Theorem poimirlem32

Description: Lemma for poimir , combining poimirlem28 , poimirlem30 , and poimirlem31 to get Equation (1) of Kulpa p. 547. (Contributed by Brendan Leahy, 21-Aug-2020)

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

Proof

Step	Hyp	Ref	Expression
1		poimir.0	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
2		poimir.i	⊢ 𝐼 = ( ( 0 [,] 1 ) ↑m ( 1 ... 𝑁 ) )
3		poimir.r	⊢ 𝑅 = ( ∏t ‘ ( ( 1 ... 𝑁 ) × { ( topGen ‘ ran (,) ) } ) )
4		poimir.1	⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝑅 ↾t 𝐼 ) Cn 𝑅 ) )
5		poimir.2	⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑧 ∈ 𝐼 ∧ ( 𝑧 ‘ 𝑛 ) = 0 ) ) → ( ( 𝐹 ‘ 𝑧 ) ‘ 𝑛 ) ≤ 0 )
6		poimir.3	⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑧 ∈ 𝐼 ∧ ( 𝑧 ‘ 𝑛 ) = 1 ) ) → 0 ≤ ( ( 𝐹 ‘ 𝑧 ) ‘ 𝑛 ) )
7	1	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝑁 ∈ ℕ )
8		fvoveq1	⊢ ( 𝑝 = ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) → ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) = ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) )
9	8	fveq1d	⊢ ( 𝑝 = ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) → ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) = ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) )
10	9	breq2d	⊢ ( 𝑝 = ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) → ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ↔ 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ) )
11		fveq1	⊢ ( 𝑝 = ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) → ( 𝑝 ‘ 𝑏 ) = ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) )
12	11	neeq1d	⊢ ( 𝑝 = ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) → ( ( 𝑝 ‘ 𝑏 ) ≠ 0 ↔ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) )
13	10 12	anbi12d	⊢ ( 𝑝 = ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) → ( ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) ↔ ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) ) )
14	13	ralbidv	⊢ ( 𝑝 = ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) → ( ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) ↔ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) ) )
15	14	rabbidv	⊢ ( 𝑝 = ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 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 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } )
16	15	uneq2d	⊢ ( 𝑝 = ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 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 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) )
17	16	supeq1d	⊢ ( 𝑝 = ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 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 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) )
18	1	nnnn0d	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
19		0elfz	⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ ( 0 ... 𝑁 ) )
20	18 19	syl	⊢ ( 𝜑 → 0 ∈ ( 0 ... 𝑁 ) )
21	20	snssd	⊢ ( 𝜑 → { 0 } ⊆ ( 0 ... 𝑁 ) )
22		ssrab2	⊢ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) } ⊆ ( 1 ... 𝑁 )
23		fz1ssfz0	⊢ ( 1 ... 𝑁 ) ⊆ ( 0 ... 𝑁 )
24	22 23	sstri	⊢ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) } ⊆ ( 0 ... 𝑁 )
25	24	a1i	⊢ ( 𝜑 → { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) } ⊆ ( 0 ... 𝑁 ) )
26	21 25	unssd	⊢ ( 𝜑 → ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) } ) ⊆ ( 0 ... 𝑁 ) )
27		ltso	⊢ < Or ℝ
28		snfi	⊢ { 0 } ∈ Fin
29		fzfi	⊢ ( 1 ... 𝑁 ) ∈ Fin
30		rabfi	⊢ ( ( 1 ... 𝑁 ) ∈ Fin → { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) } ∈ Fin )
31	29 30	ax-mp	⊢ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) } ∈ Fin
32		unfi	⊢ ( ( { 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 )
33	28 31 32	mp2an	⊢ ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) } ) ∈ Fin
34		c0ex	⊢ 0 ∈ V
35	34	snid	⊢ 0 ∈ { 0 }
36		elun1	⊢ ( 0 ∈ { 0 } → 0 ∈ ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) } ) )
37		ne0i	⊢ ( 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 ) } ) ≠ ∅ )
38	35 36 37	mp2b	⊢ ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) } ) ≠ ∅
39		0red	⊢ ( ( 𝜑 → 𝑁 ∈ ℕ ) → 0 ∈ ℝ )
40	39	snssd	⊢ ( ( 𝜑 → 𝑁 ∈ ℕ ) → { 0 } ⊆ ℝ )
41	1 40	ax-mp	⊢ { 0 } ⊆ ℝ
42		elfzelz	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → 𝑛 ∈ ℤ )
43	42	ssriv	⊢ ( 1 ... 𝑁 ) ⊆ ℤ
44		zssre	⊢ ℤ ⊆ ℝ
45	43 44	sstri	⊢ ( 1 ... 𝑁 ) ⊆ ℝ
46	22 45	sstri	⊢ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) } ⊆ ℝ
47	41 46	unssi	⊢ ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) } ) ⊆ ℝ
48	33 38 47	3pm3.2i	⊢ ( ( { 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 )
49		fisupcl	⊢ ( ( < 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 ) } ) )
50	27 48 49	mp2an	⊢ 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 ) } )
51		ssel	⊢ ( ( { 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 ) ) )
52	26 50 51	mpisyl	⊢ ( 𝜑 → sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ∈ ( 0 ... 𝑁 ) )
53	52	ad2antrr	⊢ ( ( ( 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 ) )
54		elfznn	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → 𝑛 ∈ ℕ )
55		nngt0	⊢ ( 𝑛 ∈ ℕ → 0 < 𝑛 )
56	55	adantr	⊢ ( ( 𝑛 ∈ ℕ ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) → 0 < 𝑛 )
57		simpr	⊢ ( ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) → ( 𝑝 ‘ 𝑏 ) ≠ 0 )
58	57	ralimi	⊢ ( ∀ 𝑏 ∈ ( 1 ... 𝑠 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) → ∀ 𝑏 ∈ ( 1 ... 𝑠 ) ( 𝑝 ‘ 𝑏 ) ≠ 0 )
59		elfznn	⊢ ( 𝑠 ∈ ( 1 ... 𝑁 ) → 𝑠 ∈ ℕ )
60		nnre	⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℝ )
61		nnre	⊢ ( 𝑠 ∈ ℕ → 𝑠 ∈ ℝ )
62		lenlt	⊢ ( ( 𝑛 ∈ ℝ ∧ 𝑠 ∈ ℝ ) → ( 𝑛 ≤ 𝑠 ↔ ¬ 𝑠 < 𝑛 ) )
63	60 61 62	syl2an	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ ) → ( 𝑛 ≤ 𝑠 ↔ ¬ 𝑠 < 𝑛 ) )
64		elfz1b	⊢ ( 𝑛 ∈ ( 1 ... 𝑠 ) ↔ ( 𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ ∧ 𝑛 ≤ 𝑠 ) )
65	64	biimpri	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ ∧ 𝑛 ≤ 𝑠 ) → 𝑛 ∈ ( 1 ... 𝑠 ) )
66	65	3expia	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ ) → ( 𝑛 ≤ 𝑠 → 𝑛 ∈ ( 1 ... 𝑠 ) ) )
67	63 66	sylbird	⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ ) → ( ¬ 𝑠 < 𝑛 → 𝑛 ∈ ( 1 ... 𝑠 ) ) )
68		fveq2	⊢ ( 𝑏 = 𝑛 → ( 𝑝 ‘ 𝑏 ) = ( 𝑝 ‘ 𝑛 ) )
69	68	eqeq1d	⊢ ( 𝑏 = 𝑛 → ( ( 𝑝 ‘ 𝑏 ) = 0 ↔ ( 𝑝 ‘ 𝑛 ) = 0 ) )
70	69	rspcev	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑠 ) ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) → ∃ 𝑏 ∈ ( 1 ... 𝑠 ) ( 𝑝 ‘ 𝑏 ) = 0 )
71	70	expcom	⊢ ( ( 𝑝 ‘ 𝑛 ) = 0 → ( 𝑛 ∈ ( 1 ... 𝑠 ) → ∃ 𝑏 ∈ ( 1 ... 𝑠 ) ( 𝑝 ‘ 𝑏 ) = 0 ) )
72	67 71	sylan9	⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ ) ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) → ( ¬ 𝑠 < 𝑛 → ∃ 𝑏 ∈ ( 1 ... 𝑠 ) ( 𝑝 ‘ 𝑏 ) = 0 ) )
73	72	an32s	⊢ ( ( ( 𝑛 ∈ ℕ ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) ∧ 𝑠 ∈ ℕ ) → ( ¬ 𝑠 < 𝑛 → ∃ 𝑏 ∈ ( 1 ... 𝑠 ) ( 𝑝 ‘ 𝑏 ) = 0 ) )
74		nne	⊢ ( ¬ ( 𝑝 ‘ 𝑏 ) ≠ 0 ↔ ( 𝑝 ‘ 𝑏 ) = 0 )
75	74	rexbii	⊢ ( ∃ 𝑏 ∈ ( 1 ... 𝑠 ) ¬ ( 𝑝 ‘ 𝑏 ) ≠ 0 ↔ ∃ 𝑏 ∈ ( 1 ... 𝑠 ) ( 𝑝 ‘ 𝑏 ) = 0 )
76		rexnal	⊢ ( ∃ 𝑏 ∈ ( 1 ... 𝑠 ) ¬ ( 𝑝 ‘ 𝑏 ) ≠ 0 ↔ ¬ ∀ 𝑏 ∈ ( 1 ... 𝑠 ) ( 𝑝 ‘ 𝑏 ) ≠ 0 )
77	75 76	bitr3i	⊢ ( ∃ 𝑏 ∈ ( 1 ... 𝑠 ) ( 𝑝 ‘ 𝑏 ) = 0 ↔ ¬ ∀ 𝑏 ∈ ( 1 ... 𝑠 ) ( 𝑝 ‘ 𝑏 ) ≠ 0 )
78	73 77	syl6ib	⊢ ( ( ( 𝑛 ∈ ℕ ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) ∧ 𝑠 ∈ ℕ ) → ( ¬ 𝑠 < 𝑛 → ¬ ∀ 𝑏 ∈ ( 1 ... 𝑠 ) ( 𝑝 ‘ 𝑏 ) ≠ 0 ) )
79	78	con4d	⊢ ( ( ( 𝑛 ∈ ℕ ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) ∧ 𝑠 ∈ ℕ ) → ( ∀ 𝑏 ∈ ( 1 ... 𝑠 ) ( 𝑝 ‘ 𝑏 ) ≠ 0 → 𝑠 < 𝑛 ) )
80	59 79	sylan2	⊢ ( ( ( 𝑛 ∈ ℕ ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) ∧ 𝑠 ∈ ( 1 ... 𝑁 ) ) → ( ∀ 𝑏 ∈ ( 1 ... 𝑠 ) ( 𝑝 ‘ 𝑏 ) ≠ 0 → 𝑠 < 𝑛 ) )
81	58 80	syl5	⊢ ( ( ( 𝑛 ∈ ℕ ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) ∧ 𝑠 ∈ ( 1 ... 𝑁 ) ) → ( ∀ 𝑏 ∈ ( 1 ... 𝑠 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) → 𝑠 < 𝑛 ) )
82	81	ralrimiva	⊢ ( ( 𝑛 ∈ ℕ ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) → ∀ 𝑠 ∈ ( 1 ... 𝑁 ) ( ∀ 𝑏 ∈ ( 1 ... 𝑠 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) → 𝑠 < 𝑛 ) )
83		ralunb	⊢ ( ∀ 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 ) )
84		breq1	⊢ ( 𝑠 = 0 → ( 𝑠 < 𝑛 ↔ 0 < 𝑛 ) )
85	34 84	ralsn	⊢ ( ∀ 𝑠 ∈ { 0 } 𝑠 < 𝑛 ↔ 0 < 𝑛 )
86		oveq2	⊢ ( 𝑎 = 𝑠 → ( 1 ... 𝑎 ) = ( 1 ... 𝑠 ) )
87	86	raleqdv	⊢ ( 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 ) ) )
88	87	ralrab	⊢ ( ∀ 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 ) )
89	85 88	anbi12i	⊢ ( ( ∀ 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 ) ) )
90	83 89	bitri	⊢ ( ∀ 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 ) ) )
91	56 82 90	sylanbrc	⊢ ( ( 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 )
92		breq1	⊢ ( 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 ) )
93	92	rspcva	⊢ ( ( 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 )
94	50 91 93	sylancr	⊢ ( ( 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 )
95	54 94	sylan	⊢ ( ( 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 )
96	95	3adant2	⊢ ( ( 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 )
97	96	adantl	⊢ ( ( ( 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 )
98	42	zred	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → 𝑛 ∈ ℝ )
99	98	3ad2ant1	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) → 𝑛 ∈ ℝ )
100	99	adantl	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) → 𝑛 ∈ ℝ )
101		simpr1	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) → 𝑛 ∈ ( 1 ... 𝑁 ) )
102		simpll	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) → 𝜑 )
103		simplr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ) ) → 𝑘 ∈ ℕ )
104		elfzelz	⊢ ( 𝑖 ∈ ( 0 ... 𝑘 ) → 𝑖 ∈ ℤ )
105	104	zred	⊢ ( 𝑖 ∈ ( 0 ... 𝑘 ) → 𝑖 ∈ ℝ )
106		nndivre	⊢ ( ( 𝑖 ∈ ℝ ∧ 𝑘 ∈ ℕ ) → ( 𝑖 / 𝑘 ) ∈ ℝ )
107	105 106	sylan	⊢ ( ( 𝑖 ∈ ( 0 ... 𝑘 ) ∧ 𝑘 ∈ ℕ ) → ( 𝑖 / 𝑘 ) ∈ ℝ )
108		elfzle1	⊢ ( 𝑖 ∈ ( 0 ... 𝑘 ) → 0 ≤ 𝑖 )
109	105 108	jca	⊢ ( 𝑖 ∈ ( 0 ... 𝑘 ) → ( 𝑖 ∈ ℝ ∧ 0 ≤ 𝑖 ) )
110		nnrp	⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℝ+ )
111	110	rpregt0d	⊢ ( 𝑘 ∈ ℕ → ( 𝑘 ∈ ℝ ∧ 0 < 𝑘 ) )
112		divge0	⊢ ( ( ( 𝑖 ∈ ℝ ∧ 0 ≤ 𝑖 ) ∧ ( 𝑘 ∈ ℝ ∧ 0 < 𝑘 ) ) → 0 ≤ ( 𝑖 / 𝑘 ) )
113	109 111 112	syl2an	⊢ ( ( 𝑖 ∈ ( 0 ... 𝑘 ) ∧ 𝑘 ∈ ℕ ) → 0 ≤ ( 𝑖 / 𝑘 ) )
114		elfzle2	⊢ ( 𝑖 ∈ ( 0 ... 𝑘 ) → 𝑖 ≤ 𝑘 )
115	114	adantr	⊢ ( ( 𝑖 ∈ ( 0 ... 𝑘 ) ∧ 𝑘 ∈ ℕ ) → 𝑖 ≤ 𝑘 )
116	105	adantr	⊢ ( ( 𝑖 ∈ ( 0 ... 𝑘 ) ∧ 𝑘 ∈ ℕ ) → 𝑖 ∈ ℝ )
117		1red	⊢ ( ( 𝑖 ∈ ( 0 ... 𝑘 ) ∧ 𝑘 ∈ ℕ ) → 1 ∈ ℝ )
118	110	adantl	⊢ ( ( 𝑖 ∈ ( 0 ... 𝑘 ) ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℝ+ )
119	116 117 118	ledivmuld	⊢ ( ( 𝑖 ∈ ( 0 ... 𝑘 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑖 / 𝑘 ) ≤ 1 ↔ 𝑖 ≤ ( 𝑘 · 1 ) ) )
120		nncn	⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℂ )
121	120	mulid1d	⊢ ( 𝑘 ∈ ℕ → ( 𝑘 · 1 ) = 𝑘 )
122	121	breq2d	⊢ ( 𝑘 ∈ ℕ → ( 𝑖 ≤ ( 𝑘 · 1 ) ↔ 𝑖 ≤ 𝑘 ) )
123	122	adantl	⊢ ( ( 𝑖 ∈ ( 0 ... 𝑘 ) ∧ 𝑘 ∈ ℕ ) → ( 𝑖 ≤ ( 𝑘 · 1 ) ↔ 𝑖 ≤ 𝑘 ) )
124	119 123	bitrd	⊢ ( ( 𝑖 ∈ ( 0 ... 𝑘 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑖 / 𝑘 ) ≤ 1 ↔ 𝑖 ≤ 𝑘 ) )
125	115 124	mpbird	⊢ ( ( 𝑖 ∈ ( 0 ... 𝑘 ) ∧ 𝑘 ∈ ℕ ) → ( 𝑖 / 𝑘 ) ≤ 1 )
126		elicc01	⊢ ( ( 𝑖 / 𝑘 ) ∈ ( 0 [,] 1 ) ↔ ( ( 𝑖 / 𝑘 ) ∈ ℝ ∧ 0 ≤ ( 𝑖 / 𝑘 ) ∧ ( 𝑖 / 𝑘 ) ≤ 1 ) )
127	107 113 125 126	syl3anbrc	⊢ ( ( 𝑖 ∈ ( 0 ... 𝑘 ) ∧ 𝑘 ∈ ℕ ) → ( 𝑖 / 𝑘 ) ∈ ( 0 [,] 1 ) )
128	127	ancoms	⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑖 ∈ ( 0 ... 𝑘 ) ) → ( 𝑖 / 𝑘 ) ∈ ( 0 [,] 1 ) )
129		elsni	⊢ ( 𝑗 ∈ { 𝑘 } → 𝑗 = 𝑘 )
130	129	oveq2d	⊢ ( 𝑗 ∈ { 𝑘 } → ( 𝑖 / 𝑗 ) = ( 𝑖 / 𝑘 ) )
131	130	eleq1d	⊢ ( 𝑗 ∈ { 𝑘 } → ( ( 𝑖 / 𝑗 ) ∈ ( 0 [,] 1 ) ↔ ( 𝑖 / 𝑘 ) ∈ ( 0 [,] 1 ) ) )
132	128 131	syl5ibrcom	⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑖 ∈ ( 0 ... 𝑘 ) ) → ( 𝑗 ∈ { 𝑘 } → ( 𝑖 / 𝑗 ) ∈ ( 0 [,] 1 ) ) )
133	132	impr	⊢ ( ( 𝑘 ∈ ℕ ∧ ( 𝑖 ∈ ( 0 ... 𝑘 ) ∧ 𝑗 ∈ { 𝑘 } ) ) → ( 𝑖 / 𝑗 ) ∈ ( 0 [,] 1 ) )
134	103 133	sylan	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ) ) ∧ ( 𝑖 ∈ ( 0 ... 𝑘 ) ∧ 𝑗 ∈ { 𝑘 } ) ) → ( 𝑖 / 𝑗 ) ∈ ( 0 [,] 1 ) )
135		simprr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ) ) → 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) )
136		vex	⊢ 𝑘 ∈ V
137	136	fconst	⊢ ( ( 1 ... 𝑁 ) × { 𝑘 } ) : ( 1 ... 𝑁 ) ⟶ { 𝑘 }
138	137	a1i	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ) ) → ( ( 1 ... 𝑁 ) × { 𝑘 } ) : ( 1 ... 𝑁 ) ⟶ { 𝑘 } )
139		fzfid	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ) ) → ( 1 ... 𝑁 ) ∈ Fin )
140		inidm	⊢ ( ( 1 ... 𝑁 ) ∩ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 )
141	134 135 138 139 139 140	off	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ) ) → ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 [,] 1 ) )
142	2	eleq2i	⊢ ( ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ∈ 𝐼 ↔ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ∈ ( ( 0 [,] 1 ) ↑m ( 1 ... 𝑁 ) ) )
143		ovex	⊢ ( 0 [,] 1 ) ∈ V
144		ovex	⊢ ( 1 ... 𝑁 ) ∈ V
145	143 144	elmap	⊢ ( ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ∈ ( ( 0 [,] 1 ) ↑m ( 1 ... 𝑁 ) ) ↔ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 [,] 1 ) )
146	142 145	bitri	⊢ ( ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ∈ 𝐼 ↔ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 [,] 1 ) )
147	141 146	sylibr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ) ) → ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ∈ 𝐼 )
148	147	3adantr3	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) → ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ∈ 𝐼 )
149		3anass	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ↔ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) )
150		ancom	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) ↔ ( ( 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) )
151	149 150	bitri	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ↔ ( ( 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) )
152		ffn	⊢ ( 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) → 𝑝 Fn ( 1 ... 𝑁 ) )
153	152	ad2antrl	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) → 𝑝 Fn ( 1 ... 𝑁 ) )
154		fnconstg	⊢ ( 𝑘 ∈ V → ( ( 1 ... 𝑁 ) × { 𝑘 } ) Fn ( 1 ... 𝑁 ) )
155	136 154	mp1i	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) → ( ( 1 ... 𝑁 ) × { 𝑘 } ) Fn ( 1 ... 𝑁 ) )
156		fzfid	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) → ( 1 ... 𝑁 ) ∈ Fin )
157		simplrr	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( 𝑝 ‘ 𝑛 ) = 𝑘 )
158	136	fvconst2	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → ( ( ( 1 ... 𝑁 ) × { 𝑘 } ) ‘ 𝑛 ) = 𝑘 )
159	158	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 1 ... 𝑁 ) × { 𝑘 } ) ‘ 𝑛 ) = 𝑘 )
160	153 155 156 156 140 157 159	ofval	⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ‘ 𝑛 ) = ( 𝑘 / 𝑘 ) )
161	160	anasss	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( ( 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ) → ( ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ‘ 𝑛 ) = ( 𝑘 / 𝑘 ) )
162	151 161	sylan2b	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) → ( ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ‘ 𝑛 ) = ( 𝑘 / 𝑘 ) )
163		nnne0	⊢ ( 𝑘 ∈ ℕ → 𝑘 ≠ 0 )
164	120 163	dividd	⊢ ( 𝑘 ∈ ℕ → ( 𝑘 / 𝑘 ) = 1 )
165	164	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) → ( 𝑘 / 𝑘 ) = 1 )
166	162 165	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) → ( ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ‘ 𝑛 ) = 1 )
167		ovex	⊢ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ∈ V
168		eleq1	⊢ ( 𝑧 = ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) → ( 𝑧 ∈ 𝐼 ↔ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ∈ 𝐼 ) )
169		fveq1	⊢ ( 𝑧 = ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) → ( 𝑧 ‘ 𝑛 ) = ( ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ‘ 𝑛 ) )
170	169	eqeq1d	⊢ ( 𝑧 = ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) → ( ( 𝑧 ‘ 𝑛 ) = 1 ↔ ( ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ‘ 𝑛 ) = 1 ) )
171	168 170	3anbi23d	⊢ ( 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 ) ) )
172	171	anbi2d	⊢ ( 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 ) ) ) )
173		fveq2	⊢ ( 𝑧 = ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) )
174	173	fveq1d	⊢ ( 𝑧 = ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) → ( ( 𝐹 ‘ 𝑧 ) ‘ 𝑛 ) = ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑛 ) )
175	174	breq2d	⊢ ( 𝑧 = ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) → ( 0 ≤ ( ( 𝐹 ‘ 𝑧 ) ‘ 𝑛 ) ↔ 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑛 ) ) )
176	172 175	imbi12d	⊢ ( 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 ) ) ) )
177	167 176 6	vtocl	⊢ ( ( 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 ) )
178	102 101 148 166 177	syl13anc	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) → 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑛 ) )
179		simpr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℕ )
180		simp3	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) → ( 𝑝 ‘ 𝑛 ) = 𝑘 )
181		neeq1	⊢ ( ( 𝑝 ‘ 𝑛 ) = 𝑘 → ( ( 𝑝 ‘ 𝑛 ) ≠ 0 ↔ 𝑘 ≠ 0 ) )
182	163 181	syl5ibrcom	⊢ ( 𝑘 ∈ ℕ → ( ( 𝑝 ‘ 𝑛 ) = 𝑘 → ( 𝑝 ‘ 𝑛 ) ≠ 0 ) )
183	182	imp	⊢ ( ( 𝑘 ∈ ℕ ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) → ( 𝑝 ‘ 𝑛 ) ≠ 0 )
184	179 180 183	syl2an	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝑘 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝑘 ) ) → ( 𝑝 ‘ 𝑛 ) ≠ 0 )
185		vex	⊢ 𝑛 ∈ V
186		fveq2	⊢ ( 𝑏 = 𝑛 → ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) = ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑛 ) )
187	186	breq2d	⊢ ( 𝑏 = 𝑛 → ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ↔ 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑛 ) ) )
188	68	neeq1d	⊢ ( 𝑏 = 𝑛 → ( ( 𝑝 ‘ 𝑏 ) ≠ 0 ↔ ( 𝑝 ‘ 𝑛 ) ≠ 0 ) )
189	187 188	anbi12d	⊢ ( 𝑏 = 𝑛 → ( ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) ↔ ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑛 ) ∧ ( 𝑝 ‘ 𝑛 ) ≠ 0 ) ) )
190	185 189	ralsn	⊢ ( ∀ 𝑏 ∈ { 𝑛 } ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) ↔ ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑛 ) ∧ ( 𝑝 ‘ 𝑛 ) ≠ 0 ) )
191	178 184 190	sylanbrc	⊢ ( ( ( 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 ) )
192	42	zcnd	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → 𝑛 ∈ ℂ )
193		1cnd	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → 1 ∈ ℂ )
194	192 193	subeq0ad	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → ( ( 𝑛 − 1 ) = 0 ↔ 𝑛 = 1 ) )
195	194	biimpcd	⊢ ( ( 𝑛 − 1 ) = 0 → ( 𝑛 ∈ ( 1 ... 𝑁 ) → 𝑛 = 1 ) )
196		1z	⊢ 1 ∈ ℤ
197		fzsn	⊢ ( 1 ∈ ℤ → ( 1 ... 1 ) = { 1 } )
198	196 197	ax-mp	⊢ ( 1 ... 1 ) = { 1 }
199		oveq2	⊢ ( 𝑛 = 1 → ( 1 ... 𝑛 ) = ( 1 ... 1 ) )
200		sneq	⊢ ( 𝑛 = 1 → { 𝑛 } = { 1 } )
201	198 199 200	3eqtr4a	⊢ ( 𝑛 = 1 → ( 1 ... 𝑛 ) = { 𝑛 } )
202	201	raleqdv	⊢ ( 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 ) ) )
203	202	biimprd	⊢ ( 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 ) ) )
204	195 203	syl6	⊢ ( ( 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 ) ) ) )
205		ralun	⊢ ( ( ∀ 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 ) )
206		npcan1	⊢ ( 𝑛 ∈ ℂ → ( ( 𝑛 − 1 ) + 1 ) = 𝑛 )
207	192 206	syl	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → ( ( 𝑛 − 1 ) + 1 ) = 𝑛 )
208		elfzuz	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → 𝑛 ∈ ( ℤ≥ ‘ 1 ) )
209	207 208	eqeltrd	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → ( ( 𝑛 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ 1 ) )
210		peano2zm	⊢ ( 𝑛 ∈ ℤ → ( 𝑛 − 1 ) ∈ ℤ )
211		uzid	⊢ ( ( 𝑛 − 1 ) ∈ ℤ → ( 𝑛 − 1 ) ∈ ( ℤ≥ ‘ ( 𝑛 − 1 ) ) )
212		peano2uz	⊢ ( ( 𝑛 − 1 ) ∈ ( ℤ≥ ‘ ( 𝑛 − 1 ) ) → ( ( 𝑛 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ ( 𝑛 − 1 ) ) )
213	42 210 211 212	4syl	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → ( ( 𝑛 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ ( 𝑛 − 1 ) ) )
214	207 213	eqeltrrd	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → 𝑛 ∈ ( ℤ≥ ‘ ( 𝑛 − 1 ) ) )
215		fzsplit2	⊢ ( ( ( ( 𝑛 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( 𝑛 − 1 ) ) ) → ( 1 ... 𝑛 ) = ( ( 1 ... ( 𝑛 − 1 ) ) ∪ ( ( ( 𝑛 − 1 ) + 1 ) ... 𝑛 ) ) )
216	209 214 215	syl2anc	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → ( 1 ... 𝑛 ) = ( ( 1 ... ( 𝑛 − 1 ) ) ∪ ( ( ( 𝑛 − 1 ) + 1 ) ... 𝑛 ) ) )
217	207	oveq1d	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → ( ( ( 𝑛 − 1 ) + 1 ) ... 𝑛 ) = ( 𝑛 ... 𝑛 ) )
218		fzsn	⊢ ( 𝑛 ∈ ℤ → ( 𝑛 ... 𝑛 ) = { 𝑛 } )
219	42 218	syl	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → ( 𝑛 ... 𝑛 ) = { 𝑛 } )
220	217 219	eqtrd	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → ( ( ( 𝑛 − 1 ) + 1 ) ... 𝑛 ) = { 𝑛 } )
221	220	uneq2d	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → ( ( 1 ... ( 𝑛 − 1 ) ) ∪ ( ( ( 𝑛 − 1 ) + 1 ) ... 𝑛 ) ) = ( ( 1 ... ( 𝑛 − 1 ) ) ∪ { 𝑛 } ) )
222	216 221	eqtrd	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) → ( 1 ... 𝑛 ) = ( ( 1 ... ( 𝑛 − 1 ) ) ∪ { 𝑛 } ) )
223	222	raleqdv	⊢ ( 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 ) ) )
224	205 223	syl5ibr	⊢ ( 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 ) ) )
225	224	expd	⊢ ( 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 ) ) ) )
226	225	com12	⊢ ( ∀ 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 ) ) ) )
227	226	adantl	⊢ ( ( ( 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 ) ) ) )
228	204 227	jaoi	⊢ ( ( ( 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 ) ) ) )
229	228	imdistand	⊢ ( ( ( 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 ) ) ) )
230	229	com12	⊢ ( ( 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 ) ) ) )
231		elun	⊢ ( ( 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 ) } ) )
232		ovex	⊢ ( 𝑛 − 1 ) ∈ V
233	232	elsn	⊢ ( ( 𝑛 − 1 ) ∈ { 0 } ↔ ( 𝑛 − 1 ) = 0 )
234		oveq2	⊢ ( 𝑎 = ( 𝑛 − 1 ) → ( 1 ... 𝑎 ) = ( 1 ... ( 𝑛 − 1 ) ) )
235	234	raleqdv	⊢ ( 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 ) ) )
236	235	elrab	⊢ ( ( 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 ) ) )
237	233 236	orbi12i	⊢ ( ( ( 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 ) ) ) )
238	231 237	bitri	⊢ ( ( 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 ) ) ) )
239		oveq2	⊢ ( 𝑎 = 𝑛 → ( 1 ... 𝑎 ) = ( 1 ... 𝑛 ) )
240	239	raleqdv	⊢ ( 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 ) ) )
241	240	elrab	⊢ ( 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 ) ) )
242	230 238 241	3imtr4g	⊢ ( ( 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 ) } ) )
243		elun2	⊢ ( 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 ) } ) )
244	242 243	syl6	⊢ ( ( 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 ) } ) ) )
245	101 191 244	syl2anc	⊢ ( ( ( 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 ) } ) ) )
246		fimaxre2	⊢ ( ( ( { 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 )
247	47 33 246	mp2an	⊢ ∃ 𝑖 ∈ ℝ ∀ 𝑗 ∈ ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) } ) 𝑗 ≤ 𝑖
248	47 38 247	3pm3.2i	⊢ ( ( { 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 )
249	248	suprubii	⊢ ( 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 , < ) )
250	245 249	syl6	⊢ ( ( ( 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 , < ) ) )
251		ltm1	⊢ ( 𝑛 ∈ ℝ → ( 𝑛 − 1 ) < 𝑛 )
252		peano2rem	⊢ ( 𝑛 ∈ ℝ → ( 𝑛 − 1 ) ∈ ℝ )
253	47 50	sselii	⊢ sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( 𝑝 ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ∈ ℝ
254		ltletr	⊢ ( ( ( 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 , < ) ) )
255	253 254	mp3an3	⊢ ( ( ( 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 , < ) ) )
256	252 255	mpancom	⊢ ( 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 , < ) ) )
257	251 256	mpand	⊢ ( 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 , < ) ) )
258	100 250 257	sylsyld	⊢ ( ( ( 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 , < ) ) )
259	253	ltnri	⊢ ¬ 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 , < )
260		breq1	⊢ ( 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 , < ) ) )
261	259 260	mtbii	⊢ ( 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 , < ) )
262	261	necon2ai	⊢ ( ( 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 ) )
263	258 262	syl6	⊢ ( ( ( 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 ) ) )
264		eleq1	⊢ ( 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 ) } ) ) )
265	50 264	mpbii	⊢ ( 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 ) } ) )
266	265	necon3bi	⊢ ( ¬ ( 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 ) )
267	263 266	pm2.61d1	⊢ ( ( ( 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 ) )
268	7 17 53 97 267 179	poimirlem28	⊢ ( ( 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 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) )
269		nn0ex	⊢ ℕ0 ∈ V
270		fzo0ssnn0	⊢ ( 0 ..^ 𝑘 ) ⊆ ℕ0
271		mapss	⊢ ( ( ℕ0 ∈ V ∧ ( 0 ..^ 𝑘 ) ⊆ ℕ0 ) → ( ( 0 ..^ 𝑘 ) ↑m ( 1 ... 𝑁 ) ) ⊆ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) )
272	269 270 271	mp2an	⊢ ( ( 0 ..^ 𝑘 ) ↑m ( 1 ... 𝑁 ) ) ⊆ ( ℕ0 ↑m ( 1 ... 𝑁 ) )
273		xpss1	⊢ ( ( ( 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 ) } ) )
274	272 273	ax-mp	⊢ ( ( ( 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 ) } )
275	274	sseli	⊢ ( 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 ) } ) )
276		xp1st	⊢ ( 𝑠 ∈ ( ( ( 0 ..^ 𝑘 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) → ( 1st ‘ 𝑠 ) ∈ ( ( 0 ..^ 𝑘 ) ↑m ( 1 ... 𝑁 ) ) )
277		elmapi	⊢ ( ( 1st ‘ 𝑠 ) ∈ ( ( 0 ..^ 𝑘 ) ↑m ( 1 ... 𝑁 ) ) → ( 1st ‘ 𝑠 ) : ( 1 ... 𝑁 ) ⟶ ( 0 ..^ 𝑘 ) )
278		frn	⊢ ( ( 1st ‘ 𝑠 ) : ( 1 ... 𝑁 ) ⟶ ( 0 ..^ 𝑘 ) → ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) )
279	276 277 278	3syl	⊢ ( 𝑠 ∈ ( ( ( 0 ..^ 𝑘 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) → ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) )
280	275 279	jca	⊢ ( 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 ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ) )
281	280	anim1i	⊢ ( ( 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 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) → ( ( 𝑠 ∈ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∧ ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) )
282		anass	⊢ ( ( ( 𝑠 ∈ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∧ ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ↔ ( 𝑠 ∈ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∧ ( ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ) )
283	281 282	sylib	⊢ ( ( 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 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) → ( 𝑠 ∈ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∧ ( ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ) )
284	283	reximi2	⊢ ( ∃ 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 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) → ∃ 𝑠 ∈ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ( ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) )
285	268 284	syl	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ∃ 𝑠 ∈ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ( ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) )
286	285	ralrimiva	⊢ ( 𝜑 → ∀ 𝑘 ∈ ℕ ∃ 𝑠 ∈ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ( ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) )
287		nnex	⊢ ℕ ∈ V
288	144 269	ixpconst	⊢ X 𝑛 ∈ ( 1 ... 𝑁 ) ℕ0 = ( ℕ0 ↑m ( 1 ... 𝑁 ) )
289		omelon	⊢ ω ∈ On
290		nn0ennn	⊢ ℕ0 ≈ ℕ
291		nnenom	⊢ ℕ ≈ ω
292	290 291	entr2i	⊢ ω ≈ ℕ0
293		isnumi	⊢ ( ( ω ∈ On ∧ ω ≈ ℕ0 ) → ℕ0 ∈ dom card )
294	289 292 293	mp2an	⊢ ℕ0 ∈ dom card
295	294	rgenw	⊢ ∀ 𝑛 ∈ ( 1 ... 𝑁 ) ℕ0 ∈ dom card
296		finixpnum	⊢ ( ( ( 1 ... 𝑁 ) ∈ Fin ∧ ∀ 𝑛 ∈ ( 1 ... 𝑁 ) ℕ0 ∈ dom card ) → X 𝑛 ∈ ( 1 ... 𝑁 ) ℕ0 ∈ dom card )
297	29 295 296	mp2an	⊢ X 𝑛 ∈ ( 1 ... 𝑁 ) ℕ0 ∈ dom card
298	288 297	eqeltrri	⊢ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∈ dom card
299	144 144	mapval	⊢ ( ( 1 ... 𝑁 ) ↑m ( 1 ... 𝑁 ) ) = { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) ⟶ ( 1 ... 𝑁 ) }
300		mapfi	⊢ ( ( ( 1 ... 𝑁 ) ∈ Fin ∧ ( 1 ... 𝑁 ) ∈ Fin ) → ( ( 1 ... 𝑁 ) ↑m ( 1 ... 𝑁 ) ) ∈ Fin )
301	29 29 300	mp2an	⊢ ( ( 1 ... 𝑁 ) ↑m ( 1 ... 𝑁 ) ) ∈ Fin
302	299 301	eqeltrri	⊢ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) ⟶ ( 1 ... 𝑁 ) } ∈ Fin
303		f1of	⊢ ( 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → 𝑓 : ( 1 ... 𝑁 ) ⟶ ( 1 ... 𝑁 ) )
304	303	ss2abi	⊢ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ⊆ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) ⟶ ( 1 ... 𝑁 ) }
305		ssfi	⊢ ( ( { 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 )
306	302 304 305	mp2an	⊢ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ∈ Fin
307		finnum	⊢ ( { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ∈ Fin → { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ∈ dom card )
308	306 307	ax-mp	⊢ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ∈ dom card
309		xpnum	⊢ ( ( ( 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 )
310	298 308 309	mp2an	⊢ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∈ dom card
311		ssrab2	⊢ { 𝑠 ∈ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ( ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) } ⊆ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } )
312	311	rgenw	⊢ ∀ 𝑘 ∈ ℕ { 𝑠 ∈ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ( ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) } ⊆ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } )
313		ss2iun	⊢ ( ∀ 𝑘 ∈ ℕ { 𝑠 ∈ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ( ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 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 ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) } ⊆ ∪ 𝑘 ∈ ℕ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) )
314	312 313	ax-mp	⊢ ∪ 𝑘 ∈ ℕ { 𝑠 ∈ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ( ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) } ⊆ ∪ 𝑘 ∈ ℕ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } )
315		1nn	⊢ 1 ∈ ℕ
316		ne0i	⊢ ( 1 ∈ ℕ → ℕ ≠ ∅ )
317		iunconst	⊢ ( 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 ) } ) )
318	315 316 317	mp2b	⊢ ∪ 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 ) } )
319	314 318	sseqtri	⊢ ∪ 𝑘 ∈ ℕ { 𝑠 ∈ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ( ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) } ⊆ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } )
320		ssnum	⊢ ( ( ( ( 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 ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 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 ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) } ∈ dom card )
321	310 319 320	mp2an	⊢ ∪ 𝑘 ∈ ℕ { 𝑠 ∈ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ( ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) } ∈ dom card
322		fveq2	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( 1st ‘ 𝑠 ) = ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) )
323	322	rneqd	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ran ( 1st ‘ 𝑠 ) = ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) )
324	323	sseq1d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ↔ ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ) )
325		fveq2	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( 2nd ‘ 𝑠 ) = ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) )
326	325	imaeq1d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) = ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) )
327	326	xpeq1d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) )
328	325	imaeq1d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) )
329	328	xpeq1d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) )
330	327 329	uneq12d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
331	322 330	oveq12d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
332	331	fvoveq1d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) = ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) )
333	332	fveq1d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) = ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) )
334	333	breq2d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ↔ 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ) )
335	331	fveq1d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) = ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) )
336	335	neeq1d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ↔ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) )
337	334 336	anbi12d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) ↔ ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) ) )
338	337	ralbidv	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) ↔ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) ) )
339	338	rabbidv	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } = { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } )
340	339	uneq2d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) = ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) )
341	340	supeq1d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) )
342	341	eqeq2d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ↔ 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) )
343	342	rexbidv	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ↔ ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) )
344	343	ralbidv	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 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 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) )
345	324 344	anbi12d	⊢ ( 𝑠 = ( 𝑔 ‘ 𝑘 ) → ( ( ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ↔ ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ) )
346	345	ac6num	⊢ ( ( ℕ ∈ V ∧ ∪ 𝑘 ∈ ℕ { 𝑠 ∈ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ( ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 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 ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 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 ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ) )
347	287 321 346	mp3an12	⊢ ( ∀ 𝑘 ∈ ℕ ∃ 𝑠 ∈ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ( ran ( 1st ‘ 𝑠 ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ 𝑠 ) ∘f + ( ( ( ( 2nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ 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 ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ) )
348	286 347	syl	⊢ ( 𝜑 → ∃ 𝑔 ( 𝑔 : ℕ ⟶ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∧ ∀ 𝑘 ∈ ℕ ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ) )
349	1	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ) ∧ ∀ 𝑘 ∈ ℕ ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ) → 𝑁 ∈ ℕ )
350	4	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ) ∧ ∀ 𝑘 ∈ ℕ ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ) → 𝐹 ∈ ( ( 𝑅 ↾t 𝐼 ) Cn 𝑅 ) )
351		eqid	⊢ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑛 ) = ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑛 )
352		simplr	⊢ ( ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ) ∧ ∀ 𝑘 ∈ ℕ ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ) → 𝑔 : ℕ ⟶ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) )
353		simpl	⊢ ( ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) → ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) )
354	353	ralimi	⊢ ( ∀ 𝑘 ∈ ℕ ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) → ∀ 𝑘 ∈ ℕ ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) )
355	354	adantl	⊢ ( ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ) ∧ ∀ 𝑘 ∈ ℕ ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ) → ∀ 𝑘 ∈ ℕ ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) )
356		2fveq3	⊢ ( 𝑘 = 𝑝 → ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) = ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) )
357	356	rneqd	⊢ ( 𝑘 = 𝑝 → ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) = ran ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) )
358		oveq2	⊢ ( 𝑘 = 𝑝 → ( 0 ..^ 𝑘 ) = ( 0 ..^ 𝑝 ) )
359	357 358	sseq12d	⊢ ( 𝑘 = 𝑝 → ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ↔ ran ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ⊆ ( 0 ..^ 𝑝 ) ) )
360	359	rspccva	⊢ ( ( ∀ 𝑘 ∈ ℕ ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ 𝑝 ∈ ℕ ) → ran ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ⊆ ( 0 ..^ 𝑝 ) )
361	355 360	sylan	⊢ ( ( ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ) ∧ ∀ 𝑘 ∈ ℕ ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ) ∧ 𝑝 ∈ ℕ ) → ran ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ⊆ ( 0 ..^ 𝑝 ) )
362		simpll	⊢ ( ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ) ∧ ∀ 𝑘 ∈ ℕ ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ) → 𝜑 )
363	362 5	sylan	⊢ ( ( ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ) ∧ ∀ 𝑘 ∈ ℕ ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑧 ∈ 𝐼 ∧ ( 𝑧 ‘ 𝑛 ) = 0 ) ) → ( ( 𝐹 ‘ 𝑧 ) ‘ 𝑛 ) ≤ 0 )
364		eqid	⊢ ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
365		simpr	⊢ ( ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 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 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) )
366	365	ralimi	⊢ ( ∀ 𝑘 ∈ ℕ ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 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 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) )
367	366	adantl	⊢ ( ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ) ∧ ∀ 𝑘 ∈ ℕ ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 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 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) )
368		2fveq3	⊢ ( 𝑘 = 𝑝 → ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) = ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) )
369	368	imaeq1d	⊢ ( 𝑘 = 𝑝 → ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) = ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) )
370	369	xpeq1d	⊢ ( 𝑘 = 𝑝 → ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) )
371	368	imaeq1d	⊢ ( 𝑘 = 𝑝 → ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) )
372	371	xpeq1d	⊢ ( 𝑘 = 𝑝 → ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) )
373	370 372	uneq12d	⊢ ( 𝑘 = 𝑝 → ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
374	356 373	oveq12d	⊢ ( 𝑘 = 𝑝 → ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
375		sneq	⊢ ( 𝑘 = 𝑝 → { 𝑘 } = { 𝑝 } )
376	375	xpeq2d	⊢ ( 𝑘 = 𝑝 → ( ( 1 ... 𝑁 ) × { 𝑘 } ) = ( ( 1 ... 𝑁 ) × { 𝑝 } ) )
377	374 376	oveq12d	⊢ ( 𝑘 = 𝑝 → ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) = ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) )
378	377	fveq2d	⊢ ( 𝑘 = 𝑝 → ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) = ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) )
379	378	fveq1d	⊢ ( 𝑘 = 𝑝 → ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) = ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) )
380	379	breq2d	⊢ ( 𝑘 = 𝑝 → ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ↔ 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ) )
381	374	fveq1d	⊢ ( 𝑘 = 𝑝 → ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) = ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) )
382	381	neeq1d	⊢ ( 𝑘 = 𝑝 → ( ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ↔ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) )
383	380 382	anbi12d	⊢ ( 𝑘 = 𝑝 → ( ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) ↔ ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) ) )
384	383	ralbidv	⊢ ( 𝑘 = 𝑝 → ( ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) ↔ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) ) )
385	384	rabbidv	⊢ ( 𝑘 = 𝑝 → { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } = { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } )
386	385	uneq2d	⊢ ( 𝑘 = 𝑝 → ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) = ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) )
387	386	supeq1d	⊢ ( 𝑘 = 𝑝 → sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) )
388	387	eqeq2d	⊢ ( 𝑘 = 𝑝 → ( 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ↔ 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) )
389	388	rexbidv	⊢ ( 𝑘 = 𝑝 → ( ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ↔ ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) )
390		eqeq1	⊢ ( 𝑖 = 𝑞 → ( 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ↔ 𝑞 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) )
391	390	rexbidv	⊢ ( 𝑖 = 𝑞 → ( ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ↔ ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑞 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) )
392		oveq2	⊢ ( 𝑗 = 𝑚 → ( 1 ... 𝑗 ) = ( 1 ... 𝑚 ) )
393	392	imaeq2d	⊢ ( 𝑗 = 𝑚 → ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) = ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) )
394	393	xpeq1d	⊢ ( 𝑗 = 𝑚 → ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) )
395		oveq1	⊢ ( 𝑗 = 𝑚 → ( 𝑗 + 1 ) = ( 𝑚 + 1 ) )
396	395	oveq1d	⊢ ( 𝑗 = 𝑚 → ( ( 𝑗 + 1 ) ... 𝑁 ) = ( ( 𝑚 + 1 ) ... 𝑁 ) )
397	396	imaeq2d	⊢ ( 𝑗 = 𝑚 → ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) )
398	397	xpeq1d	⊢ ( 𝑗 = 𝑚 → ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) )
399	394 398	uneq12d	⊢ ( 𝑗 = 𝑚 → ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
400	399	oveq2d	⊢ ( 𝑗 = 𝑚 → ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
401	400	fvoveq1d	⊢ ( 𝑗 = 𝑚 → ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) = ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) )
402	401	fveq1d	⊢ ( 𝑗 = 𝑚 → ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) = ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) )
403	402	breq2d	⊢ ( 𝑗 = 𝑚 → ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ↔ 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ) )
404	400	fveq1d	⊢ ( 𝑗 = 𝑚 → ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) = ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) )
405	404	neeq1d	⊢ ( 𝑗 = 𝑚 → ( ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ↔ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) )
406	403 405	anbi12d	⊢ ( 𝑗 = 𝑚 → ( ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) ↔ ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) ) )
407	406	ralbidv	⊢ ( 𝑗 = 𝑚 → ( ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) ↔ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) ) )
408	407	rabbidv	⊢ ( 𝑗 = 𝑚 → { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } = { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } )
409	408	uneq2d	⊢ ( 𝑗 = 𝑚 → ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) = ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) )
410	409	supeq1d	⊢ ( 𝑗 = 𝑚 → sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) )
411	410	eqeq2d	⊢ ( 𝑗 = 𝑚 → ( 𝑞 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ↔ 𝑞 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) )
412	411	cbvrexvw	⊢ ( ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑞 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ↔ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑞 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) )
413	391 412	bitrdi	⊢ ( 𝑖 = 𝑞 → ( ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ↔ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑞 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) )
414	389 413	rspc2v	⊢ ( ( 𝑝 ∈ ℕ ∧ 𝑞 ∈ ( 0 ... 𝑁 ) ) → ( ∀ 𝑘 ∈ ℕ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) → ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑞 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) )
415	367 414	mpan9	⊢ ( ( ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ) ∧ ∀ 𝑘 ∈ ℕ ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 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 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) )
416	349 2 3 350 363 364 352 361 415	poimirlem31	⊢ ( ( ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ) ∧ ∀ 𝑘 ∈ ℕ ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ) ∧ ( 𝑝 ∈ ℕ ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑟 ∈ { ≤ , ◡ ≤ } ) ) → ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 0 𝑟 ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑝 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( 1 ... 𝑚 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑝 ) ) “ ( ( 𝑚 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑝 } ) ) ) ‘ 𝑛 ) )
417	349 2 3 350 351 352 361 416	poimirlem30	⊢ ( ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ) ∧ ∀ 𝑘 ∈ ℕ ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ) → ∃ 𝑐 ∈ 𝐼 ∀ 𝑛 ∈ ( 1 ... 𝑁 ) ∀ 𝑣 ∈ ( 𝑅 ↾t 𝐼 ) ( 𝑐 ∈ 𝑣 → ∀ 𝑟 ∈ { ≤ , ◡ ≤ } ∃ 𝑧 ∈ 𝑣 0 𝑟 ( ( 𝐹 ‘ 𝑧 ) ‘ 𝑛 ) ) )
418	417	anasss	⊢ ( ( 𝜑 ∧ ( 𝑔 : ℕ ⟶ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∧ ∀ 𝑘 ∈ ℕ ( ran ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ⊆ ( 0 ..^ 𝑘 ) ∧ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = sup ( ( { 0 } ∪ { 𝑎 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑏 ∈ ( 1 ... 𝑎 ) ( 0 ≤ ( ( 𝐹 ‘ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∘f / ( ( 1 ... 𝑁 ) × { 𝑘 } ) ) ) ‘ 𝑏 ) ∧ ( ( ( 1st ‘ ( 𝑔 ‘ 𝑘 ) ) ∘f + ( ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 𝑔 ‘ 𝑘 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑏 ) ≠ 0 ) } ) , ℝ , < ) ) ) ) → ∃ 𝑐 ∈ 𝐼 ∀ 𝑛 ∈ ( 1 ... 𝑁 ) ∀ 𝑣 ∈ ( 𝑅 ↾t 𝐼 ) ( 𝑐 ∈ 𝑣 → ∀ 𝑟 ∈ { ≤ , ◡ ≤ } ∃ 𝑧 ∈ 𝑣 0 𝑟 ( ( 𝐹 ‘ 𝑧 ) ‘ 𝑛 ) ) )
419	348 418	exlimddv	⊢ ( 𝜑 → ∃ 𝑐 ∈ 𝐼 ∀ 𝑛 ∈ ( 1 ... 𝑁 ) ∀ 𝑣 ∈ ( 𝑅 ↾t 𝐼 ) ( 𝑐 ∈ 𝑣 → ∀ 𝑟 ∈ { ≤ , ◡ ≤ } ∃ 𝑧 ∈ 𝑣 0 𝑟 ( ( 𝐹 ‘ 𝑧 ) ‘ 𝑛 ) ) )