poimirlem29 - Mathbox for Brendan Leahy

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Theorem poimirlem29 34915

Description: Lemma for poimir 34919 connecting cubes of the tessellation to neighborhoods. (Contributed by Brendan Leahy, 21-Aug-2020.)

**Hypotheses**
Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimir.i	⊢ 𝐼 = ((0[,]1) ↑_m (1...𝑁))
poimir.r	⊢ 𝑅 = (∏_t‘((1...𝑁) × {(topGen‘ran (,))}))
poimir.1	⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
poimirlem30.x	⊢ 𝑋 = ((𝐹‘(((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)
poimirlem30.2	⊢ (𝜑 → 𝐺:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
poimirlem30.3	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran (1^st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘))
poimirlem30.4	⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ^◡ ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟𝑋)

**Assertion**
Ref	Expression
poimirlem29	⊢ (𝜑 → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝐶 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))))

Distinct variable groups: 𝑓,𝑖,𝑗,𝑘,𝑚,𝑛,𝑧 𝜑,𝑗,𝑚,𝑛 𝑗,𝐹,𝑚,𝑛 𝑗,𝑁,𝑚,𝑛 𝜑,𝑖,𝑘 𝑓,𝑁,𝑖,𝑘 𝜑,𝑧 𝑓,𝐹,𝑘,𝑧 𝑧,𝑁 𝐶,𝑖,𝑘,𝑚,𝑛,𝑧 𝑖,𝑟,𝑣,𝑗,𝑘,𝑚,𝑛,𝑧,𝜑 𝑓,𝑟,𝑣 𝑖,𝐹,𝑟,𝑣 𝑓,𝐺,𝑖,𝑗,𝑘,𝑚,𝑛,𝑟,𝑣,𝑧 𝑓,𝐼,𝑖,𝑗,𝑘,𝑚,𝑛,𝑟,𝑣,𝑧 𝑁,𝑟,𝑣 𝑅,𝑓,𝑖,𝑗,𝑘,𝑚,𝑛,𝑟,𝑣,𝑧 𝑓,𝑋,𝑖,𝑚,𝑟,𝑣,𝑧 𝐶,𝑓,𝑗,𝑣 Allowed substitution hints: 𝜑(𝑓) 𝐶(𝑟) 𝑋(𝑗,𝑘,𝑛)

Proof of Theorem poimirlem29 Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		poimir.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (∏_t‘((1...𝑁) × {(topGen‘ran (,))}))
2		fzfi 13334	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ∈ Fin
3		retop 23364	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) ∈ Top
4	3	fconst6 6564	. . . . . . . . . . . . 13 ⊢ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top
5		pttop 22184	. . . . . . . . . . . . 13 ⊢ (((1...𝑁) ∈ Fin ∧ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top) → (∏_t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Top)
6	2, 4, 5	mp2an 690	. . . . . . . . . . . 12 ⊢ (∏_t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Top
7	1, 6	eqeltri 2909	. . . . . . . . . . 11 ⊢ 𝑅 ∈ Top
8		poimir.i	. . . . . . . . . . . 12 ⊢ 𝐼 = ((0[,]1) ↑_m (1...𝑁))
9		ovex 7183	. . . . . . . . . . . 12 ⊢ ((0[,]1) ↑_m (1...𝑁)) ∈ V
10	8, 9	eqeltri 2909	. . . . . . . . . . 11 ⊢ 𝐼 ∈ V
11		elrest 16695	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝐼 ∈ V) → (𝑣 ∈ (𝑅 ↾_t 𝐼) ↔ ∃𝑧 ∈ 𝑅 𝑣 = (𝑧 ∩ 𝐼)))
12	7, 10, 11	mp2an 690	. . . . . . . . . 10 ⊢ (𝑣 ∈ (𝑅 ↾_t 𝐼) ↔ ∃𝑧 ∈ 𝑅 𝑣 = (𝑧 ∩ 𝐼))
13		eqid 2821	. . . . . . . . . . . . . 14 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
14	1, 13	ptrecube 34886	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑅 ∧ 𝐶 ∈ 𝑧) → ∃𝑐 ∈ ℝ⁺ X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ⊆ 𝑧)
15	14	ex 415	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑅 → (𝐶 ∈ 𝑧 → ∃𝑐 ∈ ℝ⁺ X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ⊆ 𝑧))
16		inss1 4205	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∩ 𝐼) ⊆ 𝑧
17		sseq1 3992	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑧 ∩ 𝐼) → (𝑣 ⊆ 𝑧 ↔ (𝑧 ∩ 𝐼) ⊆ 𝑧))
18	16, 17	mpbiri 260	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑧 ∩ 𝐼) → 𝑣 ⊆ 𝑧)
19	18	sseld 3966	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑧 ∩ 𝐼) → (𝐶 ∈ 𝑣 → 𝐶 ∈ 𝑧))
20		ssrin 4210	. . . . . . . . . . . . . . 15 ⊢ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ⊆ 𝑧 → (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ (𝑧 ∩ 𝐼))
21		sseq2 3993	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑧 ∩ 𝐼) → ((X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣 ↔ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ (𝑧 ∩ 𝐼)))
22	20, 21	syl5ibr 248	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑧 ∩ 𝐼) → (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ⊆ 𝑧 → (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣))
23	22	reximdv 3273	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑧 ∩ 𝐼) → (∃𝑐 ∈ ℝ⁺ X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ⊆ 𝑧 → ∃𝑐 ∈ ℝ⁺ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣))
24	19, 23	imim12d 81	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑧 ∩ 𝐼) → ((𝐶 ∈ 𝑧 → ∃𝑐 ∈ ℝ⁺ X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ⊆ 𝑧) → (𝐶 ∈ 𝑣 → ∃𝑐 ∈ ℝ⁺ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣)))
25	15, 24	syl5com 31	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑅 → (𝑣 = (𝑧 ∩ 𝐼) → (𝐶 ∈ 𝑣 → ∃𝑐 ∈ ℝ⁺ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣)))
26	25	rexlimiv 3280	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑅 𝑣 = (𝑧 ∩ 𝐼) → (𝐶 ∈ 𝑣 → ∃𝑐 ∈ ℝ⁺ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣))
27	12, 26	sylbi 219	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝑅 ↾_t 𝐼) → (𝐶 ∈ 𝑣 → ∃𝑐 ∈ ℝ⁺ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣))
28	27	imp 409	. . . . . . . 8 ⊢ ((𝑣 ∈ (𝑅 ↾_t 𝐼) ∧ 𝐶 ∈ 𝑣) → ∃𝑐 ∈ ℝ⁺ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣)
29	28	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝑅 ↾_t 𝐼) ∧ 𝐶 ∈ 𝑣)) → ∃𝑐 ∈ ℝ⁺ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣)
30		resttop 21762	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝐼 ∈ V) → (𝑅 ↾_t 𝐼) ∈ Top)
31	7, 10, 30	mp2an 690	. . . . . . . . . 10 ⊢ (𝑅 ↾_t 𝐼) ∈ Top
32		reex 10622	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ V
33		unitssre 12879	. . . . . . . . . . . . . 14 ⊢ (0[,]1) ⊆ ℝ
34		mapss 8447	. . . . . . . . . . . . . 14 ⊢ ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁)))
35	32, 33, 34	mp2an 690	. . . . . . . . . . . . 13 ⊢ ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁))
36	8, 35	eqsstri 4001	. . . . . . . . . . . 12 ⊢ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))
37		ovex 7183	. . . . . . . . . . . . . 14 ⊢ (1...𝑁) ∈ V
38		uniretop 23365	. . . . . . . . . . . . . . 15 ⊢ ℝ = ∪ (topGen‘ran (,))
39	1, 38	ptuniconst 22200	. . . . . . . . . . . . . 14 ⊢ (((1...𝑁) ∈ V ∧ (topGen‘ran (,)) ∈ Top) → (ℝ ↑_m (1...𝑁)) = ∪ 𝑅)
40	37, 3, 39	mp2an 690	. . . . . . . . . . . . 13 ⊢ (ℝ ↑_m (1...𝑁)) = ∪ 𝑅
41	40	restuni 21764	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Top ∧ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))) → 𝐼 = ∪ (𝑅 ↾_t 𝐼))
42	7, 36, 41	mp2an 690	. . . . . . . . . . 11 ⊢ 𝐼 = ∪ (𝑅 ↾_t 𝐼)
43	42	eltopss 21509	. . . . . . . . . 10 ⊢ (((𝑅 ↾_t 𝐼) ∈ Top ∧ 𝑣 ∈ (𝑅 ↾_t 𝐼)) → 𝑣 ⊆ 𝐼)
44	31, 43	mpan 688	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝑅 ↾_t 𝐼) → 𝑣 ⊆ 𝐼)
45	44	sselda 3967	. . . . . . . 8 ⊢ ((𝑣 ∈ (𝑅 ↾_t 𝐼) ∧ 𝐶 ∈ 𝑣) → 𝐶 ∈ 𝐼)
46		2rp 12388	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ⁺
47		rpdivcl 12408	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℝ⁺ ∧ 𝑐 ∈ ℝ⁺) → (2 / 𝑐) ∈ ℝ⁺)
48	46, 47	mpan 688	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ ℝ⁺ → (2 / 𝑐) ∈ ℝ⁺)
49	48	rpred 12425	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ℝ⁺ → (2 / 𝑐) ∈ ℝ)
50		ceicl 13205	. . . . . . . . . . . . . . 15 ⊢ ((2 / 𝑐) ∈ ℝ → -(⌊‘-(2 / 𝑐)) ∈ ℤ)
51	49, 50	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ℝ⁺ → -(⌊‘-(2 / 𝑐)) ∈ ℤ)
52		0red 10638	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ℝ⁺ → 0 ∈ ℝ)
53	51	zred 12081	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ℝ⁺ → -(⌊‘-(2 / 𝑐)) ∈ ℝ)
54	48	rpgt0d 12428	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ℝ⁺ → 0 < (2 / 𝑐))
55		ceige 13207	. . . . . . . . . . . . . . . 16 ⊢ ((2 / 𝑐) ∈ ℝ → (2 / 𝑐) ≤ -(⌊‘-(2 / 𝑐)))
56	49, 55	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ℝ⁺ → (2 / 𝑐) ≤ -(⌊‘-(2 / 𝑐)))
57	52, 49, 53, 54, 56	ltletrd 10794	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ℝ⁺ → 0 < -(⌊‘-(2 / 𝑐)))
58		elnnz 11985	. . . . . . . . . . . . . 14 ⊢ (-(⌊‘-(2 / 𝑐)) ∈ ℕ ↔ (-(⌊‘-(2 / 𝑐)) ∈ ℤ ∧ 0 < -(⌊‘-(2 / 𝑐))))
59	51, 57, 58	sylanbrc 585	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ℝ⁺ → -(⌊‘-(2 / 𝑐)) ∈ ℕ)
60		fveq2 6665	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = -(⌊‘-(2 / 𝑐)) → (ℤ_≥‘𝑖) = (ℤ_≥‘-(⌊‘-(2 / 𝑐))))
61		oveq2 7158	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = -(⌊‘-(2 / 𝑐)) → (1 / 𝑖) = (1 / -(⌊‘-(2 / 𝑐))))
62	61	oveq2d 7166	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = -(⌊‘-(2 / 𝑐)) → ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) = ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))))
63	62	eleq2d 2898	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = -(⌊‘-(2 / 𝑐)) → ((((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐))))))
64	63	ralbidv 3197	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = -(⌊‘-(2 / 𝑐)) → (∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ ∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐))))))
65	60, 64	rexeqbidv 3403	. . . . . . . . . . . . . 14 ⊢ (𝑖 = -(⌊‘-(2 / 𝑐)) → (∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ ∃𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐))))))
66	65	rspcv 3618	. . . . . . . . . . . . 13 ⊢ (-(⌊‘-(2 / 𝑐)) ∈ ℕ → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∃𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐))))))
67	59, 66	syl 17	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ℝ⁺ → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∃𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐))))))
68	67	adantl 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∃𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐))))))
69		uznnssnn 12289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (-(⌊‘-(2 / 𝑐)) ∈ ℕ → (ℤ_≥‘-(⌊‘-(2 / 𝑐))) ⊆ ℕ)
70	59, 69	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℝ⁺ → (ℤ_≥‘-(⌊‘-(2 / 𝑐))) ⊆ ℕ)
71	70	sseld 3966	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 ∈ ℝ⁺ → (𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐))) → 𝑘 ∈ ℕ))
72	71	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) → (𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐))) → 𝑘 ∈ ℕ))
73	72	imdistani 571	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ))
74	59	nnrpd 12423	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑐 ∈ ℝ⁺ → -(⌊‘-(2 / 𝑐)) ∈ ℝ⁺)
75	48, 74	lerecd 12444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 ∈ ℝ⁺ → ((2 / 𝑐) ≤ -(⌊‘-(2 / 𝑐)) ↔ (1 / -(⌊‘-(2 / 𝑐))) ≤ (1 / (2 / 𝑐))))
76	56, 75	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℝ⁺ → (1 / -(⌊‘-(2 / 𝑐))) ≤ (1 / (2 / 𝑐)))
77		rpcn 12393	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 ∈ ℝ⁺ → 𝑐 ∈ ℂ)
78		rpne0 12399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 ∈ ℝ⁺ → 𝑐 ≠ 0)
79		2cn 11706	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 2 ∈ ℂ
80		2ne0 11735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 2 ≠ 0
81		recdiv 11340	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((2 ∈ ℂ ∧ 2 ≠ 0) ∧ (𝑐 ∈ ℂ ∧ 𝑐 ≠ 0)) → (1 / (2 / 𝑐)) = (𝑐 / 2))
82	79, 80, 81	mpanl12 700	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑐 ∈ ℂ ∧ 𝑐 ≠ 0) → (1 / (2 / 𝑐)) = (𝑐 / 2))
83	77, 78, 82	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℝ⁺ → (1 / (2 / 𝑐)) = (𝑐 / 2))
84	76, 83	breqtrd 5085	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 ∈ ℝ⁺ → (1 / -(⌊‘-(2 / 𝑐))) ≤ (𝑐 / 2))
85	84	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (1 / -(⌊‘-(2 / 𝑐))) ≤ (𝑐 / 2))
86		elmapi 8422	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝐶 ∈ ((0[,]1) ↑_m (1...𝑁)) → 𝐶:(1...𝑁)⟶(0[,]1))
87	86, 8	eleq2s 2931	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝐶 ∈ 𝐼 → 𝐶:(1...𝑁)⟶(0[,]1))
88	87	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝐶:(1...𝑁)⟶(0[,]1))
89	88	ffvelrnda 6846	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (𝐶‘𝑚) ∈ (0[,]1))
90	33, 89	sseldi 3965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (𝐶‘𝑚) ∈ ℝ)
91		simp-4l 781	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝜑)
92		simplr 767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝑘 ∈ ℕ)
93	91, 92	jca 514	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (𝜑 ∧ 𝑘 ∈ ℕ))
94		poimirlem30.2	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 𝐺:ℕ⟶((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
95	94	ffvelrnda 6846	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
96		xp1st 7715	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐺‘𝑘) ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^st ‘(𝐺‘𝑘)) ∈ (ℕ₀ ↑_m (1...𝑁)))
97		elmapi 8422	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((1^st ‘(𝐺‘𝑘)) ∈ (ℕ₀ ↑_m (1...𝑁)) → (1^st ‘(𝐺‘𝑘)):(1...𝑁)⟶ℕ₀)
98	95, 96, 97	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐺‘𝑘)):(1...𝑁)⟶ℕ₀)
99	98	ffvelrnda 6846	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((1^st ‘(𝐺‘𝑘))‘𝑚) ∈ ℕ₀)
100	99	nn0red 11950	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((1^st ‘(𝐺‘𝑘))‘𝑚) ∈ ℝ)
101		simplr 767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘 ∈ ℕ)
102	100, 101	nndivred 11685	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) ∈ ℝ)
103	93, 102	sylan 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) ∈ ℝ)
104	90, 103	resubcld 11062	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) ∈ ℝ)
105	104	recnd 10663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) ∈ ℂ)
106	105	abscld 14790	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) ∈ ℝ)
107	59	nnrecred 11682	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℝ⁺ → (1 / -(⌊‘-(2 / 𝑐))) ∈ ℝ)
108	107	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (1 / -(⌊‘-(2 / 𝑐))) ∈ ℝ)
109		rphalfcl 12410	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 ∈ ℝ⁺ → (𝑐 / 2) ∈ ℝ⁺)
110	109	rpred 12425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℝ⁺ → (𝑐 / 2) ∈ ℝ)
111	110	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (𝑐 / 2) ∈ ℝ)
112		ltletr 10726	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) ∈ ℝ ∧ (1 / -(⌊‘-(2 / 𝑐))) ∈ ℝ ∧ (𝑐 / 2) ∈ ℝ) → (((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (1 / -(⌊‘-(2 / 𝑐))) ∧ (1 / -(⌊‘-(2 / 𝑐))) ≤ (𝑐 / 2)) → (abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (𝑐 / 2)))
113	106, 108, 111, 112	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (1 / -(⌊‘-(2 / 𝑐))) ∧ (1 / -(⌊‘-(2 / 𝑐))) ≤ (𝑐 / 2)) → (abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (𝑐 / 2)))
114	85, 113	mpan2d 692	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (1 / -(⌊‘-(2 / 𝑐))) → (abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (𝑐 / 2)))
115	73, 114	sylanl1 678	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (1 / -(⌊‘-(2 / 𝑐))) → (abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (𝑐 / 2)))
116		simpl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐼) → 𝜑)
117	70	sselda 3967	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → 𝑘 ∈ ℕ)
118	116, 117	anim12i 614	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐))))) → (𝜑 ∧ 𝑘 ∈ ℕ))
119	118	anassrs 470	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (𝜑 ∧ 𝑘 ∈ ℕ))
120		1re 10635	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 1 ∈ ℝ
121		snssi 4735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (1 ∈ ℝ → {1} ⊆ ℝ)
122	120, 121	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ {1} ⊆ ℝ
123		0re 10637	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 0 ∈ ℝ
124		snssi 4735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ)
125	123, 124	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ {0} ⊆ ℝ
126	122, 125	unssi 4161	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ({1} ∪ {0}) ⊆ ℝ
127		1ex 10631	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ 1 ∈ V
128	127	fconst 6560	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}):((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗))⟶{1}
129		c0ex 10629	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ 0 ∈ V
130	129	fconst 6560	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}):((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))⟶{0}
131	128, 130	pm3.2i 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}):((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗))⟶{1} ∧ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}):((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))⟶{0})
132		xp2nd 7716	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝐺‘𝑘) ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^nd ‘(𝐺‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
133	95, 132	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (2^nd ‘(𝐺‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
134		fvex 6678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (2^nd ‘(𝐺‘𝑘)) ∈ V
135		f1oeq1 6599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑓 = (2^nd ‘(𝐺‘𝑘)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^nd ‘(𝐺‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)))
136	134, 135	elab 3667	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((2^nd ‘(𝐺‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^nd ‘(𝐺‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
137	133, 136	sylib 220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (2^nd ‘(𝐺‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
138		dff1o3 6616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((2^nd ‘(𝐺‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^nd ‘(𝐺‘𝑘)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ^◡(2^nd ‘(𝐺‘𝑘))))
139	138	simprbi 499	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((2^nd ‘(𝐺‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡(2^nd ‘(𝐺‘𝑘)))
140		imain 6434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (Fun ^◡(2^nd ‘(𝐺‘𝑘)) → ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∩ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))))
141	137, 139, 140	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∩ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))))
142		elfznn0 12994	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℕ₀)
143	142	nn0red 11950	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℝ)
144	143	ltp1d 11564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 < (𝑗 + 1))
145		fzdisj 12928	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
146	144, 145	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑗 ∈ (0...𝑁) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
147	146	imaeq2d 5924	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑗 ∈ (0...𝑁) → ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((2^nd ‘(𝐺‘𝑘)) “ ∅))
148		ima0 5940	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((2^nd ‘(𝐺‘𝑘)) “ ∅) = ∅
149	147, 148	syl6eq 2872	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑗 ∈ (0...𝑁) → ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ∅)
150	141, 149	sylan9req 2877	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∩ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))) = ∅)
151		fun 6535	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}):((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗))⟶{1} ∧ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}):((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))⟶{0}) ∧ (((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∩ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))) = ∅) → ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∪ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}))
152	131, 150, 151	sylancr 589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∪ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}))
153		imaundi 6003	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∪ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)))
154		nn0p1nn 11930	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ)
155	142, 154	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ ℕ)
156		nnuz 12275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ℕ = (ℤ_≥‘1)
157	155, 156	eleqtrdi 2923	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ (ℤ_≥‘1))
158		elfzuz3 12899	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑗 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑗))
159		fzsplit2 12926	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((𝑗 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘𝑗)) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
160	157, 158, 159	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑗 ∈ (0...𝑁) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
161	160	imaeq2d 5924	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑗 ∈ (0...𝑁) → ((2^nd ‘(𝐺‘𝑘)) “ (1...𝑁)) = ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))))
162		f1ofo 6617	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((2^nd ‘(𝐺‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(𝐺‘𝑘)):(1...𝑁)–onto→(1...𝑁))
163		foima 6590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((2^nd ‘(𝐺‘𝑘)):(1...𝑁)–onto→(1...𝑁) → ((2^nd ‘(𝐺‘𝑘)) “ (1...𝑁)) = (1...𝑁))
164	137, 162, 163	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((2^nd ‘(𝐺‘𝑘)) “ (1...𝑁)) = (1...𝑁))
165	161, 164	sylan9req 2877	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑗 ∈ (0...𝑁) ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (1...𝑁))
166	165	ancoms 461	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (1...𝑁))
167	153, 166	syl5eqr 2870	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∪ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))) = (1...𝑁))
168	167	feq2d 6495	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∪ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}) ↔ ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0})))
169	152, 168	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))
170	169	ffvelrnda 6846	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ ({1} ∪ {0}))
171	126, 170	sseldi 3965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ ℝ)
172		simpllr 774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘 ∈ ℕ)
173	171, 172	nndivred 11685	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘) ∈ ℝ)
174	173	recnd 10663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘) ∈ ℂ)
175	174	absnegd 14803	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) = (abs‘((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)))
176	119, 175	sylanl1 678	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) = (abs‘((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)))
177	119, 170	sylanl1 678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ ({1} ∪ {0}))
178		elun 4125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ ({1} ∪ {0}) ↔ ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {1} ∨ (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {0}))
179	177, 178	sylib 220	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {1} ∨ (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {0}))
180		nnrecre 11673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
181		nnrp 12394	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ⁺)
182	181	rpreccld 12435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ⁺)
183	182	rpge0d 12429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑘 ∈ ℕ → 0 ≤ (1 / 𝑘))
184	180, 183	absidd 14776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑘 ∈ ℕ → (abs‘(1 / 𝑘)) = (1 / 𝑘))
185	117, 184	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (abs‘(1 / 𝑘)) = (1 / 𝑘))
186	117	nnrecred 11682	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (1 / 𝑘) ∈ ℝ)
187	107	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (1 / -(⌊‘-(2 / 𝑐))) ∈ ℝ)
188	110	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (𝑐 / 2) ∈ ℝ)
189		eluzle 12250	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐))) → -(⌊‘-(2 / 𝑐)) ≤ 𝑘)
190	189	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → -(⌊‘-(2 / 𝑐)) ≤ 𝑘)
191	59	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → -(⌊‘-(2 / 𝑐)) ∈ ℕ)
192	191	nnrpd 12423	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → -(⌊‘-(2 / 𝑐)) ∈ ℝ⁺)
193	117	nnrpd 12423	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → 𝑘 ∈ ℝ⁺)
194	192, 193	lerecd 12444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (-(⌊‘-(2 / 𝑐)) ≤ 𝑘 ↔ (1 / 𝑘) ≤ (1 / -(⌊‘-(2 / 𝑐)))))
195	190, 194	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (1 / 𝑘) ≤ (1 / -(⌊‘-(2 / 𝑐))))
196	84	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (1 / -(⌊‘-(2 / 𝑐))) ≤ (𝑐 / 2))
197	186, 187, 188, 195, 196	letrd 10791	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (1 / 𝑘) ≤ (𝑐 / 2))
198	185, 197	eqbrtrd 5081	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (abs‘(1 / 𝑘)) ≤ (𝑐 / 2))
199		elsni 4578	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {1} → (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) = 1)
200	199	fvoveq1d 7172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {1} → (abs‘((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) = (abs‘(1 / 𝑘)))
201	200	breq1d 5069	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {1} → ((abs‘((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) ≤ (𝑐 / 2) ↔ (abs‘(1 / 𝑘)) ≤ (𝑐 / 2)))
202	198, 201	syl5ibrcom 249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {1} → (abs‘((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) ≤ (𝑐 / 2)))
203	109	rpge0d 12429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑐 ∈ ℝ⁺ → 0 ≤ (𝑐 / 2))
204		nncn 11640	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
205		nnne0 11665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
206	204, 205	div0d 11409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑘 ∈ ℕ → (0 / 𝑘) = 0)
207	206	abs00bd 14645	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑘 ∈ ℕ → (abs‘(0 / 𝑘)) = 0)
208	207	breq1d 5069	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑘 ∈ ℕ → ((abs‘(0 / 𝑘)) ≤ (𝑐 / 2) ↔ 0 ≤ (𝑐 / 2)))
209	208	biimparc 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((0 ≤ (𝑐 / 2) ∧ 𝑘 ∈ ℕ) → (abs‘(0 / 𝑘)) ≤ (𝑐 / 2))
210	203, 209	sylan 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ ℕ) → (abs‘(0 / 𝑘)) ≤ (𝑐 / 2))
211	117, 210	syldan 593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (abs‘(0 / 𝑘)) ≤ (𝑐 / 2))
212		elsni 4578	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {0} → (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) = 0)
213	212	fvoveq1d 7172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {0} → (abs‘((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) = (abs‘(0 / 𝑘)))
214	213	breq1d 5069	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {0} → ((abs‘((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) ≤ (𝑐 / 2) ↔ (abs‘(0 / 𝑘)) ≤ (𝑐 / 2)))
215	211, 214	syl5ibrcom 249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {0} → (abs‘((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) ≤ (𝑐 / 2)))
216	202, 215	jaod 855	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {1} ∨ (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {0}) → (abs‘((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) ≤ (𝑐 / 2)))
217	216	adantll 712	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {1} ∨ (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {0}) → (abs‘((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) ≤ (𝑐 / 2)))
218	217	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {1} ∨ (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ {0}) → (abs‘((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) ≤ (𝑐 / 2)))
219	179, 218	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (abs‘((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) ≤ (𝑐 / 2))
220	176, 219	eqbrtrd 5081	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) ≤ (𝑐 / 2))
221	73, 106	sylanl1 678	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) ∈ ℝ)
222		simpll 765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) → 𝜑)
223	222	anim1i 616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) → (𝜑 ∧ 𝑘 ∈ ℕ))
224	173	renegcld 11061	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘) ∈ ℝ)
225	223, 224	sylanl1 678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘) ∈ ℝ)
226	225	recnd 10663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘) ∈ ℂ)
227	226	abscld 14790	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) ∈ ℝ)
228	73, 227	sylanl1 678	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) ∈ ℝ)
229	110, 110	jca 514	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 ∈ ℝ⁺ → ((𝑐 / 2) ∈ ℝ ∧ (𝑐 / 2) ∈ ℝ))
230	229	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((𝑐 / 2) ∈ ℝ ∧ (𝑐 / 2) ∈ ℝ))
231		ltleadd 11117	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) ∈ ℝ ∧ (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) ∈ ℝ) ∧ ((𝑐 / 2) ∈ ℝ ∧ (𝑐 / 2) ∈ ℝ)) → (((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (𝑐 / 2) ∧ (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) ≤ (𝑐 / 2)) → ((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) + (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2))))
232	221, 228, 230, 231	syl21anc 835	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (𝑐 / 2) ∧ (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) ≤ (𝑐 / 2)) → ((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) + (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2))))
233	220, 232	mpan2d 692	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (𝑐 / 2) → ((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) + (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2))))
234	105, 226	abstrid 14810	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) ≤ ((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) + (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))))
235	104, 225	readdcld 10664	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) ∈ ℝ)
236	235	recnd 10663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) ∈ ℂ)
237	236	abscld 14790	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) ∈ ℝ)
238	106, 227	readdcld 10664	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) + (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) ∈ ℝ)
239	110, 110	readdcld 10664	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℝ⁺ → ((𝑐 / 2) + (𝑐 / 2)) ∈ ℝ)
240	239	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((𝑐 / 2) + (𝑐 / 2)) ∈ ℝ)
241		lelttr 10725	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) ∈ ℝ ∧ ((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) + (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) ∈ ℝ ∧ ((𝑐 / 2) + (𝑐 / 2)) ∈ ℝ) → (((abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) ≤ ((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) + (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) ∧ ((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) + (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2))) → (abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2))))
242	237, 238, 240, 241	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) ≤ ((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) + (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) ∧ ((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) + (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2))) → (abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2))))
243	234, 242	mpand 693	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) + (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2)) → (abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2))))
244	73, 243	sylanl1 678	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) + (abs‘-((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2)) → (abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2))))
245	115, 233, 244	3syld 60	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (1 / -(⌊‘-(2 / 𝑐))) → (abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2))))
246	100	adantlr 713	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((1^st ‘(𝐺‘𝑘))‘𝑚) ∈ ℝ)
247	246, 171	readdcld 10664	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) ∈ ℝ)
248	247, 172	nndivred 11685	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘) ∈ ℝ)
249	119, 248	sylanl1 678	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘) ∈ ℝ)
250	245, 249	jctild 528	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (1 / -(⌊‘-(2 / 𝑐))) → (((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘) ∈ ℝ ∧ (abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2)))))
251	250	adantld 493	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ℝ ∧ (abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (1 / -(⌊‘-(2 / 𝑐)))) → (((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘) ∈ ℝ ∧ (abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2)))))
252	73	adantr 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) → (((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ))
253	87	ad3antlr 729	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) → 𝐶:(1...𝑁)⟶(0[,]1))
254	253	ffvelrnda 6846	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐶‘𝑚) ∈ (0[,]1))
255	33, 254	sseldi 3965	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐶‘𝑚) ∈ ℝ)
256	74	rpreccld 12435	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑐 ∈ ℝ⁺ → (1 / -(⌊‘-(2 / 𝑐))) ∈ ℝ⁺)
257	256	rpxrd 12426	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑐 ∈ ℝ⁺ → (1 / -(⌊‘-(2 / 𝑐))) ∈ ℝ^*)
258	257	ad3antlr 729	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (1 / -(⌊‘-(2 / 𝑐))) ∈ ℝ^*)
259	13	rexmet 23393	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
260		elbl 22992	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) ∧ (𝐶‘𝑚) ∈ ℝ ∧ (1 / -(⌊‘-(2 / 𝑐))) ∈ ℝ^*) → ((((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))) ↔ ((((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ℝ ∧ ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚)) < (1 / -(⌊‘-(2 / 𝑐))))))
261	259, 260	mp3an1 1444	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐶‘𝑚) ∈ ℝ ∧ (1 / -(⌊‘-(2 / 𝑐))) ∈ ℝ^*) → ((((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))) ↔ ((((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ℝ ∧ ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚)) < (1 / -(⌊‘-(2 / 𝑐))))))
262	255, 258, 261	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))) ↔ ((((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ℝ ∧ ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚)) < (1 / -(⌊‘-(2 / 𝑐))))))
263		elmapfn 8423	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((1^st ‘(𝐺‘𝑘)) ∈ (ℕ₀ ↑_m (1...𝑁)) → (1^st ‘(𝐺‘𝑘)) Fn (1...𝑁))
264	95, 96, 263	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐺‘𝑘)) Fn (1...𝑁))
265		vex 3498	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 𝑘 ∈ V
266		fnconstg 6562	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ V → ((1...𝑁) × {𝑘}) Fn (1...𝑁))
267	265, 266	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1...𝑁) × {𝑘}) Fn (1...𝑁))
268		fzfid 13335	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑁) ∈ Fin)
269		inidm 4195	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
270		eqidd 2822	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((1^st ‘(𝐺‘𝑘))‘𝑚) = ((1^st ‘(𝐺‘𝑘))‘𝑚))
271	265	fvconst2 6961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑚 ∈ (1...𝑁) → (((1...𝑁) × {𝑘})‘𝑚) = 𝑘)
272	271	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1...𝑁) × {𝑘})‘𝑚) = 𝑘)
273	264, 267, 268, 268, 269, 270, 272	ofval 7412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) = (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))
274	273	oveq2d 7166	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚)) = ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))(((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)))
275	222, 274	sylanl1 678	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚)) = ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))(((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)))
276	222, 102	sylanl1 678	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) ∈ ℝ)
277	13	remetdval 23391	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐶‘𝑚) ∈ ℝ ∧ (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) ∈ ℝ) → ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))(((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) = (abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))))
278	255, 276, 277	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))(((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) = (abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))))
279	275, 278	eqtrd 2856	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚)) = (abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))))
280	279	breq1d 5069	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚)) < (1 / -(⌊‘-(2 / 𝑐))) ↔ (abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (1 / -(⌊‘-(2 / 𝑐)))))
281	280	anbi2d 630	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ℝ ∧ ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚)) < (1 / -(⌊‘-(2 / 𝑐)))) ↔ ((((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ℝ ∧ (abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (1 / -(⌊‘-(2 / 𝑐))))))
282	262, 281	bitrd 281	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))) ↔ ((((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ℝ ∧ (abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (1 / -(⌊‘-(2 / 𝑐))))))
283	252, 282	sylan 582	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))) ↔ ((((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ℝ ∧ (abs‘((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘))) < (1 / -(⌊‘-(2 / 𝑐))))))
284		rpxr 12392	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑐 ∈ ℝ⁺ → 𝑐 ∈ ℝ^*)
285	284	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑐 ∈ ℝ^*)
286		elbl 22992	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) ∧ (𝐶‘𝑚) ∈ ℝ ∧ 𝑐 ∈ ℝ^*) → (((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ↔ (((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ℝ ∧ ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚)) < 𝑐)))
287	259, 286	mp3an1 1444	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐶‘𝑚) ∈ ℝ ∧ 𝑐 ∈ ℝ^*) → (((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ↔ (((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ℝ ∧ ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚)) < 𝑐)))
288	90, 285, 287	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ↔ (((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ℝ ∧ ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚)) < 𝑐)))
289		elun 4125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ ({1} ∪ {0}) ↔ (𝑧 ∈ {1} ∨ 𝑧 ∈ {0}))
290		fzofzp1 13128	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑣 ∈ (0..^𝑘) → (𝑣 + 1) ∈ (0...𝑘))
291		elsni 4578	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑧 ∈ {1} → 𝑧 = 1)
292	291	oveq2d 7166	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑧 ∈ {1} → (𝑣 + 𝑧) = (𝑣 + 1))
293	292	eleq1d 2897	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑧 ∈ {1} → ((𝑣 + 𝑧) ∈ (0...𝑘) ↔ (𝑣 + 1) ∈ (0...𝑘)))
294	290, 293	syl5ibrcom 249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑣 ∈ (0..^𝑘) → (𝑧 ∈ {1} → (𝑣 + 𝑧) ∈ (0...𝑘)))
295		elfzonn0 13076	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑣 ∈ (0..^𝑘) → 𝑣 ∈ ℕ₀)
296	295	nn0cnd 11951	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑣 ∈ (0..^𝑘) → 𝑣 ∈ ℂ)
297	296	addid1d 10834	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑣 ∈ (0..^𝑘) → (𝑣 + 0) = 𝑣)
298		elfzofz 13047	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑣 ∈ (0..^𝑘) → 𝑣 ∈ (0...𝑘))
299	297, 298	eqeltrd 2913	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑣 ∈ (0..^𝑘) → (𝑣 + 0) ∈ (0...𝑘))
300		elsni 4578	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑧 ∈ {0} → 𝑧 = 0)
301	300	oveq2d 7166	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑧 ∈ {0} → (𝑣 + 𝑧) = (𝑣 + 0))
302	301	eleq1d 2897	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑧 ∈ {0} → ((𝑣 + 𝑧) ∈ (0...𝑘) ↔ (𝑣 + 0) ∈ (0...𝑘)))
303	299, 302	syl5ibrcom 249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑣 ∈ (0..^𝑘) → (𝑧 ∈ {0} → (𝑣 + 𝑧) ∈ (0...𝑘)))
304	294, 303	jaod 855	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑣 ∈ (0..^𝑘) → ((𝑧 ∈ {1} ∨ 𝑧 ∈ {0}) → (𝑣 + 𝑧) ∈ (0...𝑘)))
305	289, 304	syl5bi 244	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑣 ∈ (0..^𝑘) → (𝑧 ∈ ({1} ∪ {0}) → (𝑣 + 𝑧) ∈ (0...𝑘)))
306	305	imp 409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑣 ∈ (0..^𝑘) ∧ 𝑧 ∈ ({1} ∪ {0})) → (𝑣 + 𝑧) ∈ (0...𝑘))
307	306	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑣 ∈ (0..^𝑘) ∧ 𝑧 ∈ ({1} ∪ {0}))) → (𝑣 + 𝑧) ∈ (0...𝑘))
308		dffn3 6520	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((1^st ‘(𝐺‘𝑘)) Fn (1...𝑁) ↔ (1^st ‘(𝐺‘𝑘)):(1...𝑁)⟶ran (1^st ‘(𝐺‘𝑘)))
309	264, 308	sylib 220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐺‘𝑘)):(1...𝑁)⟶ran (1^st ‘(𝐺‘𝑘)))
310		poimirlem30.3	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran (1^st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘))
311	309, 310	fssd 6523	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐺‘𝑘)):(1...𝑁)⟶(0..^𝑘))
312	311	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (1^st ‘(𝐺‘𝑘)):(1...𝑁)⟶(0..^𝑘))
313		fzfid 13335	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
314	307, 312, 169, 313, 313, 269	off 7418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))):(1...𝑁)⟶(0...𝑘))
315	314	ffnd 6510	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) Fn (1...𝑁))
316	265, 266	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((1...𝑁) × {𝑘}) Fn (1...𝑁))
317	264	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (1^st ‘(𝐺‘𝑘)) Fn (1...𝑁))
318	169	ffnd 6510	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})) Fn (1...𝑁))
319		eqidd 2822	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((1^st ‘(𝐺‘𝑘))‘𝑚) = ((1^st ‘(𝐺‘𝑘))‘𝑚))
320		eqidd 2822	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) = (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚))
321	317, 318, 313, 313, 269, 319, 320	ofval 7412	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑚) = (((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)))
322	271	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((1...𝑁) × {𝑘})‘𝑚) = 𝑘)
323	315, 316, 313, 313, 269, 321, 322	ofval 7412	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) = ((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘))
324	323	eleq1d 2897	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ℝ ↔ ((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘) ∈ ℝ))
325	223, 324	sylanl1 678	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ℝ ↔ ((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘) ∈ ℝ))
326	323	adantl3r 748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) = ((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘))
327	326	oveq2d 7166	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚)) = ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘)))
328	87	ad3antlr 729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝐶:(1...𝑁)⟶(0[,]1))
329	328	ffvelrnda 6846	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (𝐶‘𝑚) ∈ (0[,]1))
330	33, 329	sseldi 3965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (𝐶‘𝑚) ∈ ℝ)
331	248	adantl3r 748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘) ∈ ℝ)
332	13	remetdval 23391	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐶‘𝑚) ∈ ℝ ∧ ((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘) ∈ ℝ) → ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘)) = (abs‘((𝐶‘𝑚) − ((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘))))
333	330, 331, 332	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘)) = (abs‘((𝐶‘𝑚) − ((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘))))
334	246	recnd 10663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((1^st ‘(𝐺‘𝑘))‘𝑚) ∈ ℂ)
335	171	recnd 10663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) ∈ ℂ)
336	204	ad3antlr 729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘 ∈ ℂ)
337	205	ad3antlr 729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘 ≠ 0)
338	334, 335, 336, 337	divdird 11448	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘) = ((((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) + ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)))
339	102	recnd 10663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) ∈ ℂ)
340	339	adantlr 713	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) ∈ ℂ)
341	340, 174	subnegd 10998	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) − -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)) = ((((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) + ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)))
342	338, 341	eqtr4d 2859	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘) = ((((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) − -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)))
343	342	oveq2d 7166	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((𝐶‘𝑚) − ((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘)) = ((𝐶‘𝑚) − ((((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) − -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))))
344	343	adantl3r 748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((𝐶‘𝑚) − ((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘)) = ((𝐶‘𝑚) − ((((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) − -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))))
345	330	recnd 10663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (𝐶‘𝑚) ∈ ℂ)
346	102	adantllr 717	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) ∈ ℝ)
347	346	adantlr 713	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) ∈ ℝ)
348	347	recnd 10663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) ∈ ℂ)
349	174	adantl3r 748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘) ∈ ℂ)
350	349	negcld 10978	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘) ∈ ℂ)
351	345, 348, 350	subsubd 11019	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((𝐶‘𝑚) − ((((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) − -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) = (((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)))
352	344, 351	eqtrd 2856	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((𝐶‘𝑚) − ((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘)) = (((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘)))
353	352	fveq2d 6669	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (abs‘((𝐶‘𝑚) − ((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘))) = (abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))))
354	327, 333, 353	3eqtrd 2860	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚)) = (abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))))
355	354	adantl3r 748	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚)) = (abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))))
356	77	2halvesd 11877	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 ∈ ℝ⁺ → ((𝑐 / 2) + (𝑐 / 2)) = 𝑐)
357	356	eqcomd 2827	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑐 ∈ ℝ⁺ → 𝑐 = ((𝑐 / 2) + (𝑐 / 2)))
358	357	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑐 = ((𝑐 / 2) + (𝑐 / 2)))
359	355, 358	breq12d 5072	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚)) < 𝑐 ↔ (abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2))))
360	325, 359	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ℝ ∧ ((𝐶‘𝑚)((abs ∘ − ) ↾ (ℝ × ℝ))((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚)) < 𝑐) ↔ (((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘) ∈ ℝ ∧ (abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2)))))
361	288, 360	bitrd 281	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ↔ (((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘) ∈ ℝ ∧ (abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2)))))
362	73, 361	sylanl1 678	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ↔ (((((1^st ‘(𝐺‘𝑘))‘𝑚) + (((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚)) / 𝑘) ∈ ℝ ∧ (abs‘(((𝐶‘𝑚) − (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘)) + -((((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑚) / 𝑘))) < ((𝑐 / 2) + (𝑐 / 2)))))
363	251, 283, 362	3imtr4d 296	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))) → ((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐)))
364	363	ralimdva 3177	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) → (∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))) → ∀𝑚 ∈ (1...𝑁)((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐)))
365		simplr 767	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝑘 ∈ ℕ)
366		elfznn0 12994	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑣 ∈ (0...𝑘) → 𝑣 ∈ ℕ₀)
367	366	nn0red 11950	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑣 ∈ (0...𝑘) → 𝑣 ∈ ℝ)
368		nndivre 11672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑣 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑣 / 𝑘) ∈ ℝ)
369	367, 368	sylan 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑣 / 𝑘) ∈ ℝ)
370		elfzle1 12904	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑣 ∈ (0...𝑘) → 0 ≤ 𝑣)
371	367, 370	jca 514	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑣 ∈ (0...𝑘) → (𝑣 ∈ ℝ ∧ 0 ≤ 𝑣))
372	181	rpregt0d 12431	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
373		divge0 11503	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑣 ∈ ℝ ∧ 0 ≤ 𝑣) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝑣 / 𝑘))
374	371, 372, 373	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑣 / 𝑘))
375		elfzle2 12905	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑣 ∈ (0...𝑘) → 𝑣 ≤ 𝑘)
376	375	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑣 ≤ 𝑘)
377	367	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑣 ∈ ℝ)
378		1red 10636	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ)
379	181	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ⁺)
380	377, 378, 379	ledivmuld 12478	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑣 / 𝑘) ≤ 1 ↔ 𝑣 ≤ (𝑘 · 1)))
381	204	mulid1d 10652	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 ∈ ℕ → (𝑘 · 1) = 𝑘)
382	381	breq2d 5071	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ ℕ → (𝑣 ≤ (𝑘 · 1) ↔ 𝑣 ≤ 𝑘))
383	382	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑣 ≤ (𝑘 · 1) ↔ 𝑣 ≤ 𝑘))
384	380, 383	bitrd 281	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑣 / 𝑘) ≤ 1 ↔ 𝑣 ≤ 𝑘))
385	376, 384	mpbird 259	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑣 / 𝑘) ≤ 1)
386		elicc01 12848	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑣 / 𝑘) ∈ (0[,]1) ↔ ((𝑣 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝑣 / 𝑘) ∧ (𝑣 / 𝑘) ≤ 1))
387	369, 374, 385, 386	syl3anbrc 1339	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑣 / 𝑘) ∈ (0[,]1))
388	387	ancoms 461	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ ℕ ∧ 𝑣 ∈ (0...𝑘)) → (𝑣 / 𝑘) ∈ (0[,]1))
389		elsni 4578	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ {𝑘} → 𝑧 = 𝑘)
390	389	oveq2d 7166	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ {𝑘} → (𝑣 / 𝑧) = (𝑣 / 𝑘))
391	390	eleq1d 2897	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ {𝑘} → ((𝑣 / 𝑧) ∈ (0[,]1) ↔ (𝑣 / 𝑘) ∈ (0[,]1)))
392	388, 391	syl5ibrcom 249	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℕ ∧ 𝑣 ∈ (0...𝑘)) → (𝑧 ∈ {𝑘} → (𝑣 / 𝑧) ∈ (0[,]1)))
393	392	impr 457	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑘 ∈ ℕ ∧ (𝑣 ∈ (0...𝑘) ∧ 𝑧 ∈ {𝑘})) → (𝑣 / 𝑧) ∈ (0[,]1))
394	365, 393	sylan 582	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑣 ∈ (0...𝑘) ∧ 𝑧 ∈ {𝑘})) → (𝑣 / 𝑧) ∈ (0[,]1))
395	265	fconst 6560	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘}
396	395	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘})
397	394, 314, 396, 313, 313, 269	off 7418	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
398	397	ffnd 6510	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) Fn (1...𝑁))
399	119, 398	sylan 582	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) → (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) Fn (1...𝑁))
400	364, 399	jctild 528	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) → (∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))) → ((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐))))
401	8	eleq2i 2904	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑_m (1...𝑁)))
402		ovex 7183	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0[,]1) ∈ V
403	402, 37	elmap 8429	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑_m (1...𝑁)) ↔ (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
404	401, 403	bitri 277	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
405	397, 404	sylibr 236	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼)
406	119, 405	sylan 582	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) → (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼)
407	400, 406	jctird 529	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) → (∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))) → (((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐)) ∧ (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼)))
408		elin 4169	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ↔ ((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∧ (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼))
409		ovex 7183	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ V
410	409	elixp 8462	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ↔ ((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐)))
411	410	anbi1i 625	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∧ (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼) ↔ (((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐)) ∧ (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼))
412	408, 411	bitri 277	. . . . . . . . . . . . . . . . . 18 ⊢ ((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ↔ (((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐)) ∧ (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼))
413	407, 412	syl6ibr 254	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) → (∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))) → (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼)))
414		ssel 3961	. . . . . . . . . . . . . . . . . 18 ⊢ ((X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣 → ((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) → (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣))
415	414	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) → ((X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣 → (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣))
416	413, 415	syl6 35	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) → (∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))) → ((X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣 → (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣)))
417	416	impd 413	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) → ((∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))) ∧ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣) → (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣))
418	417	ralrimdva 3189	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → ((∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))) ∧ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣) → ∀𝑗 ∈ (0...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣))
419	418	expd 418	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))) → ((X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣 → ∀𝑗 ∈ (0...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣)))
420		poimirlem30.4	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ^◡ ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟𝑋)
421	420	3exp2 1350	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 ∈ ℕ → (𝑛 ∈ (1...𝑁) → (𝑟 ∈ { ≤ , ^◡ ≤ } → ∃𝑗 ∈ (0...𝑁)0𝑟𝑋))))
422	421	imp43 430	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ^◡ ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟𝑋)
423		r19.29 3254	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑗 ∈ (0...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣 ∧ ∃𝑗 ∈ (0...𝑁)0𝑟𝑋) → ∃𝑗 ∈ (0...𝑁)((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣 ∧ 0𝑟𝑋))
424		fveq2 6665	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 = (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) → (𝐹‘𝑧) = (𝐹‘(((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘}))))
425	424	fveq1d 6667	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 = (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) → ((𝐹‘𝑧)‘𝑛) = ((𝐹‘(((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑛))
426		poimirlem30.x	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑋 = ((𝐹‘(((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)
427	425, 426	syl6eqr 2874	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) → ((𝐹‘𝑧)‘𝑛) = 𝑋)
428	427	breq2d 5071	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = (((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) → (0𝑟((𝐹‘𝑧)‘𝑛) ↔ 0𝑟𝑋))
429	428	rspcev 3623	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣 ∧ 0𝑟𝑋) → ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
430	429	rexlimivw 3282	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑗 ∈ (0...𝑁)((((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣 ∧ 0𝑟𝑋) → ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
431	423, 430	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑗 ∈ (0...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣 ∧ ∃𝑗 ∈ (0...𝑁)0𝑟𝑋) → ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
432	431	expcom 416	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑗 ∈ (0...𝑁)0𝑟𝑋 → (∀𝑗 ∈ (0...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣 → ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
433	422, 432	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ^◡ ≤ })) → (∀𝑗 ∈ (0...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣 → ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
434	433	ralrimdvva 3194	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∀𝑗 ∈ (0...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣 → ∀𝑛 ∈ (1...𝑁)∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
435	117, 434	sylan2 594	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐))))) → (∀𝑗 ∈ (0...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣 → ∀𝑛 ∈ (1...𝑁)∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
436	435	anassrs 470	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (∀𝑗 ∈ (0...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣 → ∀𝑛 ∈ (1...𝑁)∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
437	436	adantllr 717	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (∀𝑗 ∈ (0...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝑣 → ∀𝑛 ∈ (1...𝑁)∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
438	419, 437	syl6d 75	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))) → ((X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣 → ∀𝑛 ∈ (1...𝑁)∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))))
439	438	rexlimdva 3284	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) → (∃𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))) → ((X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣 → ∀𝑛 ∈ (1...𝑁)∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))))
440	68, 439	syld 47	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ((X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣 → ∀𝑛 ∈ (1...𝑁)∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))))
441	440	com23 86	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) → ((X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣 → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∀𝑛 ∈ (1...𝑁)∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))))
442	441	impr 457	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ (𝑐 ∈ ℝ⁺ ∧ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣)) → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∀𝑛 ∈ (1...𝑁)∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
443	45, 442	sylanl2 679	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑣 ∈ (𝑅 ↾_t 𝐼) ∧ 𝐶 ∈ 𝑣)) ∧ (𝑐 ∈ ℝ⁺ ∧ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣)) → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∀𝑛 ∈ (1...𝑁)∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
444	29, 443	rexlimddv 3291	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝑅 ↾_t 𝐼) ∧ 𝐶 ∈ 𝑣)) → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∀𝑛 ∈ (1...𝑁)∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
445	444	expr 459	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑅 ↾_t 𝐼)) → (𝐶 ∈ 𝑣 → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∀𝑛 ∈ (1...𝑁)∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))))
446	445	com23 86	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑅 ↾_t 𝐼)) → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → (𝐶 ∈ 𝑣 → ∀𝑛 ∈ (1...𝑁)∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))))
447		r19.21v 3175	. . . 4 ⊢ (∀𝑛 ∈ (1...𝑁)(𝐶 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)) ↔ (𝐶 ∈ 𝑣 → ∀𝑛 ∈ (1...𝑁)∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
448	446, 447	syl6ibr 254	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑅 ↾_t 𝐼)) → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∀𝑛 ∈ (1...𝑁)(𝐶 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))))
449	448	ralrimdva 3189	. 2 ⊢ (𝜑 → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∀𝑣 ∈ (𝑅 ↾_t 𝐼)∀𝑛 ∈ (1...𝑁)(𝐶 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))))
450		ralcom 3354	. 2 ⊢ (∀𝑣 ∈ (𝑅 ↾_t 𝐼)∀𝑛 ∈ (1...𝑁)(𝐶 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)) ↔ ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝐶 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
451	449, 450	syl6ib 253	1 ⊢ (𝜑 → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝐶 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {cab 2799 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3495 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 {csn 4561 {cpr 4563 ∪ cuni 4832 class class class wbr 5059 × cxp 5548 ^◡ccnv 5549 ran crn 5551 ↾ cres 5552 “ cima 5553 ∘ ccom 5554 Fun wfun 6344 Fn wfn 6345 ⟶wf 6346 –onto→wfo 6348 –1-1-onto→wf1o 6349 ‘cfv 6350 (class class class)co 7150 ∘_f cof 7401 1^st c1st 7681 2^nd c2nd 7682 ↑_m cmap 8400 Xcixp 8455 Fincfn 8503 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 ℝ^*cxr 10668 < clt 10669 ≤ cle 10670 − cmin 10864 -cneg 10865 / cdiv 11291 ℕcn 11632 2c2 11686 ℕ₀cn0 11891 ℤcz 11975 ℤ_≥cuz 12237 ℝ⁺crp 12383 (,)cioo 12732 [,]cicc 12735 ...cfz 12886 ..^cfzo 13027 ⌊cfl 13154 abscabs 14587 ↾_t crest 16688 topGenctg 16705 ∏_tcpt 16706 ∞Metcxmet 20524 ballcbl 20526 Topctop 21495 Cn ccn 21826

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fi 8869 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-rest 16690 df-topgen 16711 df-pt 16712 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-bases 21548

This theorem is referenced by: poimirlem30 34916

W3C validator