poimirlem29 - Mathbox for Brendan Leahy

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Theorem poimirlem29 37636

Description: Lemma for poimir 37640 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 13913	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ∈ Fin
3		retop 24682	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) ∈ Top
4	3	fconst6 6732	. . . . . . . . . . . . 13 ⊢ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top
5		pttop 23502	. . . . . . . . . . . . 13 ⊢ (((1...𝑁) ∈ Fin ∧ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top) → (∏_t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Top)
6	2, 4, 5	mp2an 692	. . . . . . . . . . . 12 ⊢ (∏_t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Top
7	1, 6	eqeltri 2824	. . . . . . . . . . 11 ⊢ 𝑅 ∈ Top
8		poimir.i	. . . . . . . . . . . 12 ⊢ 𝐼 = ((0[,]1) ↑_m (1...𝑁))
9		ovex 7402	. . . . . . . . . . . 12 ⊢ ((0[,]1) ↑_m (1...𝑁)) ∈ V
10	8, 9	eqeltri 2824	. . . . . . . . . . 11 ⊢ 𝐼 ∈ V
11		elrest 17366	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝐼 ∈ V) → (𝑣 ∈ (𝑅 ↾_t 𝐼) ↔ ∃𝑧 ∈ 𝑅 𝑣 = (𝑧 ∩ 𝐼)))
12	7, 10, 11	mp2an 692	. . . . . . . . . 10 ⊢ (𝑣 ∈ (𝑅 ↾_t 𝐼) ↔ ∃𝑧 ∈ 𝑅 𝑣 = (𝑧 ∩ 𝐼))
13		eqid 2729	. . . . . . . . . . . . . 14 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
14	1, 13	ptrecube 37607	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑅 ∧ 𝐶 ∈ 𝑧) → ∃𝑐 ∈ ℝ⁺ X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ⊆ 𝑧)
15	14	ex 412	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑅 → (𝐶 ∈ 𝑧 → ∃𝑐 ∈ ℝ⁺ X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ⊆ 𝑧))
16		inss1 4196	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∩ 𝐼) ⊆ 𝑧
17		sseq1 3969	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑧 ∩ 𝐼) → (𝑣 ⊆ 𝑧 ↔ (𝑧 ∩ 𝐼) ⊆ 𝑧))
18	16, 17	mpbiri 258	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑧 ∩ 𝐼) → 𝑣 ⊆ 𝑧)
19	18	sseld 3942	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑧 ∩ 𝐼) → (𝐶 ∈ 𝑣 → 𝐶 ∈ 𝑧))
20		ssrin 4201	. . . . . . . . . . . . . . 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 3970	. . . . . . . . . . . . . . 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	imbitrrid 246	. . . . . . . . . . . . . 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 3148	. . . . . . . . . . . . 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 3127	. . . . . . . . . 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 217	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝑅 ↾_t 𝐼) → (𝐶 ∈ 𝑣 → ∃𝑐 ∈ ℝ⁺ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣))
28	27	imp 406	. . . . . . . 8 ⊢ ((𝑣 ∈ (𝑅 ↾_t 𝐼) ∧ 𝐶 ∈ 𝑣) → ∃𝑐 ∈ ℝ⁺ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣)
29	28	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝑅 ↾_t 𝐼) ∧ 𝐶 ∈ 𝑣)) → ∃𝑐 ∈ ℝ⁺ (X𝑚 ∈ (1...𝑁)((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑐) ∩ 𝐼) ⊆ 𝑣)
30		resttop 23080	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝐼 ∈ V) → (𝑅 ↾_t 𝐼) ∈ Top)
31	7, 10, 30	mp2an 692	. . . . . . . . . 10 ⊢ (𝑅 ↾_t 𝐼) ∈ Top
32		reex 11135	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ V
33		unitssre 13436	. . . . . . . . . . . . . 14 ⊢ (0[,]1) ⊆ ℝ
34		mapss 8839	. . . . . . . . . . . . . 14 ⊢ ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁)))
35	32, 33, 34	mp2an 692	. . . . . . . . . . . . 13 ⊢ ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁))
36	8, 35	eqsstri 3990	. . . . . . . . . . . 12 ⊢ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))
37		ovex 7402	. . . . . . . . . . . . . 14 ⊢ (1...𝑁) ∈ V
38		uniretop 24683	. . . . . . . . . . . . . . 15 ⊢ ℝ = ∪ (topGen‘ran (,))
39	1, 38	ptuniconst 23518	. . . . . . . . . . . . . 14 ⊢ (((1...𝑁) ∈ V ∧ (topGen‘ran (,)) ∈ Top) → (ℝ ↑_m (1...𝑁)) = ∪ 𝑅)
40	37, 3, 39	mp2an 692	. . . . . . . . . . . . 13 ⊢ (ℝ ↑_m (1...𝑁)) = ∪ 𝑅
41	40	restuni 23082	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Top ∧ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))) → 𝐼 = ∪ (𝑅 ↾_t 𝐼))
42	7, 36, 41	mp2an 692	. . . . . . . . . . 11 ⊢ 𝐼 = ∪ (𝑅 ↾_t 𝐼)
43	42	eltopss 22827	. . . . . . . . . 10 ⊢ (((𝑅 ↾_t 𝐼) ∈ Top ∧ 𝑣 ∈ (𝑅 ↾_t 𝐼)) → 𝑣 ⊆ 𝐼)
44	31, 43	mpan 690	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝑅 ↾_t 𝐼) → 𝑣 ⊆ 𝐼)
45	44	sselda 3943	. . . . . . . 8 ⊢ ((𝑣 ∈ (𝑅 ↾_t 𝐼) ∧ 𝐶 ∈ 𝑣) → 𝐶 ∈ 𝐼)
46		2rp 12932	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ⁺
47		rpdivcl 12954	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℝ⁺ ∧ 𝑐 ∈ ℝ⁺) → (2 / 𝑐) ∈ ℝ⁺)
48	46, 47	mpan 690	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ ℝ⁺ → (2 / 𝑐) ∈ ℝ⁺)
49	48	rpred 12971	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ℝ⁺ → (2 / 𝑐) ∈ ℝ)
50		ceicl 13779	. . . . . . . . . . . . . . 15 ⊢ ((2 / 𝑐) ∈ ℝ → -(⌊‘-(2 / 𝑐)) ∈ ℤ)
51	49, 50	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ℝ⁺ → -(⌊‘-(2 / 𝑐)) ∈ ℤ)
52		0red 11153	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ℝ⁺ → 0 ∈ ℝ)
53	51	zred 12614	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ℝ⁺ → -(⌊‘-(2 / 𝑐)) ∈ ℝ)
54	48	rpgt0d 12974	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ℝ⁺ → 0 < (2 / 𝑐))
55		ceige 13782	. . . . . . . . . . . . . . . 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 11310	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ℝ⁺ → 0 < -(⌊‘-(2 / 𝑐)))
58		elnnz 12515	. . . . . . . . . . . . . 14 ⊢ (-(⌊‘-(2 / 𝑐)) ∈ ℕ ↔ (-(⌊‘-(2 / 𝑐)) ∈ ℤ ∧ 0 < -(⌊‘-(2 / 𝑐))))
59	51, 57, 58	sylanbrc 583	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ℝ⁺ → -(⌊‘-(2 / 𝑐)) ∈ ℕ)
60		fveq2 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = -(⌊‘-(2 / 𝑐)) → (ℤ_≥‘𝑖) = (ℤ_≥‘-(⌊‘-(2 / 𝑐))))
61		oveq2 7377	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = -(⌊‘-(2 / 𝑐)) → (1 / 𝑖) = (1 / -(⌊‘-(2 / 𝑐))))
62	61	oveq2d 7385	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = -(⌊‘-(2 / 𝑐)) → ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) = ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / -(⌊‘-(2 / 𝑐)))))
63	62	eleq2d 2814	. . . . . . . . . . . . . . . 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 3156	. . . . . . . . . . . . . . 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 3317	. . . . . . . . . . . . . 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 3581	. . . . . . . . . . . . 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 481	. . . . . . . . . . 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 12830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (-(⌊‘-(2 / 𝑐)) ∈ ℕ → (ℤ_≥‘-(⌊‘-(2 / 𝑐))) ⊆ ℕ)
70	59, 69	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℝ⁺ → (ℤ_≥‘-(⌊‘-(2 / 𝑐))) ⊆ ℕ)
71	70	sseld 3942	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 ∈ ℝ⁺ → (𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐))) → 𝑘 ∈ ℕ))
72	71	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) → (𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐))) → 𝑘 ∈ ℕ))
73	72	imdistani 568	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ))
74	59	nnrpd 12969	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑐 ∈ ℝ⁺ → -(⌊‘-(2 / 𝑐)) ∈ ℝ⁺)
75	48, 74	lerecd 12990	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 ∈ ℝ⁺ → ((2 / 𝑐) ≤ -(⌊‘-(2 / 𝑐)) ↔ (1 / -(⌊‘-(2 / 𝑐))) ≤ (1 / (2 / 𝑐))))
76	56, 75	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℝ⁺ → (1 / -(⌊‘-(2 / 𝑐))) ≤ (1 / (2 / 𝑐)))
77		rpcn 12938	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 ∈ ℝ⁺ → 𝑐 ∈ ℂ)
78		rpne0 12944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 ∈ ℝ⁺ → 𝑐 ≠ 0)
79		2cn 12237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 2 ∈ ℂ
80		2ne0 12266	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 2 ≠ 0
81		recdiv 11864	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((2 ∈ ℂ ∧ 2 ≠ 0) ∧ (𝑐 ∈ ℂ ∧ 𝑐 ≠ 0)) → (1 / (2 / 𝑐)) = (𝑐 / 2))
82	79, 80, 81	mpanl12 702	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑐 ∈ ℂ ∧ 𝑐 ≠ 0) → (1 / (2 / 𝑐)) = (𝑐 / 2))
83	77, 78, 82	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℝ⁺ → (1 / (2 / 𝑐)) = (𝑐 / 2))
84	76, 83	breqtrd 5128	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 ∈ ℝ⁺ → (1 / -(⌊‘-(2 / 𝑐))) ≤ (𝑐 / 2))
85	84	ad4antlr 733	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (1 / -(⌊‘-(2 / 𝑐))) ≤ (𝑐 / 2))
86		elmapi 8799	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝐶 ∈ ((0[,]1) ↑_m (1...𝑁)) → 𝐶:(1...𝑁)⟶(0[,]1))
87	86, 8	eleq2s 2846	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝐶 ∈ 𝐼 → 𝐶:(1...𝑁)⟶(0[,]1))
88	87	ad4antlr 733	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝐶:(1...𝑁)⟶(0[,]1))
89	88	ffvelcdmda 7038	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (𝐶‘𝑚) ∈ (0[,]1))
90	33, 89	sselid 3941	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (𝐶‘𝑚) ∈ ℝ)
91		simp-4l 782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝜑)
92		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝑘 ∈ ℕ)
93	91, 92	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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	ffvelcdmda 7038	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
96		xp1st 7979	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8799	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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	ffvelcdmda 7038	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((1^st ‘(𝐺‘𝑘))‘𝑚) ∈ ℕ₀)
100	99	nn0red 12480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((1^st ‘(𝐺‘𝑘))‘𝑚) ∈ ℝ)
101		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘 ∈ ℕ)
102	100, 101	nndivred 12216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) ∈ ℝ)
103	93, 102	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 15381	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℝ⁺ → (1 / -(⌊‘-(2 / 𝑐))) ∈ ℝ)
108	107	ad4antlr 733	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (1 / -(⌊‘-(2 / 𝑐))) ∈ ℝ)
109		rphalfcl 12956	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 ∈ ℝ⁺ → (𝑐 / 2) ∈ ℝ⁺)
110	109	rpred 12971	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℝ⁺ → (𝑐 / 2) ∈ ℝ)
111	110	ad4antlr 733	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (𝑐 / 2) ∈ ℝ)
112		ltletr 11242	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 694	. . . . . . . . . . . . . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐼) → 𝜑)
117	70	sselda 3943	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → 𝑘 ∈ ℕ)
118	116, 117	anim12i 613	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐))))) → (𝜑 ∧ 𝑘 ∈ ℕ))
119	118	anassrs 467	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (𝜑 ∧ 𝑘 ∈ ℕ))
120		1re 11150	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 1 ∈ ℝ
121		snssi 4768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (1 ∈ ℝ → {1} ⊆ ℝ)
122	120, 121	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ {1} ⊆ ℝ
123		0re 11152	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 0 ∈ ℝ
124		snssi 4768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ)
125	123, 124	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ {0} ⊆ ℝ
126	122, 125	unssi 4150	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ({1} ∪ {0}) ⊆ ℝ
127		1ex 11146	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ 1 ∈ V
128	127	fconst 6728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}):((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗))⟶{1}
129		c0ex 11144	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ 0 ∈ V
130	129	fconst 6728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}):((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))⟶{0}
131	128, 130	pm3.2i 470	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7980	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (2^nd ‘(𝐺‘𝑘)) ∈ V
135		f1oeq1 6770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3643	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (2^nd ‘(𝐺‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
138		dff1o3 6788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((2^nd ‘(𝐺‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡(2^nd ‘(𝐺‘𝑘)))
140		imain 6585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13557	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℕ₀)
143	142	nn0red 12480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℝ)
144	143	ltp1d 12089	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 < (𝑗 + 1))
145		fzdisj 13488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6020	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑗 ∈ (0...𝑁) → ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((2^nd ‘(𝐺‘𝑘)) “ ∅))
148		ima0 6037	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((2^nd ‘(𝐺‘𝑘)) “ ∅) = ∅
149	147, 148	eqtrdi 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑗 ∈ (0...𝑁) → ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ∅)
150	141, 149	sylan9req 2785	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) ∩ ((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁))) = ∅)
151		fun 6704	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6110	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ)
155	142, 154	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ ℕ)
156		nnuz 12812	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ℕ = (ℤ_≥‘1)
157	155, 156	eleqtrdi 2838	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ (ℤ_≥‘1))
158		elfzuz3 13458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑗 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑗))
159		fzsplit2 13486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((𝑗 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘𝑗)) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
160	157, 158, 159	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑗 ∈ (0...𝑁) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
161	160	imaeq2d 6020	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑗 ∈ (0...𝑁) → ((2^nd ‘(𝐺‘𝑘)) “ (1...𝑁)) = ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))))
162		f1ofo 6789	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((2^nd ‘(𝐺‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(𝐺‘𝑘)):(1...𝑁)–onto→(1...𝑁))
163		foima 6759	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2785	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑗 ∈ (0...𝑁) ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (1...𝑁))
166	165	ancoms 458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → ((2^nd ‘(𝐺‘𝑘)) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = (1...𝑁))
167	153, 166	eqtr3id 2778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6654	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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	ffvelcdmda 7038	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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	sselid 3941	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘 ∈ ℕ)
173	171, 172	nndivred 12216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 15394	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12204	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
181		nnrp 12939	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ⁺)
182	181	rpreccld 12981	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ⁺)
183	182	rpge0d 12975	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑘 ∈ ℕ → 0 ≤ (1 / 𝑘))
184	180, 183	absidd 15365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (1 / 𝑘) ∈ ℝ)
187	107	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (1 / -(⌊‘-(2 / 𝑐))) ∈ ℝ)
188	110	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (𝑐 / 2) ∈ ℝ)
189		eluzle 12782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐))) → -(⌊‘-(2 / 𝑐)) ≤ 𝑘)
190	189	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → -(⌊‘-(2 / 𝑐)) ≤ 𝑘)
191	59	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → -(⌊‘-(2 / 𝑐)) ∈ ℕ)
192	191	nnrpd 12969	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → -(⌊‘-(2 / 𝑐)) ∈ ℝ⁺)
193	117	nnrpd 12969	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → 𝑘 ∈ ℝ⁺)
194	192, 193	lerecd 12990	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (-(⌊‘-(2 / 𝑐)) ≤ 𝑘 ↔ (1 / 𝑘) ≤ (1 / -(⌊‘-(2 / 𝑐)))))
195	190, 194	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (1 / 𝑘) ≤ (1 / -(⌊‘-(2 / 𝑐))))
196	84	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (1 / -(⌊‘-(2 / 𝑐))) ≤ (𝑐 / 2))
197	186, 187, 188, 195, 196	letrd 11307	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (1 / 𝑘) ≤ (𝑐 / 2))
198	185, 197	eqbrtrd 5124	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (abs‘(1 / 𝑘)) ≤ (𝑐 / 2))
199		elsni 4602	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7391	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5112	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12975	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑐 ∈ ℝ⁺ → 0 ≤ (𝑐 / 2))
204		nncn 12170	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
205		nnne0 12196	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
206	204, 205	div0d 11933	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑘 ∈ ℕ → (0 / 𝑘) = 0)
207	206	abs00bd 15233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑘 ∈ ℕ → (abs‘(0 / 𝑘)) = 0)
208	207	breq1d 5112	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑘 ∈ ℕ → ((abs‘(0 / 𝑘)) ≤ (𝑐 / 2) ↔ 0 ≤ (𝑐 / 2)))
209	208	biimparc 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((0 ≤ (𝑐 / 2) ∧ 𝑘 ∈ ℕ) → (abs‘(0 / 𝑘)) ≤ (𝑐 / 2))
210	203, 209	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ ℕ) → (abs‘(0 / 𝑘)) ≤ (𝑐 / 2))
211	117, 210	syldan 591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑐 ∈ ℝ⁺ ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) → (abs‘(0 / 𝑘)) ≤ (𝑐 / 2))
212		elsni 4602	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7391	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5112	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 859	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5124	. . . . . . . . . . . . . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) → 𝜑)
223	222	anim1i 615	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) → (𝜑 ∧ 𝑘 ∈ ℕ))
224	173	renegcld 11581	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 15381	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 ∈ ℝ⁺ → ((𝑐 / 2) ∈ ℝ ∧ (𝑐 / 2) ∈ ℝ))
230	229	ad4antlr 733	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((𝑐 / 2) ∈ ℝ ∧ (𝑐 / 2) ∈ ℝ))
231		ltleadd 11637	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . . . . . . . . . . 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 694	. . . . . . . . . . . . . . . . . . . . . . . . 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 15401	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 11179	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 15381	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11179	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11179	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ ℝ⁺ → ((𝑐 / 2) + (𝑐 / 2)) ∈ ℝ)
240	239	ad4antlr 733	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((𝑐 / 2) + (𝑐 / 2)) ∈ ℝ)
241		lelttr 11240	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 695	. . . . . . . . . . . . . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((1^st ‘(𝐺‘𝑘))‘𝑚) ∈ ℝ)
247	246, 171	readdcld 11179	. . . . . . . . . . . . . . . . . . . . . . . . . 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 12216	. . . . . . . . . . . . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . . . . . . . . . . . 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 525	. . . . . . . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ (ℤ_≥‘-(⌊‘-(2 / 𝑐)))) ∧ 𝑗 ∈ (0...𝑁)) → (((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ))
253	87	ad3antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) → 𝐶:(1...𝑁)⟶(0[,]1))
254	253	ffvelcdmda 7038	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐶‘𝑚) ∈ (0[,]1))
255	33, 254	sselid 3941	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐶‘𝑚) ∈ ℝ)
256	74	rpreccld 12981	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑐 ∈ ℝ⁺ → (1 / -(⌊‘-(2 / 𝑐))) ∈ ℝ⁺)
257	256	rpxrd 12972	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑐 ∈ ℝ⁺ → (1 / -(⌊‘-(2 / 𝑐))) ∈ ℝ^*)
258	257	ad3antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (1 / -(⌊‘-(2 / 𝑐))) ∈ ℝ^*)
259	13	rexmet 24712	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
260		elbl 24309	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1450	. . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . 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 8815	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3448	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 𝑘 ∈ V
266		fnconstg 6730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑁) ∈ Fin)
269		inidm 4186	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
270		eqidd 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((1^st ‘(𝐺‘𝑘))‘𝑚) = ((1^st ‘(𝐺‘𝑘))‘𝑚))
271	265	fvconst2 7160	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑚 ∈ (1...𝑁) → (((1...𝑁) × {𝑘})‘𝑚) = 𝑘)
272	271	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1...𝑁) × {𝑘})‘𝑚) = 𝑘)
273	264, 267, 268, 268, 269, 270, 272	ofval 7644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) ∈ ℝ)
277	13	remetdval 24710	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . . . . . . . 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 5112	. . . . . . . . . . . . . . . . . . . . . . . . 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 279	. . . . . . . . . . . . . . . . . . . . . . 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 580	. . . . . . . . . . . . . . . . . . . . . 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 12937	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑐 ∈ ℝ⁺ → 𝑐 ∈ ℝ^*)
285	284	ad4antlr 733	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑐 ∈ ℝ^*)
286		elbl 24309	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1450	. . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . 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 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ ({1} ∪ {0}) ↔ (𝑧 ∈ {1} ∨ 𝑧 ∈ {0}))
290		fzofzp1 13701	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑣 ∈ (0..^𝑘) → (𝑣 + 1) ∈ (0...𝑘))
291		elsni 4602	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑧 ∈ {1} → 𝑧 = 1)
292	291	oveq2d 7385	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑧 ∈ {1} → (𝑣 + 𝑧) = (𝑣 + 1))
293	292	eleq1d 2813	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑧 ∈ {1} → ((𝑣 + 𝑧) ∈ (0...𝑘) ↔ (𝑣 + 1) ∈ (0...𝑘)))
294	290, 293	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑣 ∈ (0..^𝑘) → (𝑧 ∈ {1} → (𝑣 + 𝑧) ∈ (0...𝑘)))
295		elfzonn0 13644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑣 ∈ (0..^𝑘) → 𝑣 ∈ ℕ₀)
296	295	nn0cnd 12481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑣 ∈ (0..^𝑘) → 𝑣 ∈ ℂ)
297	296	addridd 11350	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑣 ∈ (0..^𝑘) → (𝑣 + 0) = 𝑣)
298		elfzofz 13612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑣 ∈ (0..^𝑘) → 𝑣 ∈ (0...𝑘))
299	297, 298	eqeltrd 2828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑣 ∈ (0..^𝑘) → (𝑣 + 0) ∈ (0...𝑘))
300		elsni 4602	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑧 ∈ {0} → 𝑧 = 0)
301	300	oveq2d 7385	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑧 ∈ {0} → (𝑣 + 𝑧) = (𝑣 + 0))
302	301	eleq1d 2813	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑧 ∈ {0} → ((𝑣 + 𝑧) ∈ (0...𝑘) ↔ (𝑣 + 0) ∈ (0...𝑘)))
303	299, 302	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑣 ∈ (0..^𝑘) → (𝑧 ∈ {0} → (𝑣 + 𝑧) ∈ (0...𝑘)))
304	294, 303	jaod 859	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑣 ∈ (0..^𝑘) → ((𝑧 ∈ {1} ∨ 𝑧 ∈ {0}) → (𝑣 + 𝑧) ∈ (0...𝑘)))
305	289, 304	biimtrid 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑣 ∈ (0..^𝑘) → (𝑧 ∈ ({1} ∪ {0}) → (𝑣 + 𝑧) ∈ (0...𝑘)))
306	305	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑣 ∈ (0..^𝑘) ∧ 𝑧 ∈ ({1} ∪ {0})) → (𝑣 + 𝑧) ∈ (0...𝑘))
307	306	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑣 ∈ (0..^𝑘) ∧ 𝑧 ∈ ({1} ∪ {0}))) → (𝑣 + 𝑧) ∈ (0...𝑘))
308		dffn3 6682	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((1^st ‘(𝐺‘𝑘)) Fn (1...𝑁) ↔ (1^st ‘(𝐺‘𝑘)):(1...𝑁)⟶ran (1^st ‘(𝐺‘𝑘)))
309	264, 308	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6687	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐺‘𝑘)):(1...𝑁)⟶(0..^𝑘))
312	311	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (1^st ‘(𝐺‘𝑘)):(1...𝑁)⟶(0..^𝑘))
313		fzfid 13914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
314	307, 312, 169, 313, 313, 269	off 7651	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6671	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → (1^st ‘(𝐺‘𝑘)) Fn (1...𝑁))
318	169	ffnd 6671	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((1^st ‘(𝐺‘𝑘))‘𝑚) = ((1^st ‘(𝐺‘𝑘))‘𝑚))
320		eqidd 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7644	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((1...𝑁) × {𝑘})‘𝑚) = 𝑘)
323	315, 316, 313, 313, 269, 321, 322	ofval 7644	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . . . . . . . . . . . . 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 750	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝐶:(1...𝑁)⟶(0[,]1))
329	328	ffvelcdmda 7038	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (𝐶‘𝑚) ∈ (0[,]1))
330	33, 329	sselid 3941	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (𝐶‘𝑚) ∈ ℝ)
331	248	adantl3r 750	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 24710	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → ((1^st ‘(𝐺‘𝑘))‘𝑚) ∈ ℂ)
335	171	recnd 11178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘 ∈ ℂ)
337	205	ad3antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘 ≠ 0)
338	334, 335, 336, 337	divdird 11972	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) ∈ ℂ)
340	339	adantlr 715	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) ∈ ℂ)
341	340, 174	subnegd 11516	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 750	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (𝐶‘𝑚) ∈ ℂ)
346	102	adantllr 719	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) ∈ ℝ)
347	346	adantlr 715	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) ∈ ℝ)
348	347	recnd 11178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘))‘𝑚) / 𝑘) ∈ ℂ)
349	174	adantl3r 750	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11496	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11537	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 750	. . . . . . . . . . . . . . . . . . . . . . . . . 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 12404	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 ∈ ℝ⁺ → ((𝑐 / 2) + (𝑐 / 2)) = 𝑐)
357	356	eqcomd 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑐 ∈ ℝ⁺ → 𝑐 = ((𝑐 / 2) + (𝑐 / 2)))
358	357	ad4antlr 733	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝜑 ∧ 𝐶 ∈ 𝐼) ∧ 𝑐 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑐 = ((𝑐 / 2) + (𝑐 / 2)))
359	355, 358	breq12d 5115	. . . . . . . . . . . . . . . . . . . . . . . . 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 279	. . . . . . . . . . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . . . . . . . . . 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 294	. . . . . . . . . . . . . . . . . . . . 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 3145	. . . . . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) → 𝑘 ∈ ℕ)
366		elfznn0 13557	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑣 ∈ (0...𝑘) → 𝑣 ∈ ℕ₀)
367	366	nn0red 12480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑣 ∈ (0...𝑘) → 𝑣 ∈ ℝ)
368		nndivre 12203	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑣 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑣 / 𝑘) ∈ ℝ)
369	367, 368	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑣 / 𝑘) ∈ ℝ)
370		elfzle1 13464	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑣 ∈ (0...𝑘) → 0 ≤ 𝑣)
371	367, 370	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑣 ∈ (0...𝑘) → (𝑣 ∈ ℝ ∧ 0 ≤ 𝑣))
372	181	rpregt0d 12977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
373		divge0 12028	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑣 ∈ ℝ ∧ 0 ≤ 𝑣) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝑣 / 𝑘))
374	371, 372, 373	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑣 / 𝑘))
375		elfzle2 13465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑣 ∈ (0...𝑘) → 𝑣 ≤ 𝑘)
376	375	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑣 ≤ 𝑘)
377	367	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑣 ∈ ℝ)
378		1red 11151	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ)
379	181	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ⁺)
380	377, 378, 379	ledivmuld 13024	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑣 / 𝑘) ≤ 1 ↔ 𝑣 ≤ (𝑘 · 1)))
381	204	mulridd 11167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 ∈ ℕ → (𝑘 · 1) = 𝑘)
382	381	breq2d 5114	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ ℕ → (𝑣 ≤ (𝑘 · 1) ↔ 𝑣 ≤ 𝑘))
383	382	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑣 ≤ (𝑘 · 1) ↔ 𝑣 ≤ 𝑘))
384	380, 383	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑣 / 𝑘) ≤ 1 ↔ 𝑣 ≤ 𝑘))
385	376, 384	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑣 / 𝑘) ≤ 1)
386		elicc01 13403	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑣 / 𝑘) ∈ (0[,]1) ↔ ((𝑣 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝑣 / 𝑘) ∧ (𝑣 / 𝑘) ≤ 1))
387	369, 374, 385, 386	syl3anbrc 1344	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑣 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑣 / 𝑘) ∈ (0[,]1))
388	387	ancoms 458	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ ℕ ∧ 𝑣 ∈ (0...𝑘)) → (𝑣 / 𝑘) ∈ (0[,]1))
389		elsni 4602	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ {𝑘} → 𝑧 = 𝑘)
390	389	oveq2d 7385	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ {𝑘} → (𝑣 / 𝑧) = (𝑣 / 𝑘))
391	390	eleq1d 2813	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ {𝑘} → ((𝑣 / 𝑧) ∈ (0[,]1) ↔ (𝑣 / 𝑘) ∈ (0[,]1)))
392	388, 391	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℕ ∧ 𝑣 ∈ (0...𝑘)) → (𝑧 ∈ {𝑘} → (𝑣 / 𝑧) ∈ (0[,]1)))
393	392	impr 454	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑘 ∈ ℕ ∧ (𝑣 ∈ (0...𝑘) ∧ 𝑧 ∈ {𝑘})) → (𝑣 / 𝑧) ∈ (0[,]1))
394	365, 393	sylan 580	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑣 ∈ (0...𝑘) ∧ 𝑧 ∈ {𝑘})) → (𝑣 / 𝑧) ∈ (0[,]1))
395	265	fconst 6728	. . . . . . . . . . . . . . . . . . . . . . . 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 7651	. . . . . . . . . . . . . . . . . . . . . 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 6671	. . . . . . . . . . . . . . . . . . . . 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 580	. . . . . . . . . . . . . . . . . . . 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 525	. . . . . . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0[,]1) ∈ V
403	402, 37	elmap 8821	. . . . . . . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . . . . 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 580	. . . . . . . . . . . . . . . . . . 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 526	. . . . . . . . . . . . . . . . . 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 3927	. . . . . . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . . . . . . 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 8854	. . . . . . . . . . . . . . . . . . . 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 624	. . . . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . . . 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	imbitrrdi 252	. . . . . . . . . . . . . . . . 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 3937	. . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . 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 3133	. . . . . . . . . . . . . 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 415	. . . . . . . . . . . . 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 1355	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 ∈ ℕ → (𝑛 ∈ (1...𝑁) → (𝑟 ∈ { ≤ , ^◡ ≤ } → ∃𝑗 ∈ (0...𝑁)0𝑟𝑋))))
422	421	imp43 427	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ^◡ ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟𝑋)
423		r19.29 3094	. . . . . . . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . . . . . . . . . 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	eqtr4di 2782	. . . . . . . . . . . . . . . . . . . . . . 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 5114	. . . . . . . . . . . . . . . . . . . . . 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 3585	. . . . . . . . . . . . . . . . . . . . 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 3130	. . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . 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 3190	. . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . 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 719	. . . . . . . . . . . . 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 3134	. . . . . . . . . . 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 454	. . . . . . . 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 681	. . . . . . 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 3140	. . . . . 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 456	. . . . 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 3158	. . . 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	imbitrrdi 252	. . 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 3133	. 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 3263	. 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	imbitrdi 251	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 {cab 2707 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 Vcvv 3444 ∪ cun 3909 ∩ cin 3910 ⊆ wss 3911 ∅c0 4292 {csn 4585 {cpr 4587 ∪ cuni 4867 class class class wbr 5102 × cxp 5629 ^◡ccnv 5630 ran crn 5632 ↾ cres 5633 “ cima 5634 ∘ ccom 5635 Fun wfun 6493 Fn wfn 6494 ⟶wf 6495 –onto→wfo 6497 –1-1-onto→wf1o 6498 ‘cfv 6499 (class class class)co 7369 ∘_f cof 7631 1^st c1st 7945 2^nd c2nd 7946 ↑_m cmap 8776 Xcixp 8847 Fincfn 8895 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 − cmin 11381 -cneg 11382 / cdiv 11811 ℕcn 12162 2c2 12217 ℕ₀cn0 12418 ℤcz 12505 ℤ_≥cuz 12769 ℝ⁺crp 12927 (,)cioo 13282 [,]cicc 13285 ...cfz 13444 ..^cfzo 13591 ⌊cfl 13728 abscabs 15176 ↾_t crest 17359 topGenctg 17376 ∏_tcpt 17377 ∞Metcxmet 21281 ballcbl 21283 Topctop 22813 Cn ccn 23144

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fi 9338 df-sup 9369 df-inf 9370 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-n0 12419 df-z 12506 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-seq 13943 df-exp 14003 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-rest 17361 df-topgen 17382 df-pt 17383 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-top 22814 df-topon 22831 df-bases 22866

This theorem is referenced by: poimirlem30 37637

W3C validator