))
365337, 364mp1i 13 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((1...𝑁) × {𝑘}) Fn (1...𝑁))
366 eqidd 2610 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑃𝑛) = (𝑃𝑛))
367337fvconst2 6351 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {𝑘})‘𝑛) = 𝑘)
368367adantl 480 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {𝑘})‘𝑛) = 𝑘)
369363, 365, 331, 331, 332, 366, 368ofval 6781 . . . . . . . . . . . . . . 15 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = ((𝑃𝑛) / 𝑘))
370369an32s 841 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = ((𝑃𝑛) / 𝑘))
371370eqeq1d 2611 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 0 ↔ ((𝑃𝑛) / 𝑘) = 0))
372361, 371sylibrd 247 . . . . . . . . . . . 12 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑃𝑛) = 0 → ((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 0))
373 simplll 793 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → 𝜑)
374 simplr 787 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → 𝑛 ∈ (1...𝑁))
375346adantlr 746 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼)
376 ovex 6554 . . . . . . . . . . . . . 14 (𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ V
377 eleq1 2675 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑃𝑓 / ((1...𝑁) × {𝑘})) → (𝑧𝐼 ↔ (𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼))
378 fveq1 6086 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑃𝑓 / ((1...𝑁) × {𝑘})) → (𝑧𝑛) = ((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛))
379378eqeq1d 2611 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑃𝑓 / ((1...𝑁) × {𝑘})) → ((𝑧𝑛) = 0 ↔ ((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 0))
380 fveq2 6087 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑃𝑓 / ((1...𝑁) × {𝑘})) → (𝐹𝑧) = (𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘}))))
381380fveq1d 6089 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑃𝑓 / ((1...𝑁) × {𝑘})) → ((𝐹𝑧)‘𝑛) = ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛))
382381breq1d 4587 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑃𝑓 / ((1...𝑁) × {𝑘})) → (((𝐹𝑧)‘𝑛) ≤ 0 ↔ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
383379, 382imbi12d 332 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑃𝑓 / ((1...𝑁) × {𝑘})) → (((𝑧𝑛) = 0 → ((𝐹𝑧)‘𝑛) ≤ 0) ↔ (((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0)))
384377, 383imbi12d 332 . . . . . . . . . . . . . . . 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))))
385384imbi2d 328 . . . . . . . . . . . . . . 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)))))
386385imbi2d 328 . . . . . . . . . . . . . 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))))))
387 poimir.2 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧𝐼 ∧ (𝑧𝑛) = 0)) → ((𝐹𝑧)‘𝑛) ≤ 0)
3883873exp2 1276 . . . . . . . . . . . . . 14 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑧𝐼 → ((𝑧𝑛) = 0 → ((𝐹𝑧)‘𝑛) ≤ 0))))
389376, 386, 388vtocl 3231 . . . . . . . . . . . . 13 (𝜑 → (𝑛 ∈ (1...𝑁) → ((𝑃𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 → (((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))))
390373, 374, 375, 389syl3c 63 . . . . . . . . . . . 12 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (((𝑃𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
391372, 390syld 45 . . . . . . . . . . 11 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑃𝑛) = 0 → ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
392356, 391jaod 393 . . . . . . . . . 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))
393204, 392syld 45 . . . . . . . . 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))
394393reximdva 2999 . . . . . . . 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))
395394anasss 676 . . . . . . 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))
396110, 395mpd 15 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → ∃𝑗 ∈ (0...𝑁)((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0)
397 breq 4579 . . . . . . . 8 (𝑟 = ≤ → (0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ↔ 0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
398 fvex 6097 . . . . . . . . 9 ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∈ V
39934, 398brcnv 5214 . . . . . . . 8 (0 ≤ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ↔ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0)
400397, 399syl6bb 274 . . . . . . 7 (𝑟 = ≤ → (0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ↔ ((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
401400rexbidv 3033 . . . . . 6 (𝑟 = ≤ → (∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ↔ ∃𝑗 ∈ (0...𝑁)((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
402396, 401syl5ibrcom 235 . . . . 5 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (𝑟 = ≤ → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
40378, 402jaod 393 . . . 4 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → ((𝑟 = ≤ ∨ 𝑟 = ≤ ) → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
4041, 403syl5 33 . . 3 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (𝑟 ∈ { ≤ , ≤ } → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
405404exp32 628 . 2 (𝜑 → (𝑘 ∈ ℕ → (𝑛 ∈ (1...𝑁) → (𝑟 ∈ { ≤ , ≤ } → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))))
4064053imp2 1273 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 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  {cab 2595  wne 2779  wral 2895  wrex 2896  {crab 2899  Vcvv 3172  cun 3537  cin 3538  wss 3539  c0 3873  {csn 4124  {cpr 4126   cuni 4366   class class class wbr 4577   Or wor 4947   × cxp 5025  ccnv 5026  ran crn 5028  cima 5030  Fun wfun 5783   Fn wfn 5784  wf 5785  ontowfo 5787  1-1-ontowf1o 5788  cfv 5789  (class class class)co 6526  𝑓 cof 6770  1st c1st 7034  2nd c2nd 7035  𝑚 cmap 7721  Fincfn 7818  supcsup 8206  cc 9790  cr 9791  0cc0 9792  1c1 9793   + caddc 9795   · cmul 9797   < clt 9930  cle 9931  cmin 10117   / cdiv 10535  cn 10869  0cn0 11141  cz 11212  cuz 11521  +crp 11666  (,)cioo 12004  [,]cicc 12007  ...cfz 12154  ..^cfzo 12291  t crest 15852  topGenctg 15869  tcpt 15870  Topctop 20464   Cn ccn 20785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-of 6772  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fi 8177  df-sup 8208  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10536  df-nn 10870  df-n0 11142  df-z 11213  df-uz 11522  df-rp 11667  df-ioo 12008  df-icc 12011  df-fz 12155  df-fzo 12292  df-rest 15854  df-topgen 15875  df-pt 15876  df-top 20468  df-bases 20469  df-topon 20470  df-cn 20788
This theorem is referenced by:  poimirlem32  32394
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