poimirlem32 - Mathbox for Brendan Leahy

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Theorem poimirlem32 35736

Description: Lemma for poimir 35737, combining poimirlem28 35732, poimirlem30 35734, and poimirlem31 35735 to get Equation (1) 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)
poimir.3	⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → 0 ≤ ((𝐹‘𝑧)‘𝑛))

**Assertion**
Ref	Expression
poimirlem32	⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))

Distinct variable groups: 𝑧,𝑛,𝜑 𝑛,𝐹 𝑛,𝑁 𝜑,𝑧 𝑧,𝐹 𝑧,𝑁 𝑛,𝑐,𝑟,𝑣,𝑧,𝜑 𝐹,𝑐,𝑟,𝑣 𝐼,𝑐,𝑛,𝑟,𝑣,𝑧 𝑁,𝑐,𝑟,𝑣 𝑅,𝑐,𝑛,𝑟,𝑣,𝑧

Proof of Theorem poimirlem32 Dummy variables 𝑓 𝑖 𝑗 𝑘 𝑚 𝑝 𝑞 𝑠 𝑔 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		poimir.0	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑁 ∈ ℕ)
3		fvoveq1 7278	. . . . . . . . . . . . 13 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘}))) = (𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))))
4	3	fveq1d 6758	. . . . . . . . . . . 12 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏))
5	4	breq2d 5082	. . . . . . . . . . 11 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏)))
6		fveq1 6755	. . . . . . . . . . . 12 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (𝑝‘𝑏) = (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏))
7	6	neeq1d 3002	. . . . . . . . . . 11 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → ((𝑝‘𝑏) ≠ 0 ↔ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0))
8	5, 7	anbi12d 630	. . . . . . . . . 10 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → ((0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)))
9	8	ralbidv 3120	. . . . . . . . 9 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)))
10	9	rabbidv 3404	. . . . . . . 8 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} = {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)})
11	10	uneq2d 4093	. . . . . . 7 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {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‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}))
12	11	supeq1d 9135	. . . . . 6 ⊢ (𝑝 = ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → sup(({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‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
13	1	nnnn0d 12223	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
14		0elfz 13282	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ (0...𝑁))
15	13, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ (0...𝑁))
16	15	snssd 4739	. . . . . . . . 9 ⊢ (𝜑 → {0} ⊆ (0...𝑁))
17		ssrab2 4009	. . . . . . . . . . 11 ⊢ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} ⊆ (1...𝑁)
18		fz1ssfz0 13281	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ (0...𝑁)
19	17, 18	sstri 3926	. . . . . . . . . 10 ⊢ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} ⊆ (0...𝑁)
20	19	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} ⊆ (0...𝑁))
21	16, 20	unssd 4116	. . . . . . . 8 ⊢ (𝜑 → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ⊆ (0...𝑁))
22		ltso 10986	. . . . . . . . 9 ⊢ < Or ℝ
23		snfi 8788	. . . . . . . . . . 11 ⊢ {0} ∈ Fin
24		fzfi 13620	. . . . . . . . . . . 12 ⊢ (1...𝑁) ∈ Fin
25		rabfi 8973	. . . . . . . . . . . 12 ⊢ ((1...𝑁) ∈ Fin → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} ∈ Fin)
26	24, 25	ax-mp 5	. . . . . . . . . . 11 ⊢ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} ∈ Fin
27		unfi 8917	. . . . . . . . . . 11 ⊢ (({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)
28	23, 26, 27	mp2an 688	. . . . . . . . . 10 ⊢ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ∈ Fin
29		c0ex 10900	. . . . . . . . . . . 12 ⊢ 0 ∈ V
30	29	snid 4594	. . . . . . . . . . 11 ⊢ 0 ∈ {0}
31		elun1 4106	. . . . . . . . . . 11 ⊢ (0 ∈ {0} → 0 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}))
32		ne0i 4265	. . . . . . . . . . 11 ⊢ (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)}) ≠ ∅)
33	30, 31, 32	mp2b 10	. . . . . . . . . 10 ⊢ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ≠ ∅
34		0red 10909	. . . . . . . . . . . . 13 ⊢ ((𝜑 → 𝑁 ∈ ℕ) → 0 ∈ ℝ)
35	34	snssd 4739	. . . . . . . . . . . 12 ⊢ ((𝜑 → 𝑁 ∈ ℕ) → {0} ⊆ ℝ)
36	1, 35	ax-mp 5	. . . . . . . . . . 11 ⊢ {0} ⊆ ℝ
37		elfzelz 13185	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
38	37	ssriv 3921	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ⊆ ℤ
39		zssre 12256	. . . . . . . . . . . . 13 ⊢ ℤ ⊆ ℝ
40	38, 39	sstri 3926	. . . . . . . . . . . 12 ⊢ (1...𝑁) ⊆ ℝ
41	17, 40	sstri 3926	. . . . . . . . . . 11 ⊢ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} ⊆ ℝ
42	36, 41	unssi 4115	. . . . . . . . . 10 ⊢ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ⊆ ℝ
43	28, 33, 42	3pm3.2i 1337	. . . . . . . . 9 ⊢ (({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)
44		fisupcl 9158	. . . . . . . . 9 ⊢ (( < 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)}))
45	22, 43, 44	mp2an 688	. . . . . . . 8 ⊢ 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)})
46		ssel 3910	. . . . . . . 8 ⊢ (({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ⊆ (0...𝑁) → (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)}) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ∈ (0...𝑁)))
47	21, 45, 46	mpisyl 21	. . . . . . 7 ⊢ (𝜑 → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ∈ (0...𝑁))
48	47	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑝:(1...𝑁)⟶(0...𝑘)) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ∈ (0...𝑁))
49		elfznn 13214	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
50		nngt0 11934	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 < 𝑛)
51	50	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) → 0 < 𝑛)
52		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → (𝑝‘𝑏) ≠ 0)
53	52	ralimi 3086	. . . . . . . . . . . . 13 ⊢ (∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑠)(𝑝‘𝑏) ≠ 0)
54		elfznn 13214	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (1...𝑁) → 𝑠 ∈ ℕ)
55		nnre 11910	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
56		nnre 11910	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℝ)
57		lenlt 10984	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (𝑛 ≤ 𝑠 ↔ ¬ 𝑠 < 𝑛))
58	55, 56, 57	syl2an 595	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑛 ≤ 𝑠 ↔ ¬ 𝑠 < 𝑛))
59		elfz1b 13254	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (1...𝑠) ↔ (𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ ∧ 𝑛 ≤ 𝑠))
60	59	biimpri 227	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ ∧ 𝑛 ≤ 𝑠) → 𝑛 ∈ (1...𝑠))
61	60	3expia 1119	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑛 ≤ 𝑠 → 𝑛 ∈ (1...𝑠)))
62	58, 61	sylbird 259	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (¬ 𝑠 < 𝑛 → 𝑛 ∈ (1...𝑠)))
63		fveq2 6756	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = 𝑛 → (𝑝‘𝑏) = (𝑝‘𝑛))
64	63	eqeq1d 2740	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑛 → ((𝑝‘𝑏) = 0 ↔ (𝑝‘𝑛) = 0))
65	64	rspcev 3552	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ (1...𝑠) ∧ (𝑝‘𝑛) = 0) → ∃𝑏 ∈ (1...𝑠)(𝑝‘𝑏) = 0)
66	65	expcom 413	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝‘𝑛) = 0 → (𝑛 ∈ (1...𝑠) → ∃𝑏 ∈ (1...𝑠)(𝑝‘𝑏) = 0))
67	62, 66	sylan9 507	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ) ∧ (𝑝‘𝑛) = 0) → (¬ 𝑠 < 𝑛 → ∃𝑏 ∈ (1...𝑠)(𝑝‘𝑏) = 0))
68	67	an32s 648	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) ∧ 𝑠 ∈ ℕ) → (¬ 𝑠 < 𝑛 → ∃𝑏 ∈ (1...𝑠)(𝑝‘𝑏) = 0))
69		nne 2946	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (𝑝‘𝑏) ≠ 0 ↔ (𝑝‘𝑏) = 0)
70	69	rexbii 3177	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑏 ∈ (1...𝑠) ¬ (𝑝‘𝑏) ≠ 0 ↔ ∃𝑏 ∈ (1...𝑠)(𝑝‘𝑏) = 0)
71		rexnal 3165	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑏 ∈ (1...𝑠) ¬ (𝑝‘𝑏) ≠ 0 ↔ ¬ ∀𝑏 ∈ (1...𝑠)(𝑝‘𝑏) ≠ 0)
72	70, 71	bitr3i 276	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑏 ∈ (1...𝑠)(𝑝‘𝑏) = 0 ↔ ¬ ∀𝑏 ∈ (1...𝑠)(𝑝‘𝑏) ≠ 0)
73	68, 72	syl6ib 250	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) ∧ 𝑠 ∈ ℕ) → (¬ 𝑠 < 𝑛 → ¬ ∀𝑏 ∈ (1...𝑠)(𝑝‘𝑏) ≠ 0))
74	73	con4d 115	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) ∧ 𝑠 ∈ ℕ) → (∀𝑏 ∈ (1...𝑠)(𝑝‘𝑏) ≠ 0 → 𝑠 < 𝑛))
75	54, 74	sylan2 592	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) ∧ 𝑠 ∈ (1...𝑁)) → (∀𝑏 ∈ (1...𝑠)(𝑝‘𝑏) ≠ 0 → 𝑠 < 𝑛))
76	53, 75	syl5 34	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) ∧ 𝑠 ∈ (1...𝑁)) → (∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → 𝑠 < 𝑛))
77	76	ralrimiva 3107	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) → ∀𝑠 ∈ (1...𝑁)(∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → 𝑠 < 𝑛))
78		ralunb 4121	. . . . . . . . . . . 12 ⊢ (∀𝑠 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})𝑠 < 𝑛 ↔ (∀𝑠 ∈ {0}𝑠 < 𝑛 ∧ ∀𝑠 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}𝑠 < 𝑛))
79		breq1 5073	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 0 → (𝑠 < 𝑛 ↔ 0 < 𝑛))
80	29, 79	ralsn 4614	. . . . . . . . . . . . 13 ⊢ (∀𝑠 ∈ {0}𝑠 < 𝑛 ↔ 0 < 𝑛)
81		oveq2 7263	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑠 → (1...𝑎) = (1...𝑠))
82	81	raleqdv 3339	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑠 → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)))
83	82	ralrab 3623	. . . . . . . . . . . . 13 ⊢ (∀𝑠 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}𝑠 < 𝑛 ↔ ∀𝑠 ∈ (1...𝑁)(∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → 𝑠 < 𝑛))
84	80, 83	anbi12i 626	. . . . . . . . . . . 12 ⊢ ((∀𝑠 ∈ {0}𝑠 < 𝑛 ∧ ∀𝑠 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}𝑠 < 𝑛) ↔ (0 < 𝑛 ∧ ∀𝑠 ∈ (1...𝑁)(∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → 𝑠 < 𝑛)))
85	78, 84	bitri 274	. . . . . . . . . . 11 ⊢ (∀𝑠 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})𝑠 < 𝑛 ↔ (0 < 𝑛 ∧ ∀𝑠 ∈ (1...𝑁)(∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → 𝑠 < 𝑛)))
86	51, 77, 85	sylanbrc 582	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) → ∀𝑠 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})𝑠 < 𝑛)
87		breq1 5073	. . . . . . . . . . 11 ⊢ (𝑠 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) → (𝑠 < 𝑛 ↔ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) < 𝑛))
88	87	rspcva 3550	. . . . . . . . . 10 ⊢ ((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)}) ∧ ∀𝑠 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})𝑠 < 𝑛) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) < 𝑛)
89	45, 86, 88	sylancr 586	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) < 𝑛)
90	49, 89	sylan 579	. . . . . . . 8 ⊢ ((𝑛 ∈ (1...𝑁) ∧ (𝑝‘𝑛) = 0) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) < 𝑛)
91	90	3adant2 1129	. . . . . . 7 ⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 0) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) < 𝑛)
92	91	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 0)) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) < 𝑛)
93	37	zred 12355	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
94	93	3ad2ant1 1131	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘) → 𝑛 ∈ ℝ)
95	94	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → 𝑛 ∈ ℝ)
96		simpr1 1192	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → 𝑛 ∈ (1...𝑁))
97		simpll 763	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → 𝜑)
98		simplr 765	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → 𝑘 ∈ ℕ)
99		elfzelz 13185	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ (0...𝑘) → 𝑖 ∈ ℤ)
100	99	zred 12355	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (0...𝑘) → 𝑖 ∈ ℝ)
101		nndivre 11944	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ)
102	100, 101	sylan 579	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ)
103		elfzle1 13188	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ (0...𝑘) → 0 ≤ 𝑖)
104	100, 103	jca 511	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (0...𝑘) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
105		nnrp 12670	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ⁺)
106	105	rpregt0d 12707	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
107		divge0 11774	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑖 ∈ ℝ ∧ 0 ≤ 𝑖) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝑖 / 𝑘))
108	104, 106, 107	syl2an 595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑖 / 𝑘))
109		elfzle2 13189	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ (0...𝑘) → 𝑖 ≤ 𝑘)
110	109	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑖 ≤ 𝑘)
111	100	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑖 ∈ ℝ)
112		1red 10907	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ)
113	105	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ⁺)
114	111, 112, 113	ledivmuld 12754	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖 ≤ (𝑘 · 1)))
115		nncn 11911	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
116	115	mulid1d 10923	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ ℕ → (𝑘 · 1) = 𝑘)
117	116	breq2d 5082	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ ℕ → (𝑖 ≤ (𝑘 · 1) ↔ 𝑖 ≤ 𝑘))
118	117	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 ≤ (𝑘 · 1) ↔ 𝑖 ≤ 𝑘))
119	114, 118	bitrd 278	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖 ≤ 𝑘))
120	110, 119	mpbird 256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ≤ 1)
121		elicc01 13127	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 / 𝑘) ∈ (0[,]1) ↔ ((𝑖 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝑖 / 𝑘) ∧ (𝑖 / 𝑘) ≤ 1))
122	102, 108, 120, 121	syl3anbrc 1341	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ (0[,]1))
123	122	ancoms 458	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑘)) → (𝑖 / 𝑘) ∈ (0[,]1))
124		elsni 4575	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ {𝑘} → 𝑗 = 𝑘)
125	124	oveq2d 7271	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ {𝑘} → (𝑖 / 𝑗) = (𝑖 / 𝑘))
126	125	eleq1d 2823	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ {𝑘} → ((𝑖 / 𝑗) ∈ (0[,]1) ↔ (𝑖 / 𝑘) ∈ (0[,]1)))
127	123, 126	syl5ibrcom 246	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑘)) → (𝑗 ∈ {𝑘} → (𝑖 / 𝑗) ∈ (0[,]1)))
128	127	impr 454	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℕ ∧ (𝑖 ∈ (0...𝑘) ∧ 𝑗 ∈ {𝑘})) → (𝑖 / 𝑗) ∈ (0[,]1))
129	98, 128	sylan 579	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) ∧ (𝑖 ∈ (0...𝑘) ∧ 𝑗 ∈ {𝑘})) → (𝑖 / 𝑗) ∈ (0[,]1))
130		simprr 769	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → 𝑝:(1...𝑁)⟶(0...𝑘))
131		vex 3426	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑘 ∈ V
132	131	fconst 6644	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘}
133	132	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘})
134		fzfid 13621	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → (1...𝑁) ∈ Fin)
135		inidm 4149	. . . . . . . . . . . . . . . 16 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
136	129, 130, 133, 134, 134, 135	off 7529	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → (𝑝 ∘_f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
137		poimir.i	. . . . . . . . . . . . . . . . 17 ⊢ 𝐼 = ((0[,]1) ↑_m (1...𝑁))
138	137	eleq2i 2830	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑_m (1...𝑁)))
139		ovex 7288	. . . . . . . . . . . . . . . . 17 ⊢ (0[,]1) ∈ V
140		ovex 7288	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑁) ∈ V
141	139, 140	elmap 8617	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑_m (1...𝑁)) ↔ (𝑝 ∘_f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
142	138, 141	bitri 274	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (𝑝 ∘_f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
143	136, 142	sylibr 233	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → (𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼)
144	143	3adantr3 1169	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → (𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼)
145		3anass 1093	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)))
146		ancom 460	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ (1...𝑁) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) ↔ ((𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘) ∧ 𝑛 ∈ (1...𝑁)))
147	145, 146	bitri 274	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘) ↔ ((𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘) ∧ 𝑛 ∈ (1...𝑁)))
148		ffn 6584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝:(1...𝑁)⟶(0...𝑘) → 𝑝 Fn (1...𝑁))
149	148	ad2antrl 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → 𝑝 Fn (1...𝑁))
150		fnconstg 6646	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ V → ((1...𝑁) × {𝑘}) Fn (1...𝑁))
151	131, 150	mp1i 13	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ((1...𝑁) × {𝑘}) Fn (1...𝑁))
152		fzfid 13621	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → (1...𝑁) ∈ Fin)
153		simplrr 774	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑝‘𝑛) = 𝑘)
154	131	fvconst2 7061	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {𝑘})‘𝑛) = 𝑘)
155	154	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {𝑘})‘𝑛) = 𝑘)
156	149, 151, 152, 152, 135, 153, 155	ofval 7522	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑝 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = (𝑘 / 𝑘))
157	156	anasss 466	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ((𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘) ∧ 𝑛 ∈ (1...𝑁))) → ((𝑝 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = (𝑘 / 𝑘))
158	147, 157	sylan2b 593	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ((𝑝 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = (𝑘 / 𝑘))
159		nnne0 11937	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
160	115, 159	dividd 11679	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝑘 / 𝑘) = 1)
161	160	ad2antlr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → (𝑘 / 𝑘) = 1)
162	158, 161	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ((𝑝 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 1)
163		ovex 7288	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ V
164		eleq1 2826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑝 ∘_f / ((1...𝑁) × {𝑘})) → (𝑧 ∈ 𝐼 ↔ (𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼))
165		fveq1 6755	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑝 ∘_f / ((1...𝑁) × {𝑘})) → (𝑧‘𝑛) = ((𝑝 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛))
166	165	eqeq1d 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑝 ∘_f / ((1...𝑁) × {𝑘})) → ((𝑧‘𝑛) = 1 ↔ ((𝑝 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 1))
167	164, 166	3anbi23d 1437	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑝 ∘_f / ((1...𝑁) × {𝑘})) → ((𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ∧ ((𝑝 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 1)))
168	167	anbi2d 628	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑝 ∘_f / ((1...𝑁) × {𝑘})) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) ↔ (𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ∧ ((𝑝 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 1))))
169		fveq2 6756	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑝 ∘_f / ((1...𝑁) × {𝑘})) → (𝐹‘𝑧) = (𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘}))))
170	169	fveq1d 6758	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑝 ∘_f / ((1...𝑁) × {𝑘})) → ((𝐹‘𝑧)‘𝑛) = ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛))
171	170	breq2d 5082	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑝 ∘_f / ((1...𝑁) × {𝑘})) → (0 ≤ ((𝐹‘𝑧)‘𝑛) ↔ 0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
172	168, 171	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑝 ∘_f / ((1...𝑁) × {𝑘})) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → 0 ≤ ((𝐹‘𝑧)‘𝑛)) ↔ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ∧ ((𝑝 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 1)) → 0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛))))
173		poimir.3	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → 0 ≤ ((𝐹‘𝑧)‘𝑛))
174	163, 172, 173	vtocl 3488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ∧ ((𝑝 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 1)) → 0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛))
175	97, 96, 144, 162, 174	syl13anc 1370	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → 0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛))
176		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
177		simp3 1136	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘) → (𝑝‘𝑛) = 𝑘)
178		neeq1 3005	. . . . . . . . . . . . . . 15 ⊢ ((𝑝‘𝑛) = 𝑘 → ((𝑝‘𝑛) ≠ 0 ↔ 𝑘 ≠ 0))
179	159, 178	syl5ibrcom 246	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → ((𝑝‘𝑛) = 𝑘 → (𝑝‘𝑛) ≠ 0))
180	179	imp 406	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ∧ (𝑝‘𝑛) = 𝑘) → (𝑝‘𝑛) ≠ 0)
181	176, 177, 180	syl2an 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → (𝑝‘𝑛) ≠ 0)
182		vex 3426	. . . . . . . . . . . . 13 ⊢ 𝑛 ∈ V
183		fveq2 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑛 → ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛))
184	183	breq2d 5082	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑛 → (0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
185	63	neeq1d 3002	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑛 → ((𝑝‘𝑏) ≠ 0 ↔ (𝑝‘𝑛) ≠ 0))
186	184, 185	anbi12d 630	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑛 → ((0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑝‘𝑛) ≠ 0)))
187	182, 186	ralsn 4614	. . . . . . . . . . . 12 ⊢ (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑝‘𝑛) ≠ 0))
188	175, 181, 187	sylanbrc 582	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0))
189	37	zcnd 12356	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
190		1cnd 10901	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → 1 ∈ ℂ)
191	189, 190	subeq0ad 11272	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
192	191	biimpcd 248	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) = 0 → (𝑛 ∈ (1...𝑁) → 𝑛 = 1))
193		1z 12280	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℤ
194		fzsn 13227	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 ∈ ℤ → (1...1) = {1})
195	193, 194	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...1) = {1}
196		oveq2 7263	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → (1...𝑛) = (1...1))
197		sneq 4568	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → {𝑛} = {1})
198	195, 196, 197	3eqtr4a 2805	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1 → (1...𝑛) = {𝑛})
199	198	raleqdv 3339	. . . . . . . . . . . . . . . . . 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)))
200	199	biimprd 247	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → (∀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)))
201	192, 200	syl6 35	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 − 1) = 0 → (𝑛 ∈ (1...𝑁) → (∀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))))
202		ralun 4122	. . . . . . . . . . . . . . . . . . . 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))
203		npcan1 11330	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
204	189, 203	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) = 𝑛)
205		elfzuz 13181	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ_≥‘1))
206	204, 205	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) ∈ (ℤ_≥‘1))
207		peano2zm 12293	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℤ → (𝑛 − 1) ∈ ℤ)
208		uzid 12526	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ_≥‘(𝑛 − 1)))
209		peano2uz 12570	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 − 1) ∈ (ℤ_≥‘(𝑛 − 1)) → ((𝑛 − 1) + 1) ∈ (ℤ_≥‘(𝑛 − 1)))
210	37, 207, 208, 209	4syl 19	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) ∈ (ℤ_≥‘(𝑛 − 1)))
211	204, 210	eqeltrrd 2840	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ_≥‘(𝑛 − 1)))
212		fzsplit2 13210	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑛 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
213	206, 211, 212	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (1...𝑁) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
214	204	oveq1d 7270	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
215		fzsn 13227	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
216	37, 215	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛...𝑛) = {𝑛})
217	214, 216	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
218	217	uneq2d 4093	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (1...𝑁) → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
219	213, 218	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (1...𝑁) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
220	219	raleqdv 3339	. . . . . . . . . . . . . . . . . . . 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)))
221	202, 220	syl5ibr 245	. . . . . . . . . . . . . . . . . . 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)))
222	221	expd 415	. . . . . . . . . . . . . . . . . 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))))
223	222	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → (𝑛 ∈ (1...𝑁) → (∀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))))
224	223	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)) → (𝑛 ∈ (1...𝑁) → (∀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))))
225	201, 224	jaoi 853	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0))) → (𝑛 ∈ (1...𝑁) → (∀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))))
226	225	imdistand 570	. . . . . . . . . . . . . 14 ⊢ (((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0))) → ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ {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))))
227	226	com12 32	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ {𝑛} (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))) → (𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0))))
228		elun 4079	. . . . . . . . . . . . . 14 ⊢ ((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)}))
229		ovex 7288	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 − 1) ∈ V
230	229	elsn 4573	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 − 1) ∈ {0} ↔ (𝑛 − 1) = 0)
231		oveq2 7263	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝑛 − 1) → (1...𝑎) = (1...(𝑛 − 1)))
232	231	raleqdv 3339	. . . . . . . . . . . . . . . 16 ⊢ (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)))
233	232	elrab 3617	. . . . . . . . . . . . . . 15 ⊢ ((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)))
234	230, 233	orbi12i 911	. . . . . . . . . . . . . 14 ⊢ (((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))))
235	228, 234	bitri 274	. . . . . . . . . . . . 13 ⊢ ((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))))
236		oveq2 7263	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑛 → (1...𝑎) = (1...𝑛))
237	236	raleqdv 3339	. . . . . . . . . . . . . 14 ⊢ (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)))
238	237	elrab 3617	. . . . . . . . . . . . 13 ⊢ (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)))
239	227, 235, 238	3imtr4g 295	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)) → ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) → 𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}))
240		elun2 4107	. . . . . . . . . . . 12 ⊢ (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)}))
241	239, 240	syl6 35	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)) → ((𝑛 − 1) ∈ ({0} ∪ {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)})))
242	96, 188, 241	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ((𝑛 − 1) ∈ ({0} ∪ {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)})))
243		fimaxre2 11850	. . . . . . . . . . . . 13 ⊢ ((({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) → ∃𝑖 ∈ ℝ ∀𝑗 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})𝑗 ≤ 𝑖)
244	42, 28, 243	mp2an 688	. . . . . . . . . . . 12 ⊢ ∃𝑖 ∈ ℝ ∀𝑗 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})𝑗 ≤ 𝑖
245	42, 33, 244	3pm3.2i 1337	. . . . . . . . . . 11 ⊢ (({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)}) ≠ ∅ ∧ ∃𝑖 ∈ ℝ ∀𝑗 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})𝑗 ≤ 𝑖)
246	245	suprubii 11880	. . . . . . . . . 10 ⊢ (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, < ))
247	242, 246	syl6 35	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ((𝑛 − 1) ∈ ({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, < )))
248		ltm1 11747	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
249		peano2rem 11218	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
250	42, 45	sselii 3914	. . . . . . . . . . . 12 ⊢ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ∈ ℝ
251		ltletr 10997	. . . . . . . . . . . 12 ⊢ (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ∈ ℝ) → (((𝑛 − 1) < 𝑛 ∧ 𝑛 ≤ 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, < )))
252	250, 251	mp3an3 1448	. . . . . . . . . . 11 ⊢ (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → (((𝑛 − 1) < 𝑛 ∧ 𝑛 ≤ 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, < )))
253	249, 252	mpancom 684	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ → (((𝑛 − 1) < 𝑛 ∧ 𝑛 ≤ 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, < )))
254	248, 253	mpand 691	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ → (𝑛 ≤ 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, < )))
255	95, 247, 254	sylsyld 61	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) → (𝑛 − 1) < sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < )))
256	250	ltnri 11014	. . . . . . . . . 10 ⊢ ¬ sup(({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, < )
257		breq1 5073	. . . . . . . . . 10 ⊢ (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, < ) < 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, < )))
258	256, 257	mtbii 325	. . . . . . . . 9 ⊢ (sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) = (𝑛 − 1) → ¬ (𝑛 − 1) < sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ))
259	258	necon2ai 2972	. . . . . . . 8 ⊢ ((𝑛 − 1) < sup(({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, < ) ≠ (𝑛 − 1))
260	255, 259	syl6 35	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ((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, < ) ≠ (𝑛 − 1)))
261		eleq1 2826	. . . . . . . . 9 ⊢ (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, < ) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ↔ (𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})))
262	45, 261	mpbii 232	. . . . . . . 8 ⊢ (sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) = (𝑛 − 1) → (𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}))
263	262	necon3bi 2969	. . . . . . 7 ⊢ (¬ (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, < ) ≠ (𝑛 − 1))
264	260, 263	pm2.61d1 180	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ≠ (𝑛 − 1))
265	2, 12, 48, 92, 264, 176	poimirlem28 35732	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∃𝑠 ∈ (((0..^𝑘) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
266		nn0ex 12169	. . . . . . . . . . . 12 ⊢ ℕ₀ ∈ V
267		fzo0ssnn0 13396	. . . . . . . . . . . 12 ⊢ (0..^𝑘) ⊆ ℕ₀
268		mapss 8635	. . . . . . . . . . . 12 ⊢ ((ℕ₀ ∈ V ∧ (0..^𝑘) ⊆ ℕ₀) → ((0..^𝑘) ↑_m (1...𝑁)) ⊆ (ℕ₀ ↑_m (1...𝑁)))
269	266, 267, 268	mp2an 688	. . . . . . . . . . 11 ⊢ ((0..^𝑘) ↑_m (1...𝑁)) ⊆ (ℕ₀ ↑_m (1...𝑁))
270		xpss1 5599	. . . . . . . . . . 11 ⊢ (((0..^𝑘) ↑_m (1...𝑁)) ⊆ (ℕ₀ ↑_m (1...𝑁)) → (((0..^𝑘) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ⊆ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
271	269, 270	ax-mp 5	. . . . . . . . . 10 ⊢ (((0..^𝑘) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ⊆ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
272	271	sseli 3913	. . . . . . . . 9 ⊢ (𝑠 ∈ (((0..^𝑘) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → 𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
273		xp1st 7836	. . . . . . . . . 10 ⊢ (𝑠 ∈ (((0..^𝑘) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^st ‘𝑠) ∈ ((0..^𝑘) ↑_m (1...𝑁)))
274		elmapi 8595	. . . . . . . . . 10 ⊢ ((1^st ‘𝑠) ∈ ((0..^𝑘) ↑_m (1...𝑁)) → (1^st ‘𝑠):(1...𝑁)⟶(0..^𝑘))
275		frn 6591	. . . . . . . . . 10 ⊢ ((1^st ‘𝑠):(1...𝑁)⟶(0..^𝑘) → ran (1^st ‘𝑠) ⊆ (0..^𝑘))
276	273, 274, 275	3syl 18	. . . . . . . . 9 ⊢ (𝑠 ∈ (((0..^𝑘) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ran (1^st ‘𝑠) ⊆ (0..^𝑘))
277	272, 276	jca 511	. . . . . . . 8 ⊢ (𝑠 ∈ (((0..^𝑘) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ran (1^st ‘𝑠) ⊆ (0..^𝑘)))
278	277	anim1i 614	. . . . . . 7 ⊢ ((𝑠 ∈ (((0..^𝑘) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → ((𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ran (1^st ‘𝑠) ⊆ (0..^𝑘)) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
279		anass 468	. . . . . . 7 ⊢ (((𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ran (1^st ‘𝑠) ⊆ (0..^𝑘)) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) ↔ (𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))))
280	278, 279	sylib 217	. . . . . 6 ⊢ ((𝑠 ∈ (((0..^𝑘) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → (𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))))
281	280	reximi2 3171	. . . . 5 ⊢ (∃𝑠 ∈ (((0..^𝑘) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) → ∃𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
282	265, 281	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∃𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
283	282	ralrimiva 3107	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ∃𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
284		nnex 11909	. . . 4 ⊢ ℕ ∈ V
285	140, 266	ixpconst 8653	. . . . . . 7 ⊢ X𝑛 ∈ (1...𝑁)ℕ₀ = (ℕ₀ ↑_m (1...𝑁))
286		omelon 9334	. . . . . . . . . 10 ⊢ ω ∈ On
287		nn0ennn 13627	. . . . . . . . . . 11 ⊢ ℕ₀ ≈ ℕ
288		nnenom 13628	. . . . . . . . . . 11 ⊢ ℕ ≈ ω
289	287, 288	entr2i 8750	. . . . . . . . . 10 ⊢ ω ≈ ℕ₀
290		isnumi 9635	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ ω ≈ ℕ₀) → ℕ₀ ∈ dom card)
291	286, 289, 290	mp2an 688	. . . . . . . . 9 ⊢ ℕ₀ ∈ dom card
292	291	rgenw 3075	. . . . . . . 8 ⊢ ∀𝑛 ∈ (1...𝑁)ℕ₀ ∈ dom card
293		finixpnum 35689	. . . . . . . 8 ⊢ (((1...𝑁) ∈ Fin ∧ ∀𝑛 ∈ (1...𝑁)ℕ₀ ∈ dom card) → X𝑛 ∈ (1...𝑁)ℕ₀ ∈ dom card)
294	24, 292, 293	mp2an 688	. . . . . . 7 ⊢ X𝑛 ∈ (1...𝑁)ℕ₀ ∈ dom card
295	285, 294	eqeltrri 2836	. . . . . 6 ⊢ (ℕ₀ ↑_m (1...𝑁)) ∈ dom card
296	140, 140	mapval 8585	. . . . . . . . 9 ⊢ ((1...𝑁) ↑_m (1...𝑁)) = {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)}
297		mapfi 9045	. . . . . . . . . 10 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ↑_m (1...𝑁)) ∈ Fin)
298	24, 24, 297	mp2an 688	. . . . . . . . 9 ⊢ ((1...𝑁) ↑_m (1...𝑁)) ∈ Fin
299	296, 298	eqeltrri 2836	. . . . . . . 8 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} ∈ Fin
300		f1of 6700	. . . . . . . . 9 ⊢ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁))
301	300	ss2abi 3996	. . . . . . . 8 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)}
302		ssfi 8918	. . . . . . . 8 ⊢ (({𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)}) → {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin)
303	299, 301, 302	mp2an 688	. . . . . . 7 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin
304		finnum 9637	. . . . . . 7 ⊢ ({𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin → {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ dom card)
305	303, 304	ax-mp 5	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ dom card
306		xpnum 9640	. . . . . 6 ⊢ (((ℕ₀ ↑_m (1...𝑁)) ∈ dom card ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ dom card) → ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ dom card)
307	295, 305, 306	mp2an 688	. . . . 5 ⊢ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ dom card
308		ssrab2 4009	. . . . . . . 8 ⊢ {𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
309	308	rgenw 3075	. . . . . . 7 ⊢ ∀𝑘 ∈ ℕ {𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
310		ss2iun 4939	. . . . . . 7 ⊢ (∀𝑘 ∈ ℕ {𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ∪ 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆ ∪ 𝑘 ∈ ℕ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
311	309, 310	ax-mp 5	. . . . . 6 ⊢ ∪ 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆ ∪ 𝑘 ∈ ℕ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
312		1nn 11914	. . . . . . 7 ⊢ 1 ∈ ℕ
313		ne0i 4265	. . . . . . 7 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
314		iunconst 4930	. . . . . . 7 ⊢ (ℕ ≠ ∅ → ∪ 𝑘 ∈ ℕ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) = ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
315	312, 313, 314	mp2b 10	. . . . . 6 ⊢ ∪ 𝑘 ∈ ℕ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) = ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
316	311, 315	sseqtri 3953	. . . . 5 ⊢ ∪ 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
317		ssnum 9726	. . . . 5 ⊢ ((((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ dom card ∧ ∪ 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → ∪ 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ∈ dom card)
318	307, 316, 317	mp2an 688	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ∈ dom card
319		fveq2 6756	. . . . . . . 8 ⊢ (𝑠 = (𝑔‘𝑘) → (1^st ‘𝑠) = (1^st ‘(𝑔‘𝑘)))
320	319	rneqd 5836	. . . . . . 7 ⊢ (𝑠 = (𝑔‘𝑘) → ran (1^st ‘𝑠) = ran (1^st ‘(𝑔‘𝑘)))
321	320	sseq1d 3948	. . . . . 6 ⊢ (𝑠 = (𝑔‘𝑘) → (ran (1^st ‘𝑠) ⊆ (0..^𝑘) ↔ ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘)))
322		fveq2 6756	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = (𝑔‘𝑘) → (2^nd ‘𝑠) = (2^nd ‘(𝑔‘𝑘)))
323	322	imaeq1d 5957	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = (𝑔‘𝑘) → ((2^nd ‘𝑠) “ (1...𝑗)) = ((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)))
324	323	xpeq1d 5609	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = (𝑔‘𝑘) → (((2^nd ‘𝑠) “ (1...𝑗)) × {1}) = (((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}))
325	322	imaeq1d 5957	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = (𝑔‘𝑘) → ((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)))
326	325	xpeq1d 5609	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = (𝑔‘𝑘) → (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))
327	324, 326	uneq12d 4094	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = (𝑔‘𝑘) → ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
328	319, 327	oveq12d 7273	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝑔‘𝑘) → ((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))))
329	328	fvoveq1d 7277	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = (𝑔‘𝑘) → (𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))) = (𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))))
330	329	fveq1d 6758	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = (𝑔‘𝑘) → ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏))
331	330	breq2d 5082	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑔‘𝑘) → (0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏)))
332	328	fveq1d 6758	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = (𝑔‘𝑘) → (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) = (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏))
333	332	neeq1d 3002	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑔‘𝑘) → ((((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0 ↔ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0))
334	331, 333	anbi12d 630	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑔‘𝑘) → ((0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)))
335	334	ralbidv 3120	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑔‘𝑘) → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)))
336	335	rabbidv 3404	. . . . . . . . . . 11 ⊢ (𝑠 = (𝑔‘𝑘) → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)} = {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)})
337	336	uneq2d 4093	. . . . . . . . . 10 ⊢ (𝑠 = (𝑔‘𝑘) → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}) = ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}))
338	337	supeq1d 9135	. . . . . . . . 9 ⊢ (𝑠 = (𝑔‘𝑘) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
339	338	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑠 = (𝑔‘𝑘) → (𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ 𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
340	339	rexbidv 3225	. . . . . . 7 ⊢ (𝑠 = (𝑔‘𝑘) → (∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
341	340	ralbidv 3120	. . . . . 6 ⊢ (𝑠 = (𝑔‘𝑘) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
342	321, 341	anbi12d 630	. . . . 5 ⊢ (𝑠 = (𝑔‘𝑘) → ((ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) ↔ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))))
343	342	ac6num 10166	. . . 4 ⊢ ((ℕ ∈ V ∧ ∪ 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ∈ dom card ∧ ∀𝑘 ∈ ℕ ∃𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → ∃𝑔(𝑔:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))))
344	284, 318, 343	mp3an12 1449	. . 3 ⊢ (∀𝑘 ∈ ℕ ∃𝑠 ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(ran (1^st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘𝑠) ∘_f + ((((2^nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → ∃𝑔(𝑔:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))))
345	283, 344	syl 17	. 2 ⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))))
346	1	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ 𝑔:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → 𝑁 ∈ ℕ)
347		poimir.r	. . . 4 ⊢ 𝑅 = (∏_t‘((1...𝑁) × {(topGen‘ran (,))}))
348		poimir.1	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
349	348	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ 𝑔:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → 𝐹 ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
350		eqid 2738	. . . 4 ⊢ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑛) = ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑛)
351		simplr 765	. . . 4 ⊢ (((𝜑 ∧ 𝑔:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → 𝑔:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
352		simpl 482	. . . . . . 7 ⊢ ((ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘))
353	352	ralimi 3086	. . . . . 6 ⊢ (∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → ∀𝑘 ∈ ℕ ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘))
354	353	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑔:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → ∀𝑘 ∈ ℕ ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘))
355		2fveq3 6761	. . . . . . . 8 ⊢ (𝑘 = 𝑝 → (1^st ‘(𝑔‘𝑘)) = (1^st ‘(𝑔‘𝑝)))
356	355	rneqd 5836	. . . . . . 7 ⊢ (𝑘 = 𝑝 → ran (1^st ‘(𝑔‘𝑘)) = ran (1^st ‘(𝑔‘𝑝)))
357		oveq2 7263	. . . . . . 7 ⊢ (𝑘 = 𝑝 → (0..^𝑘) = (0..^𝑝))
358	356, 357	sseq12d 3950	. . . . . 6 ⊢ (𝑘 = 𝑝 → (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ↔ ran (1^st ‘(𝑔‘𝑝)) ⊆ (0..^𝑝)))
359	358	rspccva 3551	. . . . 5 ⊢ ((∀𝑘 ∈ ℕ ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ 𝑝 ∈ ℕ) → ran (1^st ‘(𝑔‘𝑝)) ⊆ (0..^𝑝))
360	354, 359	sylan 579	. . . 4 ⊢ ((((𝜑 ∧ 𝑔:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) ∧ 𝑝 ∈ ℕ) → ran (1^st ‘(𝑔‘𝑝)) ⊆ (0..^𝑝))
361		simpll 763	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → 𝜑)
362		poimir.2	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → ((𝐹‘𝑧)‘𝑛) ≤ 0)
363	361, 362	sylan 579	. . . . 5 ⊢ ((((𝜑 ∧ 𝑔:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → ((𝐹‘𝑧)‘𝑛) ≤ 0)
364		eqid 2738	. . . . 5 ⊢ ((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) = ((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
365		simpr 484	. . . . . . . 8 ⊢ ((ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
366	365	ralimi 3086	. . . . . . 7 ⊢ (∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → ∀𝑘 ∈ ℕ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
367	366	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → ∀𝑘 ∈ ℕ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
368		2fveq3 6761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑝 → (2^nd ‘(𝑔‘𝑘)) = (2^nd ‘(𝑔‘𝑝)))
369	368	imaeq1d 5957	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑝 → ((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) = ((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)))
370	369	xpeq1d 5609	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑝 → (((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) = (((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}))
371	368	imaeq1d 5957	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑝 → ((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)))
372	371	xpeq1d 5609	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑝 → (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))
373	370, 372	uneq12d 4094	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑝 → ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
374	355, 373	oveq12d 7273	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑝 → ((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))))
375		sneq 4568	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑝 → {𝑘} = {𝑝})
376	375	xpeq2d 5610	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑝 → ((1...𝑁) × {𝑘}) = ((1...𝑁) × {𝑝}))
377	374, 376	oveq12d 7273	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑝 → (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) = (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))
378	377	fveq2d 6760	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑝 → (𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))) = (𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝}))))
379	378	fveq1d 6758	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑝 → ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏))
380	379	breq2d 5082	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑝 → (0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏)))
381	374	fveq1d 6758	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑝 → (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) = (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏))
382	381	neeq1d 3002	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑝 → ((((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0 ↔ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0))
383	380, 382	anbi12d 630	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑝 → ((0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)))
384	383	ralbidv 3120	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑝 → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)))
385	384	rabbidv 3404	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑝 → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)} = {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)})
386	385	uneq2d 4093	. . . . . . . . . 10 ⊢ (𝑘 = 𝑝 → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}) = ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}))
387	386	supeq1d 9135	. . . . . . . . 9 ⊢ (𝑘 = 𝑝 → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
388	387	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑘 = 𝑝 → (𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ 𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
389	388	rexbidv 3225	. . . . . . 7 ⊢ (𝑘 = 𝑝 → (∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
390		eqeq1 2742	. . . . . . . . 9 ⊢ (𝑖 = 𝑞 → (𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ 𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
391	390	rexbidv 3225	. . . . . . . 8 ⊢ (𝑖 = 𝑞 → (∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ ∃𝑗 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
392		oveq2 7263	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑚 → (1...𝑗) = (1...𝑚))
393	392	imaeq2d 5958	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑚 → ((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) = ((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)))
394	393	xpeq1d 5609	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑚 → (((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) = (((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}))
395		oveq1 7262	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑚 → (𝑗 + 1) = (𝑚 + 1))
396	395	oveq1d 7270	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑚 → ((𝑗 + 1)...𝑁) = ((𝑚 + 1)...𝑁))
397	396	imaeq2d 5958	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑚 → ((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)))
398	397	xpeq1d 5609	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑚 → (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))
399	394, 398	uneq12d 4094	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑚 → ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
400	399	oveq2d 7271	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑚 → ((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))))
401	400	fvoveq1d 7277	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑚 → (𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝}))) = (𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝}))))
402	401	fveq1d 6758	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑚 → ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) = ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏))
403	402	breq2d 5082	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑚 → (0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ↔ 0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏)))
404	400	fveq1d 6758	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑚 → (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) = (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏))
405	404	neeq1d 3002	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑚 → ((((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0 ↔ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0))
406	403, 405	anbi12d 630	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑚 → ((0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)))
407	406	ralbidv 3120	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑚 → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)))
408	407	rabbidv 3404	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑚 → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)} = {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)})
409	408	uneq2d 4093	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑚 → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}) = ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}))
410	409	supeq1d 9135	. . . . . . . . . 10 ⊢ (𝑗 = 𝑚 → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
411	410	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑗 = 𝑚 → (𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ 𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
412	411	cbvrexvw 3373	. . . . . . . 8 ⊢ (∃𝑗 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ ∃𝑚 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
413	391, 412	bitrdi 286	. . . . . . 7 ⊢ (𝑖 = 𝑞 → (∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ ∃𝑚 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
414	389, 413	rspc2v 3562	. . . . . 6 ⊢ ((𝑝 ∈ ℕ ∧ 𝑞 ∈ (0...𝑁)) → (∀𝑘 ∈ ℕ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) → ∃𝑚 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
415	367, 414	mpan9 506	. . . . 5 ⊢ ((((𝜑 ∧ 𝑔:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) ∧ (𝑝 ∈ ℕ ∧ 𝑞 ∈ (0...𝑁))) → ∃𝑚 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
416	346, 137, 347, 349, 363, 364, 351, 360, 415	poimirlem31 35735	. . . 4 ⊢ ((((𝜑 ∧ 𝑔:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) ∧ (𝑝 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ^◡ ≤ })) → ∃𝑚 ∈ (0...𝑁)0𝑟((𝐹‘(((1^st ‘(𝑔‘𝑝)) ∘_f + ((((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑝})))‘𝑛))
417	346, 137, 347, 349, 350, 351, 360, 416	poimirlem30 35734	. . 3 ⊢ (((𝜑 ∧ 𝑔:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
418	417	anasss 466	. 2 ⊢ ((𝜑 ∧ (𝑔:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑘 ∈ ℕ (ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1^st ‘(𝑔‘𝑘)) ∘_f + ((((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))) → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
419	345, 418	exlimddv 1939	1 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∃wex 1783 ∈ wcel 2108 {cab 2715 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 {crab 3067 Vcvv 3422 ∪ cun 3881 ⊆ wss 3883 ∅c0 4253 {csn 4558 {cpr 4560 ∪ ciun 4921 class class class wbr 5070 Or wor 5493 × cxp 5578 ^◡ccnv 5579 dom cdm 5580 ran crn 5581 “ cima 5583 Oncon0 6251 Fn wfn 6413 ⟶wf 6414 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 ∘_f cof 7509 ωcom 7687 1^st c1st 7802 2^nd c2nd 7803 ↑_m cmap 8573 Xcixp 8643 ≈ cen 8688 Fincfn 8691 supcsup 9129 cardccrd 9624 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 / cdiv 11562 ℕcn 11903 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 ℝ⁺crp 12659 (,)cioo 13008 [,]cicc 13011 ...cfz 13168 ..^cfzo 13311 ↾_t crest 17048 topGenctg 17065 ∏_tcpt 17066 Cn ccn 22283

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-oadd 8271 df-omul 8272 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-acn 9631 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-dvds 15892 df-rest 17050 df-topgen 17071 df-pt 17072 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-top 21951 df-topon 21968 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-lp 22195 df-cn 22286 df-cnp 22287 df-t1 22373 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-hmph 22815 df-ii 23946

This theorem is referenced by: poimir 35737

W3C validator