poimirlem32 - Mathbox for Brendan Leahy

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Theorem poimirlem32 37961

Description: Lemma for poimir 37962, combining poimirlem28 37957, poimirlem30 37959, and poimirlem31 37960 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 7379	. . . . . . . . . . . . 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 6831	. . . . . . . . . . . 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 5086	. . . . . . . . . . 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 6828	. . . . . . . . . . . 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 2989	. . . . . . . . . . 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 633	. . . . . . . . . 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 3158	. . . . . . . . 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 3394	. . . . . . . 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 4100	. . . . . . 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 9348	. . . . . 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 12487	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
14		0elfz 13567	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ (0...𝑁))
15	13, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ (0...𝑁))
16	15	snssd 4720	. . . . . . . . 9 ⊢ (𝜑 → {0} ⊆ (0...𝑁))
17		ssrab2 4013	. . . . . . . . . . 11 ⊢ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} ⊆ (1...𝑁)
18		fz1ssfz0 13566	. . . . . . . . . . 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 4123	. . . . . . . 8 ⊢ (𝜑 → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ⊆ (0...𝑁))
22		ltso 11215	. . . . . . . . 9 ⊢ < Or ℝ
23		snfi 8979	. . . . . . . . . . 11 ⊢ {0} ∈ Fin
24		fzfi 13923	. . . . . . . . . . . 12 ⊢ (1...𝑁) ∈ Fin
25		rabfi 9170	. . . . . . . . . . . 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 9094	. . . . . . . . . . 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 693	. . . . . . . . . 10 ⊢ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ∈ Fin
29		c0ex 11127	. . . . . . . . . . . 12 ⊢ 0 ∈ V
30	29	snid 4596	. . . . . . . . . . 11 ⊢ 0 ∈ {0}
31		elun1 4113	. . . . . . . . . . 11 ⊢ (0 ∈ {0} → 0 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}))
32		ne0i 4271	. . . . . . . . . . 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 11136	. . . . . . . . . . . . 13 ⊢ ((𝜑 → 𝑁 ∈ ℕ) → 0 ∈ ℝ)
35	34	snssd 4720	. . . . . . . . . . . 12 ⊢ ((𝜑 → 𝑁 ∈ ℕ) → {0} ⊆ ℝ)
36	1, 35	ax-mp 5	. . . . . . . . . . 11 ⊢ {0} ⊆ ℝ
37		elfzelz 13467	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
38	37	ssriv 3921	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ⊆ ℤ
39		zssre 12520	. . . . . . . . . . . . 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 4122	. . . . . . . . . 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 1341	. . . . . . . . 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 9372	. . . . . . . . 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 693	. . . . . . . 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 3911	. . . . . . . 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 727	. . . . . 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 13496	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
50		nngt0 12197	. . . . . . . . . . . 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 3072	. . . . . . . . . . . . 13 ⊢ (∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑠)(𝑝‘𝑏) ≠ 0)
54		elfznn 13496	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (1...𝑁) → 𝑠 ∈ ℕ)
55		nnre 12170	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
56		nnre 12170	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℝ)
57		lenlt 11213	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (𝑛 ≤ 𝑠 ↔ ¬ 𝑠 < 𝑛))
58	55, 56, 57	syl2an 597	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑛 ≤ 𝑠 ↔ ¬ 𝑠 < 𝑛))
59		elfz1b 13536	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (1...𝑠) ↔ (𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ ∧ 𝑛 ≤ 𝑠))
60	59	biimpri 228	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ ∧ 𝑛 ≤ 𝑠) → 𝑛 ∈ (1...𝑠))
61	60	3expia 1122	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑛 ≤ 𝑠 → 𝑛 ∈ (1...𝑠)))
62	58, 61	sylbird 260	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (¬ 𝑠 < 𝑛 → 𝑛 ∈ (1...𝑠)))
63		fveq2 6829	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = 𝑛 → (𝑝‘𝑏) = (𝑝‘𝑛))
64	63	eqeq1d 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑛 → ((𝑝‘𝑏) = 0 ↔ (𝑝‘𝑛) = 0))
65	64	rspcev 3562	. . . . . . . . . . . . . . . . . . 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 653	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) ∧ 𝑠 ∈ ℕ) → (¬ 𝑠 < 𝑛 → ∃𝑏 ∈ (1...𝑠)(𝑝‘𝑏) = 0))
69		nne 2934	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (𝑝‘𝑏) ≠ 0 ↔ (𝑝‘𝑏) = 0)
70	69	rexbii 3082	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑏 ∈ (1...𝑠) ¬ (𝑝‘𝑏) ≠ 0 ↔ ∃𝑏 ∈ (1...𝑠)(𝑝‘𝑏) = 0)
71		rexnal 3087	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑏 ∈ (1...𝑠) ¬ (𝑝‘𝑏) ≠ 0 ↔ ¬ ∀𝑏 ∈ (1...𝑠)(𝑝‘𝑏) ≠ 0)
72	70, 71	bitr3i 277	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑏 ∈ (1...𝑠)(𝑝‘𝑏) = 0 ↔ ¬ ∀𝑏 ∈ (1...𝑠)(𝑝‘𝑏) ≠ 0)
73	68, 72	imbitrdi 251	. . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . 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 3127	. . . . . . . . . . 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 4128	. . . . . . . . . . . 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 5077	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 0 → (𝑠 < 𝑛 ↔ 0 < 𝑛))
80	29, 79	ralsn 4615	. . . . . . . . . . . . 13 ⊢ (∀𝑠 ∈ {0}𝑠 < 𝑛 ↔ 0 < 𝑛)
81		oveq2 7364	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑠 → (1...𝑎) = (1...𝑠))
82	81	raleqdv 3293	. . . . . . . . . . . . . 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 3637	. . . . . . . . . . . . 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 629	. . . . . . . . . . . 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 275	. . . . . . . . . . 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 584	. . . . . . . . . 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 5077	. . . . . . . . . . 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 3560	. . . . . . . . . 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 588	. . . . . . . . 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 581	. . . . . . . 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 1132	. . . . . . 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 12622	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
94	93	3ad2ant1 1134	. . . . . . . . . 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 1196	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → 𝑛 ∈ (1...𝑁))
97		simpll 767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → 𝜑)
98		simplr 769	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → 𝑘 ∈ ℕ)
99		elfzelz 13467	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ (0...𝑘) → 𝑖 ∈ ℤ)
100	99	zred 12622	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (0...𝑘) → 𝑖 ∈ ℝ)
101		nndivre 12207	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ)
102	100, 101	sylan 581	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ)
103		elfzle1 13470	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ (0...𝑘) → 0 ≤ 𝑖)
104	100, 103	jca 511	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (0...𝑘) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
105		nnrp 12943	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ⁺)
106	105	rpregt0d 12981	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
107		divge0 12014	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑖 ∈ ℝ ∧ 0 ≤ 𝑖) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝑖 / 𝑘))
108	104, 106, 107	syl2an 597	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑖 / 𝑘))
109		elfzle2 13471	. . . . . . . . . . . . . . . . . . . . . . 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 11134	. . . . . . . . . . . . . . . . . . . . . . . 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 13028	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖 ≤ (𝑘 · 1)))
115		nncn 12171	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
116	115	mulridd 11151	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ ℕ → (𝑘 · 1) = 𝑘)
117	116	breq2d 5086	. . . . . . . . . . . . . . . . . . . . . . . 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 279	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖 ≤ 𝑘))
120	110, 119	mpbird 257	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ≤ 1)
121		elicc01 13408	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 / 𝑘) ∈ (0[,]1) ↔ ((𝑖 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝑖 / 𝑘) ∧ (𝑖 / 𝑘) ≤ 1))
122	102, 108, 120, 121	syl3anbrc 1345	. . . . . . . . . . . . . . . . . . . 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 4574	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ {𝑘} → 𝑗 = 𝑘)
125	124	oveq2d 7372	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ {𝑘} → (𝑖 / 𝑗) = (𝑖 / 𝑘))
126	125	eleq1d 2820	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ {𝑘} → ((𝑖 / 𝑗) ∈ (0[,]1) ↔ (𝑖 / 𝑘) ∈ (0[,]1)))
127	123, 126	syl5ibrcom 247	. . . . . . . . . . . . . . . . . 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 581	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) ∧ (𝑖 ∈ (0...𝑘) ∧ 𝑗 ∈ {𝑘})) → (𝑖 / 𝑗) ∈ (0[,]1))
130		simprr 773	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → 𝑝:(1...𝑁)⟶(0...𝑘))
131		vex 3431	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑘 ∈ V
132	131	fconst 6715	. . . . . . . . . . . . . . . . 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 13924	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → (1...𝑁) ∈ Fin)
135		inidm 4157	. . . . . . . . . . . . . . . 16 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
136	129, 130, 133, 134, 134, 135	off 7638	. . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑_m (1...𝑁)))
139		ovex 7389	. . . . . . . . . . . . . . . . 17 ⊢ (0[,]1) ∈ V
140		ovex 7389	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑁) ∈ V
141	139, 140	elmap 8808	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑_m (1...𝑁)) ↔ (𝑝 ∘_f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
142	138, 141	bitri 275	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (𝑝 ∘_f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
143	136, 142	sylibr 234	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → (𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼)
144	143	3adantr3 1173	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → (𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼)
145		3anass 1095	. . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . 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 6657	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝:(1...𝑁)⟶(0...𝑘) → 𝑝 Fn (1...𝑁))
149	148	ad2antrl 729	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → 𝑝 Fn (1...𝑁))
150		fnconstg 6717	. . . . . . . . . . . . . . . . . 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 13924	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → (1...𝑁) ∈ Fin)
153		simplrr 778	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑝‘𝑛) = 𝑘)
154	131	fvconst2 7148	. . . . . . . . . . . . . . . . . 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 7631	. . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . 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 12200	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
160	115, 159	dividd 11918	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝑘 / 𝑘) = 1)
161	160	ad2antlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → (𝑘 / 𝑘) = 1)
162	158, 161	eqtrd 2770	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ((𝑝 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 1)
163		ovex 7389	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ V
164		eleq1 2823	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑝 ∘_f / ((1...𝑁) × {𝑘})) → (𝑧 ∈ 𝐼 ↔ (𝑝 ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼))
165		fveq1 6828	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑝 ∘_f / ((1...𝑁) × {𝑘})) → (𝑧‘𝑛) = ((𝑝 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛))
166	165	eqeq1d 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑝 ∘_f / ((1...𝑁) × {𝑘})) → ((𝑧‘𝑛) = 1 ↔ ((𝑝 ∘_f / ((1...𝑁) × {𝑘}))‘𝑛) = 1))
167	164, 166	3anbi23d 1442	. . . . . . . . . . . . . . . 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 631	. . . . . . . . . . . . . . 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 6829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑝 ∘_f / ((1...𝑁) × {𝑘})) → (𝐹‘𝑧) = (𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘}))))
170	169	fveq1d 6831	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑝 ∘_f / ((1...𝑁) × {𝑘})) → ((𝐹‘𝑧)‘𝑛) = ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛))
171	170	breq2d 5086	. . . . . . . . . . . . . . 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 3501	. . . . . . . . . . . . 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 1375	. . . . . . . . . . . 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 1139	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘) → (𝑝‘𝑛) = 𝑘)
178		neeq1 2992	. . . . . . . . . . . . . . 15 ⊢ ((𝑝‘𝑛) = 𝑘 → ((𝑝‘𝑛) ≠ 0 ↔ 𝑘 ≠ 0))
179	159, 178	syl5ibrcom 247	. . . . . . . . . . . . . 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 597	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → (𝑝‘𝑛) ≠ 0)
182		vex 3431	. . . . . . . . . . . . 13 ⊢ 𝑛 ∈ V
183		fveq2 6829	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑛 → ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛))
184	183	breq2d 5086	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑛 → (0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)))
185	63	neeq1d 2989	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑛 → ((𝑝‘𝑏) ≠ 0 ↔ (𝑝‘𝑛) ≠ 0))
186	184, 185	anbi12d 633	. . . . . . . . . . . . 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 4615	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 12623	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
190		1cnd 11128	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → 1 ∈ ℂ)
191	189, 190	subeq0ad 11504	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
192	191	biimpcd 249	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) = 0 → (𝑛 ∈ (1...𝑁) → 𝑛 = 1))
193		1z 12546	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℤ
194		fzsn 13509	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 ∈ ℤ → (1...1) = {1})
195	193, 194	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...1) = {1}
196		oveq2 7364	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → (1...𝑛) = (1...1))
197		sneq 4567	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → {𝑛} = {1})
198	195, 196, 197	3eqtr4a 2796	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1 → (1...𝑛) = {𝑛})
199	198	raleqdv 3293	. . . . . . . . . . . . . . . . . 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 248	. . . . . . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . . . . . . . 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 11564	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
204	189, 203	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) = 𝑛)
205		elfzuz 13463	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ_≥‘1))
206	204, 205	eqeltrd 2835	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) ∈ (ℤ_≥‘1))
207		peano2zm 12559	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℤ → (𝑛 − 1) ∈ ℤ)
208		uzid 12792	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ_≥‘(𝑛 − 1)))
209		peano2uz 12840	. . . . . . . . . . . . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ_≥‘(𝑛 − 1)))
212		fzsplit2 13492	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑛 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
213	206, 211, 212	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (1...𝑁) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
214	204	oveq1d 7371	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
215		fzsn 13509	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
216	37, 215	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛...𝑛) = {𝑛})
217	214, 216	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
218	217	uneq2d 4100	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (1...𝑁) → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
219	213, 218	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (1...𝑁) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
220	219	raleqdv 3293	. . . . . . . . . . . . . . . . . . . 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	imbitrrid 246	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → ((∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝 ∘_f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)))
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 858	. . . . . . . . . . . . . . 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 4085	. . . . . . . . . . . . . 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 7389	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 − 1) ∈ V
230	229	elsn 4572	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 − 1) ∈ {0} ↔ (𝑛 − 1) = 0)
231		oveq2 7364	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝑛 − 1) → (1...𝑎) = (1...(𝑛 − 1)))
232	231	raleqdv 3293	. . . . . . . . . . . . . . . 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 3631	. . . . . . . . . . . . . . 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 915	. . . . . . . . . . . . . 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 275	. . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑛 → (1...𝑎) = (1...𝑛))
237	236	raleqdv 3293	. . . . . . . . . . . . . 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 3631	. . . . . . . . . . . . 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 296	. . . . . . . . . . . 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 4114	. . . . . . . . . . . 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 585	. . . . . . . . . 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 12090	. . . . . . . . . . . . 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 693	. . . . . . . . . . . 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 1341	. . . . . . . . . . 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 12120	. . . . . . . . . 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 11986	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
249		peano2rem 11450	. . . . . . . . . . 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 11227	. . . . . . . . . . . 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 1453	. . . . . . . . . . 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 689	. . . . . . . . . 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 696	. . . . . . . . 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 11244	. . . . . . . . . 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 5077	. . . . . . . . . 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 326	. . . . . . . . 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 2959	. . . . . . . 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 2823	. . . . . . . . 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 233	. . . . . . . 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 2956	. . . . . . 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 37957	. . . . 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 12432	. . . . . . . . . . . 12 ⊢ ℕ₀ ∈ V
267		fzo0ssnn0 13690	. . . . . . . . . . . 12 ⊢ (0..^𝑘) ⊆ ℕ₀
268		mapss 8826	. . . . . . . . . . . 12 ⊢ ((ℕ₀ ∈ V ∧ (0..^𝑘) ⊆ ℕ₀) → ((0..^𝑘) ↑_m (1...𝑁)) ⊆ (ℕ₀ ↑_m (1...𝑁)))
269	266, 267, 268	mp2an 693	. . . . . . . . . . 11 ⊢ ((0..^𝑘) ↑_m (1...𝑁)) ⊆ (ℕ₀ ↑_m (1...𝑁))
270		xpss1 5639	. . . . . . . . . . 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 7963	. . . . . . . . . 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 8785	. . . . . . . . . 10 ⊢ ((1^st ‘𝑠) ∈ ((0..^𝑘) ↑_m (1...𝑁)) → (1^st ‘𝑠):(1...𝑁)⟶(0..^𝑘))
275		frn 6664	. . . . . . . . . 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 616	. . . . . . 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 218	. . . . . 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 3068	. . . . 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 3127	. . 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 12169	. . . 4 ⊢ ℕ ∈ V
285	140, 266	ixpconst 8844	. . . . . . 7 ⊢ X𝑛 ∈ (1...𝑁)ℕ₀ = (ℕ₀ ↑_m (1...𝑁))
286		omelon 9556	. . . . . . . . . 10 ⊢ ω ∈ On
287		nn0ennn 13930	. . . . . . . . . . 11 ⊢ ℕ₀ ≈ ℕ
288		nnenom 13931	. . . . . . . . . . 11 ⊢ ℕ ≈ ω
289	287, 288	entr2i 8945	. . . . . . . . . 10 ⊢ ω ≈ ℕ₀
290		isnumi 9859	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ ω ≈ ℕ₀) → ℕ₀ ∈ dom card)
291	286, 289, 290	mp2an 693	. . . . . . . . 9 ⊢ ℕ₀ ∈ dom card
292	291	rgenw 3053	. . . . . . . 8 ⊢ ∀𝑛 ∈ (1...𝑁)ℕ₀ ∈ dom card
293		finixpnum 37914	. . . . . . . 8 ⊢ (((1...𝑁) ∈ Fin ∧ ∀𝑛 ∈ (1...𝑁)ℕ₀ ∈ dom card) → X𝑛 ∈ (1...𝑁)ℕ₀ ∈ dom card)
294	24, 292, 293	mp2an 693	. . . . . . 7 ⊢ X𝑛 ∈ (1...𝑁)ℕ₀ ∈ dom card
295	285, 294	eqeltrri 2832	. . . . . 6 ⊢ (ℕ₀ ↑_m (1...𝑁)) ∈ dom card
296	140, 140	mapval 8774	. . . . . . . . 9 ⊢ ((1...𝑁) ↑_m (1...𝑁)) = {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)}
297		mapfi 9247	. . . . . . . . . 10 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ↑_m (1...𝑁)) ∈ Fin)
298	24, 24, 297	mp2an 693	. . . . . . . . 9 ⊢ ((1...𝑁) ↑_m (1...𝑁)) ∈ Fin
299	296, 298	eqeltrri 2832	. . . . . . . 8 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} ∈ Fin
300		f1of 6769	. . . . . . . . 9 ⊢ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁))
301	300	ss2abi 3999	. . . . . . . 8 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)}
302		ssfi 9096	. . . . . . . 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 693	. . . . . . 7 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin
304		finnum 9861	. . . . . . 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 9864	. . . . . 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 693	. . . . 5 ⊢ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ dom card
308		ssrab2 4013	. . . . . . . 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 3053	. . . . . . 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 4942	. . . . . . 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 12174	. . . . . . 7 ⊢ 1 ∈ ℕ
313		ne0i 4271	. . . . . . 7 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
314		iunconst 4933	. . . . . . 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 3965	. . . . 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 9950	. . . . 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 693	. . . 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 6829	. . . . . . . 8 ⊢ (𝑠 = (𝑔‘𝑘) → (1^st ‘𝑠) = (1^st ‘(𝑔‘𝑘)))
320	319	rneqd 5882	. . . . . . 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 6829	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = (𝑔‘𝑘) → (2^nd ‘𝑠) = (2^nd ‘(𝑔‘𝑘)))
323	322	imaeq1d 6013	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = (𝑔‘𝑘) → ((2^nd ‘𝑠) “ (1...𝑗)) = ((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)))
324	323	xpeq1d 5649	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = (𝑔‘𝑘) → (((2^nd ‘𝑠) “ (1...𝑗)) × {1}) = (((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}))
325	322	imaeq1d 6013	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = (𝑔‘𝑘) → ((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)))
326	325	xpeq1d 5649	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = (𝑔‘𝑘) → (((2^nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))
327	324, 326	uneq12d 4101	. . . . . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . . . . . 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 7378	. . . . . . . . . . . . . . . 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 6831	. . . . . . . . . . . . . . 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 5086	. . . . . . . . . . . . . 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 6831	. . . . . . . . . . . . . . 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 2989	. . . . . . . . . . . . . 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 633	. . . . . . . . . . . . 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 3158	. . . . . . . . . . . 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 3394	. . . . . . . . . . 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 4100	. . . . . . . . . 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 9348	. . . . . . . . 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 2746	. . . . . . . 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 3159	. . . . . . 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 3158	. . . . . 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 633	. . . . 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 10390	. . . 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 1454	. . 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 727	. . . 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 727	. . . 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 2735	. . . 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 769	. . . 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 3072	. . . . . 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 6834	. . . . . . . 8 ⊢ (𝑘 = 𝑝 → (1^st ‘(𝑔‘𝑘)) = (1^st ‘(𝑔‘𝑝)))
356	355	rneqd 5882	. . . . . . 7 ⊢ (𝑘 = 𝑝 → ran (1^st ‘(𝑔‘𝑘)) = ran (1^st ‘(𝑔‘𝑝)))
357		oveq2 7364	. . . . . . 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 3561	. . . . 5 ⊢ ((∀𝑘 ∈ ℕ ran (1^st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ 𝑝 ∈ ℕ) → ran (1^st ‘(𝑔‘𝑝)) ⊆ (0..^𝑝))
360	354, 359	sylan 581	. . . 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 767	. . . . . 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 581	. . . . 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 2735	. . . . 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 3072	. . . . . . 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 6834	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑝 → (2^nd ‘(𝑔‘𝑘)) = (2^nd ‘(𝑔‘𝑝)))
369	368	imaeq1d 6013	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑝 → ((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) = ((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)))
370	369	xpeq1d 5649	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑝 → (((2^nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) = (((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}))
371	368	imaeq1d 6013	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑝 → ((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)))
372	371	xpeq1d 5649	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑝 → (((2^nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))
373	370, 372	uneq12d 4101	. . . . . . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . . . . . . 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 4567	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑝 → {𝑘} = {𝑝})
376	375	xpeq2d 5650	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑝 → ((1...𝑁) × {𝑘}) = ((1...𝑁) × {𝑝}))
377	374, 376	oveq12d 7374	. . . . . . . . . . . . . . . . 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 6833	. . . . . . . . . . . . . . . 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 6831	. . . . . . . . . . . . . . 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 5086	. . . . . . . . . . . . . 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 6831	. . . . . . . . . . . . . . 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 2989	. . . . . . . . . . . . . 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 633	. . . . . . . . . . . . 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 3158	. . . . . . . . . . . 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 3394	. . . . . . . . . . 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 4100	. . . . . . . . . 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 9348	. . . . . . . . 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 2746	. . . . . . . 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 3159	. . . . . . 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 2739	. . . . . . . . 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 3159	. . . . . . . 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 7364	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑚 → (1...𝑗) = (1...𝑚))
393	392	imaeq2d 6014	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑚 → ((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) = ((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)))
394	393	xpeq1d 5649	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑚 → (((2^nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) = (((2^nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}))
395		oveq1 7363	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑚 → (𝑗 + 1) = (𝑚 + 1))
396	395	oveq1d 7371	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑚 → ((𝑗 + 1)...𝑁) = ((𝑚 + 1)...𝑁))
397	396	imaeq2d 6014	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑚 → ((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)))
398	397	xpeq1d 5649	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑚 → (((2^nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))
399	394, 398	uneq12d 4101	. . . . . . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . . . . . . 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 7378	. . . . . . . . . . . . . . . . 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 6831	. . . . . . . . . . . . . . . 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 5086	. . . . . . . . . . . . . . 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 6831	. . . . . . . . . . . . . . . 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 2989	. . . . . . . . . . . . . . 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 633	. . . . . . . . . . . . . 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 3158	. . . . . . . . . . . . 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 3394	. . . . . . . . . . . 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 4100	. . . . . . . . . . 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 9348	. . . . . . . . . 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 2746	. . . . . . . . 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 3214	. . . . . . . 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 287	. . . . . . 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 3573	. . . . . 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 37960	. . . 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 37959	. . 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 1937	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 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∃wex 1781 ∈ wcel 2114 {cab 2713 ≠ wne 2930 ∀wral 3049 ∃wrex 3059 {crab 3387 Vcvv 3427 ∪ cun 3883 ⊆ wss 3885 ∅c0 4263 {csn 4557 {cpr 4559 ∪ ciun 4923 class class class wbr 5074 Or wor 5527 × cxp 5618 ^◡ccnv 5619 dom cdm 5620 ran crn 5621 “ cima 5623 Oncon0 6312 Fn wfn 6482 ⟶wf 6483 –1-1-onto→wf1o 6486 ‘cfv 6487 (class class class)co 7356 ∘_f cof 7618 ωcom 7806 1^st c1st 7929 2^nd c2nd 7930 ↑_m cmap 8762 Xcixp 8834 ≈ cen 8879 Fincfn 8882 supcsup 9342 cardccrd 9848 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 + caddc 11030 · cmul 11032 < clt 11168 ≤ cle 11169 − cmin 11366 / cdiv 11796 ℕcn 12163 ℕ₀cn0 12426 ℤcz 12513 ℤ_≥cuz 12777 ℝ⁺crp 12931 (,)cioo 13287 [,]cicc 13290 ...cfz 13450 ..^cfzo 13597 ↾_t crest 17372 topGenctg 17389 ∏_tcpt 17390 Cn ccn 23177

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-disj 5042 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-oadd 8398 df-omul 8399 df-er 8632 df-map 8764 df-pm 8765 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fi 9313 df-sup 9344 df-inf 9345 df-oi 9414 df-dju 9814 df-card 9852 df-acn 9855 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-n0 12427 df-xnn0 12500 df-z 12514 df-uz 12778 df-q 12888 df-rp 12932 df-xneg 13052 df-xadd 13053 df-xmul 13054 df-ioo 13291 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-seq 13953 df-exp 14013 df-fac 14225 df-bc 14254 df-hash 14282 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15439 df-sum 15638 df-dvds 16211 df-rest 17374 df-topgen 17395 df-pt 17396 df-psmet 21333 df-xmet 21334 df-met 21335 df-bl 21336 df-mopn 21337 df-top 22847 df-topon 22864 df-bases 22899 df-cld 22972 df-ntr 22973 df-cls 22974 df-lp 23089 df-cn 23180 df-cnp 23181 df-t1 23267 df-haus 23268 df-cmp 23340 df-tx 23515 df-hmeo 23708 df-hmph 23709 df-ii 24832

This theorem is referenced by: poimir 37962

W3C validator