poimirlem31 - Mathbox for Brendan Leahy

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Theorem poimirlem31 37969

Description: Lemma for poimir 37971, assigning values to the vertices of the tessellation that meet the hypotheses of both poimirlem30 37968 and poimirlem28 37966. Equation (2) of [Kulpa] p. 547. (Contributed by Brendan Leahy, 21-Aug-2020.)

**Hypotheses**
Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimir.i	⊢ 𝐼 = ((0[,]1) ↑_m (1...𝑁))
poimir.r	⊢ 𝑅 = (∏_t‘((1...𝑁) × {(topGen‘ran (,))}))
poimir.1	⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
poimir.2	⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → ((𝐹‘𝑧)‘𝑛) ≤ 0)
poimirlem31.p	⊢ 𝑃 = ((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
poimirlem31.3	⊢ (𝜑 → 𝐺:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
poimirlem31.4	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran (1^st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘))
poimirlem31.5	⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ))

**Assertion**
Ref	Expression
poimirlem31	⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ^◡ ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃 ∘_f / ((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.

Step	Hyp	Ref	Expression
1		elpri 4592	. . . 4 ⊢ (𝑟 ∈ { ≤ , ^◡ ≤ } → (𝑟 = ≤ ∨ 𝑟 = ^◡ ≤ ))
2		simprr 773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → 𝑛 ∈ (1...𝑁))
3		fz1ssfz0 13574	. . . . . . . . . 10 ⊢ (1...𝑁) ⊆ (0...𝑁)
4	3	sseli 3918	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (0...𝑁))
5	4	anim2i 618	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁)))
6		eleq1 2825	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑛 → (𝑖 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...𝑁)))
7	6	anbi2d 631	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑛 → ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁)) ↔ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁))))
8	7	anbi2d 631	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑛 → ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) ↔ (𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁)))))
9		eqeq1 2741	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑛 → (𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) ↔ 𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < )))
10	9	rexbidv 3162	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑛 → (∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) ↔ ∃𝑗 ∈ (0...𝑁)𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < )))
11	8, 10	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑖 = 𝑛 → (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < )) ↔ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 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‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ))
13	11, 12	chvarvv 1991	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ))
14		elfzle1 13478	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑁) → 1 ≤ 𝑛)
15		1re 11141	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
16		elfzelz 13475	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
17	16	zred 12630	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
18		lenlt 11221	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (1 ≤ 𝑛 ↔ ¬ 𝑛 < 1))
19	15, 17, 18	sylancr 588	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑁) → (1 ≤ 𝑛 ↔ ¬ 𝑛 < 1))
20	14, 19	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...𝑁) → ¬ 𝑛 < 1)
21		elsni 4585	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ {0} → 𝑛 = 0)
22		0lt1 11669	. . . . . . . . . . . . 13 ⊢ 0 < 1
23	21, 22	eqbrtrdi 5125	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {0} → 𝑛 < 1)
24	20, 23	nsyl 140	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...𝑁) → ¬ 𝑛 ∈ {0})
25		ltso 11223	. . . . . . . . . . . . . . 15 ⊢ < Or ℝ
26		snfi 8987	. . . . . . . . . . . . . . . . 17 ⊢ {0} ∈ Fin
27		fzfi 13931	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑁) ∈ Fin
28		rabfi 9178	. . . . . . . . . . . . . . . . . 18 ⊢ ((1...𝑁) ∈ Fin → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)} ∈ Fin)
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)} ∈ Fin
30		unfi 9102	. . . . . . . . . . . . . . . . 17 ⊢ (({0} ∈ Fin ∧ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)} ∈ Fin) → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ∈ Fin)
31	26, 29, 30	mp2an 693	. . . . . . . . . . . . . . . 16 ⊢ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ∈ Fin
32		c0ex 11135	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
33	32	snid 4607	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ {0}
34		elun1 4123	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ {0} → 0 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}))
35		ne0i 4282	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ≠ ∅)
36	33, 34, 35	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ≠ ∅
37		0re 11143	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
38		snssi 4730	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ)
39	37, 38	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ {0} ⊆ ℝ
40		ssrab2 4021	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)} ⊆ (1...𝑁)
41	16	ssriv 3926	. . . . . . . . . . . . . . . . . . 19 ⊢ (1...𝑁) ⊆ ℤ
42		zssre 12528	. . . . . . . . . . . . . . . . . . 19 ⊢ ℤ ⊆ ℝ
43	41, 42	sstri 3932	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑁) ⊆ ℝ
44	40, 43	sstri 3932	. . . . . . . . . . . . . . . . 17 ⊢ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)} ⊆ ℝ
45	39, 44	unssi 4132	. . . . . . . . . . . . . . . 16 ⊢ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ⊆ ℝ
46	31, 36, 45	3pm3.2i 1341	. . . . . . . . . . . . . . 15 ⊢ (({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ∈ Fin ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ≠ ∅ ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ⊆ ℝ)
47		fisupcl 9380	. . . . . . . . . . . . . . 15 ⊢ (( < Or ℝ ∧ (({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ∈ Fin ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ≠ ∅ ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ⊆ ℝ)) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}))
48	25, 46, 47	mp2an 693	. . . . . . . . . . . . . 14 ⊢ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)})
49		eleq1 2825	. . . . . . . . . . . . . 14 ⊢ (𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → (𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ↔ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)})))
50	48, 49	mpbiri 258	. . . . . . . . . . . . 13 ⊢ (𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → 𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}))
51		elun 4094	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ↔ (𝑛 ∈ {0} ∨ 𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}))
52	50, 51	sylib 218	. . . . . . . . . . . 12 ⊢ (𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → (𝑛 ∈ {0} ∨ 𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}))
53		oveq2 7372	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑛 → (1...𝑎) = (1...𝑛))
54	53	raleqdv 3296	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑛 → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
55	54	elrab 3635	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)} ↔ (𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
56		elfzuz 13471	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ_≥‘1))
57		eluzfz2 13483	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℤ_≥‘1) → 𝑛 ∈ (1...𝑛))
58	56, 57	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (1...𝑛))
59		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) → 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏))
60	59	ralimi 3075	. . . . . . . . . . . . . . 15 ⊢ (∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏))
61		fveq2 6838	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑛 → ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛))
62	61	breq2d 5098	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑛 → (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
63	62	rspcva 3563	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (1...𝑛) ∧ ∀𝑏 ∈ (1...𝑛)0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏)) → 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛))
64	58, 60, 63	syl2an 597	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) → 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛))
65	55, 64	sylbi 217	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)} → 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛))
66	65	orim2i 911	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ {0} ∨ 𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) → (𝑛 ∈ {0} ∨ 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
67	52, 66	syl 17	. . . . . . . . . . 11 ⊢ (𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → (𝑛 ∈ {0} ∨ 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
68		orel1 889	. . . . . . . . . . 11 ⊢ (¬ 𝑛 ∈ {0} → ((𝑛 ∈ {0} ∨ 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)) → 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
69	24, 67, 68	syl2im 40	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
70	69	reximdv 3153	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...𝑁) → (∃𝑗 ∈ (0...𝑁)𝑛 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → ∃𝑗 ∈ (0...𝑁)0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
71	13, 70	syl5 34	. . . . . . . 8 ⊢ (𝑛 ∈ (1...𝑁) → ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
72	5, 71	sylan2i 607	. . . . . . 7 ⊢ (𝑛 ∈ (1...𝑁) → ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → ∃𝑗 ∈ (0...𝑁)0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
73	2, 72	mpcom 38	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → ∃𝑗 ∈ (0...𝑁)0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛))
74		breq 5088	. . . . . . 7 ⊢ (𝑟 = ≤ → (0𝑟((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ↔ 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
75	74	rexbidv 3162	. . . . . 6 ⊢ (𝑟 = ≤ → (∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ↔ ∃𝑗 ∈ (0...𝑁)0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
76	73, 75	syl5ibrcom 247	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (𝑟 = ≤ → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
77		poimir.0	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ)
78	77	nnzd 12547	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℤ)
79		elfzm1b 13553	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
80	16, 78, 79	syl2anr 598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
81	80	biimpd 229	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ (0...(𝑁 − 1))))
82	81	ex 412	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ (0...(𝑁 − 1)))))
83	82	pm2.43d 53	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ (0...(𝑁 − 1))))
84	77	nncnd 12187	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℂ)
85		npcan1 11572	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
86	84, 85	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
87		nnm1nn0 12475	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
88	77, 87	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
89	88	nn0zd 12546	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
90		uzid 12800	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)))
91		peano2uz 12848	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
92	89, 90, 91	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
93	86, 92	eqeltrrd 2838	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
94		fzss2 13515	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑁 − 1)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
95	93, 94	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...(𝑁 − 1)) ⊆ (0...𝑁))
96	95	sseld 3921	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑛 − 1) ∈ (0...(𝑁 − 1)) → (𝑛 − 1) ∈ (0...𝑁)))
97	83, 96	syld 47	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ (0...𝑁)))
98	97	anim2d 613	. . . . . . . . 9 ⊢ (𝜑 → ((𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁))))
99	98	imp 406	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁)))
100		ovex 7397	. . . . . . . . 9 ⊢ (𝑛 − 1) ∈ V
101		eleq1 2825	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑛 − 1) → (𝑖 ∈ (0...𝑁) ↔ (𝑛 − 1) ∈ (0...𝑁)))
102	101	anbi2d 631	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑛 − 1) → ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁)) ↔ (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁))))
103	102	anbi2d 631	. . . . . . . . . 10 ⊢ (𝑖 = (𝑛 − 1) → ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) ↔ (𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁)))))
104		eqeq1 2741	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑛 − 1) → (𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) ↔ (𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < )))
105	104	rexbidv 3162	. . . . . . . . . 10 ⊢ (𝑖 = (𝑛 − 1) → (∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) ↔ ∃𝑗 ∈ (0...𝑁)(𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < )))
106	103, 105	imbi12d 344	. . . . . . . . 9 ⊢ (𝑖 = (𝑛 − 1) → (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < )) ↔ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)(𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ))))
107	100, 106, 12	vtocl 3504	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑛 − 1) ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)(𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ))
108	99, 107	syldan 592	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → ∃𝑗 ∈ (0...𝑁)(𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ))
109		eleq1 2825	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ↔ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)})))
110	48, 109	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → (𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}))
111		elun 4094	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ↔ ((𝑛 − 1) ∈ {0} ∨ (𝑛 − 1) ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}))
112	100	elsn 4583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) ∈ {0} ↔ (𝑛 − 1) = 0)
113		oveq2 7372	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (𝑛 − 1) → (1...𝑎) = (1...(𝑛 − 1)))
114	113	raleqdv 3296	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝑛 − 1) → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
115	114	elrab 3635	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)} ↔ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
116	112, 115	orbi12i 915	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 − 1) ∈ {0} ∨ (𝑛 − 1) ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ↔ ((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0))))
117	111, 116	bitri 275	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ↔ ((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0))))
118	110, 117	sylib 218	. . . . . . . . . . . . . 14 ⊢ ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → ((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0))))
119	118	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → ((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))))
120		ltm1 11994	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
121		peano2rem 11458	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
122		ltnle 11222	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
123	121, 122	mpancom 689	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
124	120, 123	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ → ¬ 𝑛 ≤ (𝑛 − 1))
125	17, 124	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...𝑁) → ¬ 𝑛 ≤ (𝑛 − 1))
126		breq2 5090	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → (𝑛 ≤ (𝑛 − 1) ↔ 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < )))
127	126	notbid 318	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → (¬ 𝑛 ≤ (𝑛 − 1) ↔ ¬ 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < )))
128	125, 127	syl5ibcom 245	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → ¬ 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < )))
129		elun2 4124	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)} → 𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}))
130		fimaxre2 12098	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ⊆ ℝ ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)})𝑦 ≤ 𝑥)
131	45, 31, 130	mp2an 693	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)})𝑦 ≤ 𝑥
132	45, 36, 131	3pm3.2i 1341	. . . . . . . . . . . . . . . . . . 19 ⊢ (({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ⊆ ℝ ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)})𝑦 ≤ 𝑥)
133	132	suprubii 12128	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}) → 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ))
134	129, 133	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)} → 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ))
135	134	con3i 154	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → ¬ 𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)})
136		ianor 984	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) ↔ (¬ 𝑛 ∈ (1...𝑁) ∨ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
137	136, 55	xchnxbir 333	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)} ↔ (¬ 𝑛 ∈ (1...𝑁) ∨ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
138	135, 137	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → (¬ 𝑛 ∈ (1...𝑁) ∨ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
139	128, 138	syl6 35	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → (¬ 𝑛 ∈ (1...𝑁) ∨ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0))))
140		pm2.63 943	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (1...𝑁) ∨ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) → ((¬ 𝑛 ∈ (1...𝑁) ∨ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) → ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
141	140	orcs 876	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → ((¬ 𝑛 ∈ (1...𝑁) ∨ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) → ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
142	139, 141	syld 47	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
143	119, 142	jcad 512	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → (((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0))) ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0))))
144		andir 1011	. . . . . . . . . . . . . 14 ⊢ ((((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0))) ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) ↔ (((𝑛 − 1) = 0 ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) ∨ (((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0))))
145	16	zcnd 12631	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
146		ax-1cn 11093	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℂ
147		0cn 11133	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℂ
148		subadd 11393	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 0 ∈ ℂ) → ((𝑛 − 1) = 0 ↔ (1 + 0) = 𝑛))
149	146, 147, 148	mp3an23 1456	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) = 0 ↔ (1 + 0) = 𝑛))
150	145, 149	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = 0 ↔ (1 + 0) = 𝑛))
151	150	biimpa 476	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ (1...𝑁) ∧ (𝑛 − 1) = 0) → (1 + 0) = 𝑛)
152		1p0e1 12297	. . . . . . . . . . . . . . . . . 18 ⊢ (1 + 0) = 1
153	151, 152	eqtr3di 2787	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (1...𝑁) ∧ (𝑛 − 1) = 0) → 𝑛 = 1)
154		1z 12554	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℤ
155		fzsn 13517	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 ∈ ℤ → (1...1) = {1})
156	154, 155	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1...1) = {1}
157		oveq2 7372	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1 → (1...𝑛) = (1...1))
158		sneq 4578	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1 → {𝑛} = {1})
159	156, 157, 158	3eqtr4a 2798	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → (1...𝑛) = {𝑛})
160	159	raleqdv 3296	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1 → (∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
161	160	notbid 318	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → (¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) ↔ ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
162	161	biimpd 229	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → (¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) → ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
163	153, 162	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ (1...𝑁) ∧ (𝑛 − 1) = 0) → (¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) → ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
164	163	expimpd 453	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) = 0 ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) → ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
165		ralun 4139	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) → ∀𝑏 ∈ ((1...(𝑛 − 1)) ∪ {𝑛})(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0))
166		npcan1 11572	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
167	145, 166	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) = 𝑛)
168	167, 56	eqeltrd 2837	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) ∈ (ℤ_≥‘1))
169		peano2zm 12567	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℤ → (𝑛 − 1) ∈ ℤ)
170		uzid 12800	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ_≥‘(𝑛 − 1)))
171		peano2uz 12848	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 − 1) ∈ (ℤ_≥‘(𝑛 − 1)) → ((𝑛 − 1) + 1) ∈ (ℤ_≥‘(𝑛 − 1)))
172	16, 169, 170, 171	4syl 19	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) ∈ (ℤ_≥‘(𝑛 − 1)))
173	167, 172	eqeltrrd 2838	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ_≥‘(𝑛 − 1)))
174		fzsplit2 13500	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑛 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
175	168, 173, 174	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (1...𝑁) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
176	167	oveq1d 7379	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
177		fzsn 13517	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
178	16, 177	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛...𝑛) = {𝑛})
179	176, 178	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
180	179	uneq2d 4109	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (1...𝑁) → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
181	175, 180	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (1...𝑁) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
182	181	raleqdv 3296	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ ((1...(𝑛 − 1)) ∪ {𝑛})(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
183	165, 182	imbitrrid 246	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → ((∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
184	183	expdimp 452	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
185	184	con3d 152	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) → (¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) → ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
186	185	adantrl 717	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0))) → (¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) → ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
187	186	expimpd 453	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (1...𝑁) → ((((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) → ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
188	164, 187	jaod 860	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → ((((𝑛 − 1) = 0 ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) ∨ (((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0))) → ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
189	144, 188	biimtrid 242	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑁) → ((((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0))) ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) → ¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)))
190		fveq2 6838	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑛 → (𝑃‘𝑏) = (𝑃‘𝑛))
191	190	neeq1d 2992	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑛 → ((𝑃‘𝑏) ≠ 0 ↔ (𝑃‘𝑛) ≠ 0))
192	62, 191	anbi12d 633	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑛 → ((0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃‘𝑛) ≠ 0)))
193	192	ralsng 4620	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃‘𝑛) ≠ 0)))
194	193	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → (¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) ↔ ¬ (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃‘𝑛) ≠ 0)))
195		ianor 984	. . . . . . . . . . . . . . 15 ⊢ (¬ (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃‘𝑛) ≠ 0) ↔ (¬ 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ ¬ (𝑃‘𝑛) ≠ 0))
196		nne 2937	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑃‘𝑛) ≠ 0 ↔ (𝑃‘𝑛) = 0)
197	196	orbi2i 913	. . . . . . . . . . . . . . 15 ⊢ ((¬ 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ ¬ (𝑃‘𝑛) ≠ 0) ↔ (¬ 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃‘𝑛) = 0))
198	195, 197	bitri 275	. . . . . . . . . . . . . 14 ⊢ (¬ (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑃‘𝑛) ≠ 0) ↔ (¬ 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃‘𝑛) = 0))
199	194, 198	bitrdi 287	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑁) → (¬ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0) ↔ (¬ 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃‘𝑛) = 0)))
200	189, 199	sylibd 239	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...𝑁) → ((((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0))) ∧ ¬ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)) → (¬ 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃‘𝑛) = 0)))
201	143, 200	syld 47	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → (¬ 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃‘𝑛) = 0)))
202	201	ad2antlr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → (¬ 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃‘𝑛) = 0)))
203		poimir.1	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
204		poimir.r	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑅 = (∏_t‘((1...𝑁) × {(topGen‘ran (,))}))
205		retop 24723	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (topGen‘ran (,)) ∈ Top
206	205	fconst6 6728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top
207		pttop 23544	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...𝑁) ∈ Fin ∧ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top) → (∏_t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Top)
208	27, 206, 207	mp2an 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∏_t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Top
209	204, 208	eqeltri 2833	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑅 ∈ Top
210		poimir.i	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐼 = ((0[,]1) ↑_m (1...𝑁))
211		reex 11126	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℝ ∈ V
212		unitssre 13449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0[,]1) ⊆ ℝ
213		mapss 8834	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁)))
214	211, 212, 213	mp2an 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁))
215	210, 214	eqsstri 3969	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))
216		uniretop 24724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℝ = ∪ (topGen‘ran (,))
217	204, 216	ptuniconst 23560	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...𝑁) ∈ Fin ∧ (topGen‘ran (,)) ∈ Top) → (ℝ ↑_m (1...𝑁)) = ∪ 𝑅)
218	27, 205, 217	mp2an 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℝ ↑_m (1...𝑁)) = ∪ 𝑅
219	218	restuni 23124	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Top ∧ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))) → 𝐼 = ∪ (𝑅 ↾_t 𝐼))
220	209, 215, 219	mp2an 693	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐼 = ∪ (𝑅 ↾_t 𝐼)
221	220, 218	cnf 23208	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅) → 𝐹:𝐼⟶(ℝ ↑_m (1...𝑁)))
222	203, 221	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹:𝐼⟶(ℝ ↑_m (1...𝑁)))
223	222	ad2antrr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝐹:𝐼⟶(ℝ ↑_m (1...𝑁)))
224		simplr 769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝑘 ∈ ℕ)
225		elfzelz 13475	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℤ)
226	225	zred 12630	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℝ)
227	226	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
228		nnre 12178	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
229	228	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
230		nnne0 12208	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
231	230	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ≠ 0)
232	227, 229, 231	redivcld 11980	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑥 / 𝑘) ∈ ℝ)
233		elfzle1 13478	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ (0...𝑘) → 0 ≤ 𝑥)
234	226, 233	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ (0...𝑘) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
235		nnrp 12951	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ⁺)
236	235	rpregt0d 12989	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
237		divge0 12022	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝑥 / 𝑘))
238	234, 236, 237	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑥 / 𝑘))
239		elfzle2 13479	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ (0...𝑘) → 𝑥 ≤ 𝑘)
240	239	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑥 ≤ 𝑘)
241		1red 11142	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ)
242	235	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ⁺)
243	227, 241, 242	ledivmuld 13036	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑥 / 𝑘) ≤ 1 ↔ 𝑥 ≤ (𝑘 · 1)))
244		nncn 12179	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
245	244	mulridd 11159	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ ℕ → (𝑘 · 1) = 𝑘)
246	245	breq2d 5098	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ ℕ → (𝑥 ≤ (𝑘 · 1) ↔ 𝑥 ≤ 𝑘))
247	246	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑥 ≤ (𝑘 · 1) ↔ 𝑥 ≤ 𝑘))
248	243, 247	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑥 / 𝑘) ≤ 1 ↔ 𝑥 ≤ 𝑘))
249	240, 248	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑥 / 𝑘) ≤ 1)
250		elicc01 13416	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 / 𝑘) ∈ (0[,]1) ↔ ((𝑥 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝑥 / 𝑘) ∧ (𝑥 / 𝑘) ≤ 1))
251	232, 238, 249, 250	syl3anbrc 1345	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑥 / 𝑘) ∈ (0[,]1))
252	251	ancoms 458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ (0...𝑘)) → (𝑥 / 𝑘) ∈ (0[,]1))
253		elsni 4585	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ {𝑘} → 𝑦 = 𝑘)
254	253	oveq2d 7380	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ {𝑘} → (𝑥 / 𝑦) = (𝑥 / 𝑘))
255	254	eleq1d 2822	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ {𝑘} → ((𝑥 / 𝑦) ∈ (0[,]1) ↔ (𝑥 / 𝑘) ∈ (0[,]1)))
256	252, 255	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℕ ∧ 𝑥 ∈ (0...𝑘)) → (𝑦 ∈ {𝑘} → (𝑥 / 𝑦) ∈ (0[,]1)))
257	256	impr 454	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℕ ∧ (𝑥 ∈ (0...𝑘) ∧ 𝑦 ∈ {𝑘})) → (𝑥 / 𝑦) ∈ (0[,]1))
258	224, 257	sylan 581	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑥 ∈ (0...𝑘) ∧ 𝑦 ∈ {𝑘})) → (𝑥 / 𝑦) ∈ (0[,]1))
259		elun 4094	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ({1} ∪ {0}) ↔ (𝑦 ∈ {1} ∨ 𝑦 ∈ {0}))
260		fzofzp1 13716	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ (0..^𝑘) → (𝑥 + 1) ∈ (0...𝑘))
261		elsni 4585	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ {1} → 𝑦 = 1)
262	261	oveq2d 7380	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ {1} → (𝑥 + 𝑦) = (𝑥 + 1))
263	262	eleq1d 2822	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ {1} → ((𝑥 + 𝑦) ∈ (0...𝑘) ↔ (𝑥 + 1) ∈ (0...𝑘)))
264	260, 263	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ (0..^𝑘) → (𝑦 ∈ {1} → (𝑥 + 𝑦) ∈ (0...𝑘)))
265		elfzonn0 13659	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 ∈ (0..^𝑘) → 𝑥 ∈ ℕ₀)
266	265	nn0cnd 12497	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 ∈ (0..^𝑘) → 𝑥 ∈ ℂ)
267	266	addridd 11343	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ (0..^𝑘) → (𝑥 + 0) = 𝑥)
268		elfzofz 13627	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ (0..^𝑘) → 𝑥 ∈ (0...𝑘))
269	267, 268	eqeltrd 2837	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ (0..^𝑘) → (𝑥 + 0) ∈ (0...𝑘))
270		elsni 4585	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ {0} → 𝑦 = 0)
271	270	oveq2d 7380	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ {0} → (𝑥 + 𝑦) = (𝑥 + 0))
272	271	eleq1d 2822	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ {0} → ((𝑥 + 𝑦) ∈ (0...𝑘) ↔ (𝑥 + 0) ∈ (0...𝑘)))
273	269, 272	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ (0..^𝑘) → (𝑦 ∈ {0} → (𝑥 + 𝑦) ∈ (0...𝑘)))
274	264, 273	jaod 860	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ (0..^𝑘) → ((𝑦 ∈ {1} ∨ 𝑦 ∈ {0}) → (𝑥 + 𝑦) ∈ (0...𝑘)))
275	259, 274	biimtrid 242	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (0..^𝑘) → (𝑦 ∈ ({1} ∪ {0}) → (𝑥 + 𝑦) ∈ (0...𝑘)))
276	275	imp 406	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ (0..^𝑘) ∧ 𝑦 ∈ ({1} ∪ {0})) → (𝑥 + 𝑦) ∈ (0...𝑘))
277	276	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑥 ∈ (0..^𝑘) ∧ 𝑦 ∈ ({1} ∪ {0}))) → (𝑥 + 𝑦) ∈ (0...𝑘))
278		poimirlem31.3	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐺:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
279	278	ffvelcdmda 7034	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
280		xp1st 7971	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺‘𝑘) ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^st ‘(𝐺‘𝑘)) ∈ (ℕ₀ ↑_m (1...𝑁)))
281		elmapfn 8809	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1^st ‘(𝐺‘𝑘)) ∈ (ℕ₀ ↑_m (1...𝑁)) → (1^st ‘(𝐺‘𝑘)) Fn (1...𝑁))
282	279, 280, 281	3syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐺‘𝑘)) Fn (1...𝑁))
283		poimirlem31.4	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran (1^st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘))
284		df-f 6500	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1^st ‘(𝐺‘𝑘)):(1...𝑁)⟶(0..^𝑘) ↔ ((1^st ‘(𝐺‘𝑘)) Fn (1...𝑁) ∧ ran (1^st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘)))
285	282, 283, 284	sylanbrc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐺‘𝑘)):(1...𝑁)⟶(0..^𝑘))
286	285	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (1^st ‘(𝐺‘𝑘)):(1...𝑁)⟶(0..^𝑘))
287		1ex 11137	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ V
288	287	fconst 6724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}):((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗))⟶{1}
289	32	fconst 6724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}):((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))⟶{0}
290	288, 289	pm3.2i 470	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}):((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗))⟶{1} ∧ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}):((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))⟶{0})
291		xp2nd 7972	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐺‘𝑘) ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^nd ‘(𝐺‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
292	279, 291	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (2^nd ‘(𝐺‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
293		fvex 6851	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (2^nd ‘(𝐺‘𝑘)) ∈ V
294		f1oeq1 6766	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 = (2^nd ‘(𝐺‘𝑘)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^nd ‘(𝐺‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)))
295	293, 294	elab 3623	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((2^nd ‘(𝐺‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^nd ‘(𝐺‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
296	292, 295	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (2^nd ‘(𝐺‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
297		dff1o3 6784	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((2^nd ‘(𝐺‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^nd ‘(𝐺‘𝑘)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ^◡(2^nd ‘(𝐺‘𝑘))))
298	297	simprbi 497	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((2^nd ‘(𝐺‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡(2^nd ‘(𝐺‘𝑘)))
299		imain 6581	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (Fun ^◡(2^nd ‘(𝐺‘𝑘)) → ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∩ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))))
300	296, 298, 299	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∩ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))))
301		elfznn0 13571	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℕ₀)
302	301	nn0red 12496	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℝ)
303	302	ltp1d 12083	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 < (𝑗 + 1))
304		fzdisj 13502	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
305	303, 304	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (0...𝑁) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
306	305	imaeq2d 6023	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0...𝑁) → ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((2^nd ‘(𝐺‘𝑘)) “ ∅))
307		ima0 6040	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((2^nd ‘(𝐺‘𝑘)) “ ∅) = ∅
308	306, 307	eqtrdi 2788	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (0...𝑁) → ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ∅)
309	300, 308	sylan9req 2793	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∩ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))) = ∅)
310		fun 6700	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}):((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗))⟶{1} ∧ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}):((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))⟶{0}) ∧ (((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∩ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))) = ∅) → ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∪ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}))
311	290, 309, 310	sylancr 588	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∪ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}))
312		imaundi 6111	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∪ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)))
313		nn0p1nn 12473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ)
314	301, 313	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ ℕ)
315		nnuz 12824	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ℕ = (ℤ_≥‘1)
316	314, 315	eleqtrdi 2847	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ (ℤ_≥‘1))
317		elfzuz3 13472	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑗))
318		fzsplit2 13500	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑗 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘𝑗)) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
319	316, 317, 318	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (0...𝑁) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
320	319	imaeq2d 6023	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (0...𝑁) → ((2^nd ‘(𝐺‘𝑘)) “ (1...𝑁)) = ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))))
321		f1ofo 6785	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((2^nd ‘(𝐺‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(𝐺‘𝑘)):(1...𝑁)–onto→(1...𝑁))
322		foima 6755	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((2^nd ‘(𝐺‘𝑘)):(1...𝑁)–onto→(1...𝑁) → ((2^nd ‘(𝐺‘𝑘)) “ (1...𝑁)) = (1...𝑁))
323	296, 321, 322	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((2^nd ‘(𝐺‘𝑘)) “ (1...𝑁)) = (1...𝑁))
324	320, 323	sylan9req 2793	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑗 ∈ (0...𝑁) ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (1...𝑁))
325	324	ancoms 458	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (1...𝑁))
326	312, 325	eqtr3id 2786	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∪ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))) = (1...𝑁))
327	326	feq2d 6650	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∪ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}) ↔ ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0})))
328	311, 327	mpbid 232	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))
329		fzfid 13932	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
330		inidm 4168	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
331	277, 286, 328, 329, 329, 330	off 7646	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝑘))
332		poimirlem31.p	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑃 = ((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
333	332	feq1i 6657	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃:(1...𝑁)⟶(0...𝑘) ↔ ((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝑘))
334	331, 333	sylibr 234	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝑃:(1...𝑁)⟶(0...𝑘))
335		vex 3434	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑘 ∈ V
336	335	fconst 6724	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘}
337	336	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘})
338	258, 334, 337, 329, 329, 330	off 7646	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (𝑃 ∘_f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
339	210	eleq2i 2829	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (𝑃 ∘_f / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑_m (1...𝑁)))
340		ovex 7397	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0[,]1) ∈ V
341		ovex 7397	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑁) ∈ V
342	340, 341	elmap 8816	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∘_f / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑_m (1...𝑁)) ↔ (𝑃 ∘_f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
343	339, 342	bitri 275	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (𝑃 ∘_f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
344	338, 343	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (𝑃 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼)
345	223, 344	ffvelcdmd 7035	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘}))) ∈ (ℝ ↑_m (1...𝑁)))
346		elmapi 8793	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘}))) ∈ (ℝ ↑_m (1...𝑁)) → (𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘}))):(1...𝑁)⟶ℝ)
347	345, 346	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘}))):(1...𝑁)⟶ℝ)
348	347	ffvelcdmda 7034	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∈ ℝ)
349	348	an32s 653	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∈ ℝ)
350		0red 11144	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → 0 ∈ ℝ)
351	349, 350	ltnled 11290	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) < 0 ↔ ¬ 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
352		ltle 11231	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∈ ℝ ∧ 0 ∈ ℝ) → (((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) < 0 → ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
353	349, 37, 352	sylancl 587	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) < 0 → ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
354	351, 353	sylbird 260	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (¬ 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) → ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
355	244, 230	div0d 11927	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (0 / 𝑘) = 0)
356		oveq1 7371	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃‘𝑛) = 0 → ((𝑃‘𝑛) / 𝑘) = (0 / 𝑘))
357	356	eqeq1d 2739	. . . . . . . . . . . . . . 15 ⊢ ((𝑃‘𝑛) = 0 → (((𝑃‘𝑛) / 𝑘) = 0 ↔ (0 / 𝑘) = 0))
358	355, 357	syl5ibrcom 247	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → ((𝑃‘𝑛) = 0 → ((𝑃‘𝑛) / 𝑘) = 0))
359	358	ad3antlr 732	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑃‘𝑛) = 0 → ((𝑃‘𝑛) / 𝑘) = 0))
360	334	ffnd 6667	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝑃 Fn (1...𝑁))
361		fnconstg 6726	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ V → ((1...𝑁) × {𝑘}) Fn (1...𝑁))
362	335, 361	mp1i 13	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((1...𝑁) × {𝑘}) Fn (1...𝑁))
363		eqidd 2738	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑃‘𝑛) = (𝑃‘𝑛))
364	335	fvconst2 7156	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {𝑘})‘𝑛) = 𝑘)
365	364	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {𝑘})‘𝑛) = 𝑘)
366	360, 362, 329, 329, 330, 363, 365	ofval 7639	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑃 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = ((𝑃‘𝑛) / 𝑘))
367	366	an32s 653	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑃 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = ((𝑃‘𝑛) / 𝑘))
368	367	eqeq1d 2739	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (((𝑃 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 0 ↔ ((𝑃‘𝑛) / 𝑘) = 0))
369	359, 368	sylibrd 259	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑃‘𝑛) = 0 → ((𝑃 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 0))
370		simplll 775	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → 𝜑)
371		simplr 769	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → 𝑛 ∈ (1...𝑁))
372	344	adantlr 716	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (𝑃 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼)
373		ovex 7397	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∘_f / ((1...𝑁) × {𝑘})) ∈ V
374		eleq1 2825	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑃 ∘_f / ((1...𝑁) × {𝑘})) → (𝑧 ∈ 𝐼 ↔ (𝑃 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼))
375		fveq1 6837	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑃 ∘_f / ((1...𝑁) × {𝑘})) → (𝑧‘𝑛) = ((𝑃 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛))
376	375	eqeq1d 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑃 ∘_f / ((1...𝑁) × {𝑘})) → ((𝑧‘𝑛) = 0 ↔ ((𝑃 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 0))
377		fveq2 6838	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝑃 ∘_f / ((1...𝑁) × {𝑘})) → (𝐹‘𝑧) = (𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘}))))
378	377	fveq1d 6840	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑃 ∘_f / ((1...𝑁) × {𝑘})) → ((𝐹‘𝑧)‘𝑛) = ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛))
379	378	breq1d 5096	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑃 ∘_f / ((1...𝑁) × {𝑘})) → (((𝐹‘𝑧)‘𝑛) ≤ 0 ↔ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
380	376, 379	imbi12d 344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑃 ∘_f / ((1...𝑁) × {𝑘})) → (((𝑧‘𝑛) = 0 → ((𝐹‘𝑧)‘𝑛) ≤ 0) ↔ (((𝑃 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0)))
381	374, 380	imbi12d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑃 ∘_f / ((1...𝑁) × {𝑘})) → ((𝑧 ∈ 𝐼 → ((𝑧‘𝑛) = 0 → ((𝐹‘𝑧)‘𝑛) ≤ 0)) ↔ ((𝑃 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 → (((𝑃 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))))
382	381	imbi2d 340	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑃 ∘_f / ((1...𝑁) × {𝑘})) → ((𝑛 ∈ (1...𝑁) → (𝑧 ∈ 𝐼 → ((𝑧‘𝑛) = 0 → ((𝐹‘𝑧)‘𝑛) ≤ 0))) ↔ (𝑛 ∈ (1...𝑁) → ((𝑃 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 → (((𝑃 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0)))))
383	382	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑃 ∘_f / ((1...𝑁) × {𝑘})) → ((𝜑 → (𝑛 ∈ (1...𝑁) → (𝑧 ∈ 𝐼 → ((𝑧‘𝑛) = 0 → ((𝐹‘𝑧)‘𝑛) ≤ 0)))) ↔ (𝜑 → (𝑛 ∈ (1...𝑁) → ((𝑃 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 → (((𝑃 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))))))
384		poimir.2	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → ((𝐹‘𝑧)‘𝑛) ≤ 0)
385	384	3exp2 1356	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑧 ∈ 𝐼 → ((𝑧‘𝑛) = 0 → ((𝐹‘𝑧)‘𝑛) ≤ 0))))
386	373, 383, 385	vtocl 3504	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → ((𝑃 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 → (((𝑃 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))))
387	370, 371, 372, 386	syl3c 66	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → (((𝑃 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 0 → ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
388	369, 387	syld 47	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑃‘𝑛) = 0 → ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
389	354, 388	jaod 860	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((¬ 0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∨ (𝑃‘𝑛) = 0) → ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
390	202, 389	syld 47	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
391	390	reximdva 3151	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑗 ∈ (0...𝑁)(𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → ∃𝑗 ∈ (0...𝑁)((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
392	391	anasss 466	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (∃𝑗 ∈ (0...𝑁)(𝑛 − 1) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ) → ∃𝑗 ∈ (0...𝑁)((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
393	108, 392	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → ∃𝑗 ∈ (0...𝑁)((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0)
394		breq 5088	. . . . . . . 8 ⊢ (𝑟 = ^◡ ≤ → (0𝑟((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ↔ 0^◡ ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
395		fvex 6851	. . . . . . . . 9 ⊢ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∈ V
396	32, 395	brcnv 5835	. . . . . . . 8 ⊢ (0^◡ ≤ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ↔ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0)
397	394, 396	bitrdi 287	. . . . . . 7 ⊢ (𝑟 = ^◡ ≤ → (0𝑟((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ↔ ((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
398	397	rexbidv 3162	. . . . . 6 ⊢ (𝑟 = ^◡ ≤ → (∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ↔ ∃𝑗 ∈ (0...𝑁)((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ≤ 0))
399	393, 398	syl5ibrcom 247	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (𝑟 = ^◡ ≤ → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
400	76, 399	jaod 860	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → ((𝑟 = ≤ ∨ 𝑟 = ^◡ ≤ ) → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
401	1, 400	syl5 34	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁))) → (𝑟 ∈ { ≤ , ^◡ ≤ } → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
402	401	exp32 420	. 2 ⊢ (𝜑 → (𝑘 ∈ ℕ → (𝑛 ∈ (1...𝑁) → (𝑟 ∈ { ≤ , ^◡ ≤ } → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))))
403	402	3imp2 1351	1 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ^◡ ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {cab 2715 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 {crab 3390 Vcvv 3430 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 ∅c0 4274 {csn 4568 {cpr 4570 ∪ cuni 4851 class class class wbr 5086 Or wor 5535 × cxp 5626 ^◡ccnv 5627 ran crn 5629 “ cima 5631 Fun wfun 6490 Fn wfn 6491 ⟶wf 6492 –onto→wfo 6494 –1-1-onto→wf1o 6495 ‘cfv 6496 (class class class)co 7364 ∘_f cof 7626 1^st c1st 7937 2^nd c2nd 7938 ↑_m cmap 8770 Fincfn 8890 supcsup 9350 ℂcc 11033 ℝcr 11034 0cc0 11035 1c1 11036 + caddc 11038 · cmul 11040 < clt 11176 ≤ cle 11177 − cmin 11374 / cdiv 11804 ℕcn 12171 ℕ₀cn0 12434 ℤcz 12521 ℤ_≥cuz 12785 ℝ⁺crp 12939 (,)cioo 13295 [,]cicc 13298 ...cfz 13458 ..^cfzo 13605 ↾_t crest 17380 topGenctg 17397 ∏_tcpt 17398 Topctop 22855 Cn ccn 23186

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5306 ax-pr 5374 ax-un 7686 ax-cnex 11091 ax-resscn 11092 ax-1cn 11093 ax-icn 11094 ax-addcl 11095 ax-addrcl 11096 ax-mulcl 11097 ax-mulrcl 11098 ax-mulcom 11099 ax-addass 11100 ax-mulass 11101 ax-distr 11102 ax-i2m1 11103 ax-1ne0 11104 ax-1rid 11105 ax-rnegex 11106 ax-rrecex 11107 ax-cnre 11108 ax-pre-lttri 11109 ax-pre-lttrn 11110 ax-pre-ltadd 11111 ax-pre-mulgt0 11112 ax-pre-sup 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5523 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5581 df-we 5583 df-xp 5634 df-rel 5635 df-cnv 5636 df-co 5637 df-dm 5638 df-rn 5639 df-res 5640 df-ima 5641 df-pred 6263 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7321 df-ov 7367 df-oprab 7368 df-mpo 7369 df-of 7628 df-om 7815 df-1st 7939 df-2nd 7940 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fi 9321 df-sup 9352 df-pnf 11178 df-mnf 11179 df-xr 11180 df-ltxr 11181 df-le 11182 df-sub 11376 df-neg 11377 df-div 11805 df-nn 12172 df-n0 12435 df-z 12522 df-uz 12786 df-rp 12940 df-ioo 13299 df-icc 13302 df-fz 13459 df-fzo 13606 df-rest 17382 df-topgen 17403 df-pt 17404 df-top 22856 df-topon 22873 df-bases 22908 df-cn 23189

This theorem is referenced by: poimirlem32 37970

W3C validator