Theorem poimirlem30 35544

Description: Lemma for poimir 35547 combining poimirlem29 35543 with bwth 22307. (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‘(((1st ‘(𝐺‘𝑘)) ∘f + ((((2nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(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 (1st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘))
poimirlem30.4	⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ◡ ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟𝑋)

Assertion

Ref	Expression
poimirlem30	⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))

Distinct variable groups:   𝑓,𝑗,𝑘,𝑛,𝑧   𝜑,𝑗,𝑛   𝑗,𝐹,𝑛   𝑗,𝑁,𝑛   𝜑,𝑘   𝑓,𝑁,𝑘   𝜑,𝑧   𝑓,𝐹,𝑘,𝑧   𝑧,𝑁   𝑗,𝑐,𝑘,𝑛,𝑟,𝑣,𝑧,𝜑   𝑓,𝑐,𝐹,𝑟,𝑣   𝐺,𝑐,𝑓,𝑗,𝑘,𝑛,𝑟,𝑣,𝑧   𝐼,𝑐,𝑓,𝑗,𝑘,𝑛,𝑟,𝑣,𝑧   𝑁,𝑐,𝑟,𝑣   𝑅,𝑐,𝑓,𝑗,𝑘,𝑛,𝑟,𝑣,𝑧   𝑋,𝑐,𝑓,𝑟,𝑣,𝑧

Allowed substitution hints:   𝜑(𝑓)   𝑋(𝑗,𝑘,𝑛)

Proof of Theorem poimirlem30

Dummy variables 𝑖 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfzonn0 13287	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑘) → 𝑖 ∈ ℕ0)
2	1	nn0red 12151	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑘) → 𝑖 ∈ ℝ)
3		nndivre 11871	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ)
4	2, 3	sylan 583	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ)
5		elfzole1 13251	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑘) → 0 ≤ 𝑖)
6	2, 5	jca 515	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑘) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
7		nnrp 12597	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
8	7	rpregt0d 12634	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
9		divge0 11701	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ℝ ∧ 0 ≤ 𝑖) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝑖 / 𝑘))
10	6, 8, 9	syl2an 599	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑖 / 𝑘))
11		elfzo0le 13286	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑘) → 𝑖 ≤ 𝑘)
12	11	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 𝑖 ≤ 𝑘)
13	2	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 𝑖 ∈ ℝ)
14		1red 10834	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ)
15	7	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ+)
16	13, 14, 15	ledivmuld 12681	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖 ≤ (𝑘 · 1)))
17		nncn 11838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
18	17	mulid1d 10850	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → (𝑘 · 1) = 𝑘)
19	18	breq2d 5065	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → (𝑖 ≤ (𝑘 · 1) ↔ 𝑖 ≤ 𝑘))
20	19	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 ≤ (𝑘 · 1) ↔ 𝑖 ≤ 𝑘))
21	16, 20	bitrd 282	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖 ≤ 𝑘))
22	12, 21	mpbird 260	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ≤ 1)
23		elicc01 13054	. . . . . . . . . . . . . 14 ⊢ ((𝑖 / 𝑘) ∈ (0[,]1) ↔ ((𝑖 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝑖 / 𝑘) ∧ (𝑖 / 𝑘) ≤ 1))
24	4, 10, 22, 23	syl3anbrc 1345	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ (0[,]1))
25	24	ancoms 462	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0..^𝑘)) → (𝑖 / 𝑘) ∈ (0[,]1))
26		elsni 4558	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ {𝑘} → 𝑗 = 𝑘)
27	26	oveq2d 7229	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ {𝑘} → (𝑖 / 𝑗) = (𝑖 / 𝑘))
28	27	eleq1d 2822	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ {𝑘} → ((𝑖 / 𝑗) ∈ (0[,]1) ↔ (𝑖 / 𝑘) ∈ (0[,]1)))
29	25, 28	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0..^𝑘)) → (𝑗 ∈ {𝑘} → (𝑖 / 𝑗) ∈ (0[,]1)))
30	29	impr 458	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ (𝑖 ∈ (0..^𝑘) ∧ 𝑗 ∈ {𝑘})) → (𝑖 / 𝑗) ∈ (0[,]1))
31	30	adantll 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑖 ∈ (0..^𝑘) ∧ 𝑗 ∈ {𝑘})) → (𝑖 / 𝑗) ∈ (0[,]1))
32		poimirlem30.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ℕ⟶((ℕ0 ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
33	32	ffvelrnda 6904	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ((ℕ0 ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
34		xp1st 7793	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑘) ∈ ((ℕ0 ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(𝐺‘𝑘)) ∈ (ℕ0 ↑m (1...𝑁)))
35		elmapfn 8546	. . . . . . . . . . 11 ⊢ ((1st ‘(𝐺‘𝑘)) ∈ (ℕ0 ↑m (1...𝑁)) → (1st ‘(𝐺‘𝑘)) Fn (1...𝑁))
36	33, 34, 35	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐺‘𝑘)) Fn (1...𝑁))
37		poimirlem30.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran (1st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘))
38		df-f 6384	. . . . . . . . . 10 ⊢ ((1st ‘(𝐺‘𝑘)):(1...𝑁)⟶(0..^𝑘) ↔ ((1st ‘(𝐺‘𝑘)) Fn (1...𝑁) ∧ ran (1st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘)))
39	36, 37, 38	sylanbrc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐺‘𝑘)):(1...𝑁)⟶(0..^𝑘))
40		vex 3412	. . . . . . . . . . 11 ⊢ 𝑘 ∈ V
41	40	fconst 6605	. . . . . . . . . 10 ⊢ ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘}
42	41	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘})
43		fzfid 13546	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑁) ∈ Fin)
44		inidm 4133	. . . . . . . . 9 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
45	31, 39, 42, 43, 43, 44	off 7486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
46		poimir.i	. . . . . . . . . 10 ⊢ 𝐼 = ((0[,]1) ↑m (1...𝑁))
47	46	eleq2i 2829	. . . . . . . . 9 ⊢ (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑m (1...𝑁)))
48		ovex 7246	. . . . . . . . . 10 ⊢ (0[,]1) ∈ V
49		ovex 7246	. . . . . . . . . 10 ⊢ (1...𝑁) ∈ V
50	48, 49	elmap 8552	. . . . . . . . 9 ⊢ (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑m (1...𝑁)) ↔ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
51	47, 50	bitri 278	. . . . . . . 8 ⊢ (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
52	45, 51	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ 𝐼)
53	52	fmpttd 6932	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))):ℕ⟶𝐼)
54	53	frnd 6553	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ⊆ 𝐼)
55		ominf 8890	. . . . . . 7 ⊢ ¬ ω ∈ Fin
56		nnenom 13553	. . . . . . . . 9 ⊢ ℕ ≈ ω
57		enfi 8865	. . . . . . . . 9 ⊢ (ℕ ≈ ω → (ℕ ∈ Fin ↔ ω ∈ Fin))
58	56, 57	ax-mp 5	. . . . . . . 8 ⊢ (ℕ ∈ Fin ↔ ω ∈ Fin)
59		iunid 4969	. . . . . . . . . . 11 ⊢ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))){𝑐} = ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))
60	59	imaeq2i 5927	. . . . . . . . . 10 ⊢ (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))){𝑐}) = (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))))
61		imaiun 7058	. . . . . . . . . 10 ⊢ (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))){𝑐}) = ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐})
62		ovex 7246	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ V
63		eqid 2737	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) = (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))
64	62, 63	fnmpti 6521	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) Fn ℕ
65		dffn3 6558	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) Fn ℕ ↔ (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))):ℕ⟶ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))))
66	64, 65	mpbi 233	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))):ℕ⟶ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))
67		fimacnv 6567	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))):ℕ⟶ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) → (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) = ℕ)
68	66, 67	ax-mp 5	. . . . . . . . . 10 ⊢ (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) = ℕ
69	60, 61, 68	3eqtr3ri 2774	. . . . . . . . 9 ⊢ ℕ = ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐})
70	69	eleq1i 2828	. . . . . . . 8 ⊢ (ℕ ∈ Fin ↔ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
71	58, 70	bitr3i 280	. . . . . . 7 ⊢ (ω ∈ Fin ↔ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
72	55, 71	mtbi 325	. . . . . 6 ⊢ ¬ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin
73		ralnex 3158	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ¬ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐)
74	73	rexbii 3170	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ∃𝑖 ∈ ℕ ¬ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐)
75		rexnal 3160	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ ℕ ¬ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ¬ ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐)
76	74, 75	bitri 278	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ¬ ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐)
77	76	ralbii 3088	. . . . . . . . 9 ⊢ (∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ¬ ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐)
78		ralnex 3158	. . . . . . . . 9 ⊢ (∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ¬ ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ¬ ∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐)
79	77, 78	bitri 278	. . . . . . . 8 ⊢ (∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ¬ ∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐)
80		nnuz 12477	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ≥‘1)
81		elnnuz 12478	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ ↔ 𝑖 ∈ (ℤ≥‘1))
82		fzouzsplit 13277	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (ℤ≥‘1) → (ℤ≥‘1) = ((1..^𝑖) ∪ (ℤ≥‘𝑖)))
83	81, 82	sylbi 220	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℕ → (ℤ≥‘1) = ((1..^𝑖) ∪ (ℤ≥‘𝑖)))
84	80, 83	syl5eq 2790	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℕ → ℕ = ((1..^𝑖) ∪ (ℤ≥‘𝑖)))
85	84	difeq1d 4036	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ → (ℕ ∖ (1..^𝑖)) = (((1..^𝑖) ∪ (ℤ≥‘𝑖)) ∖ (1..^𝑖)))
86		uncom 4067	. . . . . . . . . . . . . . . 16 ⊢ ((1..^𝑖) ∪ (ℤ≥‘𝑖)) = ((ℤ≥‘𝑖) ∪ (1..^𝑖))
87	86	difeq1i 4033	. . . . . . . . . . . . . . 15 ⊢ (((1..^𝑖) ∪ (ℤ≥‘𝑖)) ∖ (1..^𝑖)) = (((ℤ≥‘𝑖) ∪ (1..^𝑖)) ∖ (1..^𝑖))
88		difun2 4395	. . . . . . . . . . . . . . 15 ⊢ (((ℤ≥‘𝑖) ∪ (1..^𝑖)) ∖ (1..^𝑖)) = ((ℤ≥‘𝑖) ∖ (1..^𝑖))
89	87, 88	eqtri 2765	. . . . . . . . . . . . . 14 ⊢ (((1..^𝑖) ∪ (ℤ≥‘𝑖)) ∖ (1..^𝑖)) = ((ℤ≥‘𝑖) ∖ (1..^𝑖))
90	85, 89	eqtrdi 2794	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ℕ → (ℕ ∖ (1..^𝑖)) = ((ℤ≥‘𝑖) ∖ (1..^𝑖)))
91		difss 4046	. . . . . . . . . . . . 13 ⊢ ((ℤ≥‘𝑖) ∖ (1..^𝑖)) ⊆ (ℤ≥‘𝑖)
92	90, 91	eqsstrdi 3955	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ℕ → (ℕ ∖ (1..^𝑖)) ⊆ (ℤ≥‘𝑖))
93		ssralv 3967	. . . . . . . . . . . 12 ⊢ ((ℕ ∖ (1..^𝑖)) ⊆ (ℤ≥‘𝑖) → (∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑘 ∈ (ℕ ∖ (1..^𝑖)) ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐))
94	92, 93	syl 17	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ → (∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑘 ∈ (ℕ ∖ (1..^𝑖)) ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐))
95		impexp 454	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ℕ ∧ ¬ 𝑘 ∈ (1..^𝑖)) → ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐) ↔ (𝑘 ∈ ℕ → (¬ 𝑘 ∈ (1..^𝑖) → ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐)))
96		eldif 3876	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℕ ∖ (1..^𝑖)) ↔ (𝑘 ∈ ℕ ∧ ¬ 𝑘 ∈ (1..^𝑖)))
97	96	imbi1i 353	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (ℕ ∖ (1..^𝑖)) → ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐) ↔ ((𝑘 ∈ ℕ ∧ ¬ 𝑘 ∈ (1..^𝑖)) → ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐))
98		con34b 319	. . . . . . . . . . . . . . . 16 ⊢ ((((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖)) ↔ (¬ 𝑘 ∈ (1..^𝑖) → ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐))
99	98	imbi2i 339	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ → (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖))) ↔ (𝑘 ∈ ℕ → (¬ 𝑘 ∈ (1..^𝑖) → ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐)))
100	95, 97, 99	3bitr4i 306	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (ℕ ∖ (1..^𝑖)) → ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐) ↔ (𝑘 ∈ ℕ → (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖))))
101	100	albii 1827	. . . . . . . . . . . . 13 ⊢ (∀𝑘(𝑘 ∈ (ℕ ∖ (1..^𝑖)) → ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐) ↔ ∀𝑘(𝑘 ∈ ℕ → (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖))))
102		df-ral 3066	. . . . . . . . . . . . 13 ⊢ (∀𝑘 ∈ (ℕ ∖ (1..^𝑖)) ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ∀𝑘(𝑘 ∈ (ℕ ∖ (1..^𝑖)) → ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐))
103		vex 3412	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
104	63	mptiniseg 6102	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ V → (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) = {𝑘 ∈ ℕ ∣ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐})
105	103, 104	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) = {𝑘 ∈ ℕ ∣ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐}
106	105	sseq1i 3929	. . . . . . . . . . . . . 14 ⊢ ((◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ⊆ (1..^𝑖) ↔ {𝑘 ∈ ℕ ∣ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐} ⊆ (1..^𝑖))
107		rabss 3985	. . . . . . . . . . . . . 14 ⊢ ({𝑘 ∈ ℕ ∣ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐} ⊆ (1..^𝑖) ↔ ∀𝑘 ∈ ℕ (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖)))
108		df-ral 3066	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ ℕ (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖)) ↔ ∀𝑘(𝑘 ∈ ℕ → (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖))))
109	106, 107, 108	3bitri 300	. . . . . . . . . . . . 13 ⊢ ((◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ⊆ (1..^𝑖) ↔ ∀𝑘(𝑘 ∈ ℕ → (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖))))
110	101, 102, 109	3bitr4i 306	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ (ℕ ∖ (1..^𝑖)) ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ⊆ (1..^𝑖))
111		fzofi 13547	. . . . . . . . . . . . 13 ⊢ (1..^𝑖) ∈ Fin
112		ssfi 8851	. . . . . . . . . . . . 13 ⊢ (((1..^𝑖) ∈ Fin ∧ (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ⊆ (1..^𝑖)) → (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
113	111, 112	mpan 690	. . . . . . . . . . . 12 ⊢ ((◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ⊆ (1..^𝑖) → (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
114	110, 113	sylbi 220	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ (ℕ ∖ (1..^𝑖)) ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
115	94, 114	syl6 35	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℕ → (∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin))
116	115	rexlimiv 3199	. . . . . . . . 9 ⊢ (∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
117	116	ralimi 3083	. . . . . . . 8 ⊢ (∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
118	79, 117	sylbir 238	. . . . . . 7 ⊢ (¬ ∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
119		iunfi 8964	. . . . . . . 8 ⊢ ((ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∈ Fin ∧ ∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin) → ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
120	119	ex 416	. . . . . . 7 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∈ Fin → (∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin → ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin))
121	118, 120	syl5 34	. . . . . 6 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∈ Fin → (¬ ∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin))
122	72, 121	mt3i 151	. . . . 5 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐)
123		ssrexv 3968	. . . . 5 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ⊆ 𝐼 → (∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐))
124	54, 122, 123	syl2im 40	. . . 4 ⊢ (𝜑 → (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐))
125		unitssre 13087	. . . . . . . . . . . 12 ⊢ (0[,]1) ⊆ ℝ
126		elmapi 8530	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ((0[,]1) ↑m (1...𝑁)) → 𝑐:(1...𝑁)⟶(0[,]1))
127	126, 46	eleq2s 2856	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝐼 → 𝑐:(1...𝑁)⟶(0[,]1))
128	127	ffvelrnda 6904	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) → (𝑐‘𝑚) ∈ (0[,]1))
129	125, 128	sseldi 3899	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) → (𝑐‘𝑚) ∈ ℝ)
130		nnrp 12597	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℝ+)
131	130	rpreccld 12638	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ → (1 / 𝑖) ∈ ℝ+)
132		eqid 2737	. . . . . . . . . . . . 13 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
133	132	rexmet 23688	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
134		blcntr 23311	. . . . . . . . . . . 12 ⊢ ((((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) ∧ (𝑐‘𝑚) ∈ ℝ ∧ (1 / 𝑖) ∈ ℝ+) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
135	133, 134	mp3an1 1450	. . . . . . . . . . 11 ⊢ (((𝑐‘𝑚) ∈ ℝ ∧ (1 / 𝑖) ∈ ℝ+) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
136	129, 131, 135	syl2an 599	. . . . . . . . . 10 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) ∧ 𝑖 ∈ ℕ) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
137	136	an32s 652	. . . . . . . . 9 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
138		fveq1 6716	. . . . . . . . . 10 ⊢ (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) = (𝑐‘𝑚))
139	138	eleq1d 2822	. . . . . . . . 9 ⊢ (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → ((((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
140	137, 139	syl5ibrcom 250	. . . . . . . 8 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
141	140	ralrimdva 3110	. . . . . . 7 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
142	141	reximdv 3192	. . . . . 6 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → (∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
143	142	ralimdva 3100	. . . . 5 ⊢ (𝑐 ∈ 𝐼 → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
144	143	reximia 3165	. . . 4 ⊢ (∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
145	124, 144	syl6 35	. . 3 ⊢ (𝜑 → (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
146		poimir.r	. . . . . . . 8 ⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))
147	49, 48	ixpconst 8588	. . . . . . . . 9 ⊢ X𝑛 ∈ (1...𝑁)(0[,]1) = ((0[,]1) ↑m (1...𝑁))
148	46, 147	eqtr4i 2768	. . . . . . . 8 ⊢ 𝐼 = X𝑛 ∈ (1...𝑁)(0[,]1)
149	146, 148	oveq12i 7225	. . . . . . 7 ⊢ (𝑅 ↾t 𝐼) = ((∏t‘((1...𝑁) × {(topGen‘ran (,))})) ↾t X𝑛 ∈ (1...𝑁)(0[,]1))
150		fzfid 13546	. . . . . . . . 9 ⊢ (⊤ → (1...𝑁) ∈ Fin)
151		retop 23659	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) ∈ Top
152	151	fconst6 6609	. . . . . . . . . 10 ⊢ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top
153	152	a1i 11	. . . . . . . . 9 ⊢ (⊤ → ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top)
154	48	a1i 11	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ (1...𝑁)) → (0[,]1) ∈ V)
155	150, 153, 154	ptrest 35513	. . . . . . . 8 ⊢ (⊤ → ((∏t‘((1...𝑁) × {(topGen‘ran (,))})) ↾t X𝑛 ∈ (1...𝑁)(0[,]1)) = (∏t‘(𝑛 ∈ (1...𝑁) ↦ ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1)))))
156	155	mptru 1550	. . . . . . 7 ⊢ ((∏t‘((1...𝑁) × {(topGen‘ran (,))})) ↾t X𝑛 ∈ (1...𝑁)(0[,]1)) = (∏t‘(𝑛 ∈ (1...𝑁) ↦ ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1))))
157		fvex 6730	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) ∈ V
158	157	fvconst2 7019	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
159	158	oveq1d 7228	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑁) → ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1)) = ((topGen‘ran (,)) ↾t (0[,]1)))
160	159	mpteq2ia 5146	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...𝑁) ↦ ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1))) = (𝑛 ∈ (1...𝑁) ↦ ((topGen‘ran (,)) ↾t (0[,]1)))
161		fconstmpt 5611	. . . . . . . . 9 ⊢ ((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))}) = (𝑛 ∈ (1...𝑁) ↦ ((topGen‘ran (,)) ↾t (0[,]1)))
162	160, 161	eqtr4i 2768	. . . . . . . 8 ⊢ (𝑛 ∈ (1...𝑁) ↦ ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1))) = ((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))})
163	162	fveq2i 6720	. . . . . . 7 ⊢ (∏t‘(𝑛 ∈ (1...𝑁) ↦ ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1)))) = (∏t‘((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))}))
164	149, 156, 163	3eqtri 2769	. . . . . 6 ⊢ (𝑅 ↾t 𝐼) = (∏t‘((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))}))
165		fzfi 13545	. . . . . . 7 ⊢ (1...𝑁) ∈ Fin
166		dfii2 23779	. . . . . . . . 9 ⊢ II = ((topGen‘ran (,)) ↾t (0[,]1))
167		iicmp 23783	. . . . . . . . 9 ⊢ II ∈ Comp
168	166, 167	eqeltrri 2835	. . . . . . . 8 ⊢ ((topGen‘ran (,)) ↾t (0[,]1)) ∈ Comp
169	168	fconst6 6609	. . . . . . 7 ⊢ ((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))}):(1...𝑁)⟶Comp
170		ptcmpfi 22710	. . . . . . 7 ⊢ (((1...𝑁) ∈ Fin ∧ ((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))}):(1...𝑁)⟶Comp) → (∏t‘((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))})) ∈ Comp)
171	165, 169, 170	mp2an 692	. . . . . 6 ⊢ (∏t‘((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))})) ∈ Comp
172	164, 171	eqeltri 2834	. . . . 5 ⊢ (𝑅 ↾t 𝐼) ∈ Comp
173		rehaus 23696	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) ∈ Haus
174	173	fconst6 6609	. . . . . . . . . . 11 ⊢ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Haus
175		pthaus 22535	. . . . . . . . . . 11 ⊢ (((1...𝑁) ∈ Fin ∧ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Haus) → (∏t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Haus)
176	165, 174, 175	mp2an 692	. . . . . . . . . 10 ⊢ (∏t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Haus
177	146, 176	eqeltri 2834	. . . . . . . . 9 ⊢ 𝑅 ∈ Haus
178		haustop 22228	. . . . . . . . 9 ⊢ (𝑅 ∈ Haus → 𝑅 ∈ Top)
179	177, 178	ax-mp 5	. . . . . . . 8 ⊢ 𝑅 ∈ Top
180		reex 10820	. . . . . . . . . 10 ⊢ ℝ ∈ V
181		mapss 8570	. . . . . . . . . 10 ⊢ ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑m (1...𝑁)) ⊆ (ℝ ↑m (1...𝑁)))
182	180, 125, 181	mp2an 692	. . . . . . . . 9 ⊢ ((0[,]1) ↑m (1...𝑁)) ⊆ (ℝ ↑m (1...𝑁))
183	46, 182	eqsstri 3935	. . . . . . . 8 ⊢ 𝐼 ⊆ (ℝ ↑m (1...𝑁))
184		uniretop 23660	. . . . . . . . . . 11 ⊢ ℝ = ∪ (topGen‘ran (,))
185	146, 184	ptuniconst 22495	. . . . . . . . . 10 ⊢ (((1...𝑁) ∈ Fin ∧ (topGen‘ran (,)) ∈ Top) → (ℝ ↑m (1...𝑁)) = ∪ 𝑅)
186	165, 151, 185	mp2an 692	. . . . . . . . 9 ⊢ (ℝ ↑m (1...𝑁)) = ∪ 𝑅
187	186	restuni 22059	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝐼 ⊆ (ℝ ↑m (1...𝑁))) → 𝐼 = ∪ (𝑅 ↾t 𝐼))
188	179, 183, 187	mp2an 692	. . . . . . 7 ⊢ 𝐼 = ∪ (𝑅 ↾t 𝐼)
189	188	bwth 22307	. . . . . 6 ⊢ (((𝑅 ↾t 𝐼) ∈ Comp ∧ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ⊆ 𝐼 ∧ ¬ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∈ Fin) → ∃𝑐 ∈ 𝐼 𝑐 ∈ ((limPt‘(𝑅 ↾t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))))
190	189	3expia 1123	. . . . 5 ⊢ (((𝑅 ↾t 𝐼) ∈ Comp ∧ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ⊆ 𝐼) → (¬ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ 𝐼 𝑐 ∈ ((limPt‘(𝑅 ↾t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))))))
191	172, 54, 190	sylancr 590	. . . 4 ⊢ (𝜑 → (¬ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ 𝐼 𝑐 ∈ ((limPt‘(𝑅 ↾t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))))))
192		cmptop 22292	. . . . . . . . 9 ⊢ ((𝑅 ↾t 𝐼) ∈ Comp → (𝑅 ↾t 𝐼) ∈ Top)
193	172, 192	ax-mp 5	. . . . . . . 8 ⊢ (𝑅 ↾t 𝐼) ∈ Top
194	188	islp3 22043	. . . . . . . 8 ⊢ (((𝑅 ↾t 𝐼) ∈ Top ∧ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ⊆ 𝐼 ∧ 𝑐 ∈ 𝐼) → (𝑐 ∈ ((limPt‘(𝑅 ↾t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) ↔ ∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)))
195	193, 194	mp3an1 1450	. . . . . . 7 ⊢ ((ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ⊆ 𝐼 ∧ 𝑐 ∈ 𝐼) → (𝑐 ∈ ((limPt‘(𝑅 ↾t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) ↔ ∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)))
196	54, 195	sylan 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑐 ∈ ((limPt‘(𝑅 ↾t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) ↔ ∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)))
197		fzfid 13546	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → (1...𝑁) ∈ Fin)
198	152	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top)
199		nnrecre 11872	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ → (1 / 𝑖) ∈ ℝ)
200	199	rexrd 10883	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℕ → (1 / 𝑖) ∈ ℝ*)
201		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
202	132, 201	tgioo 23693	. . . . . . . . . . . . . . . . . 18 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
203	202	blopn 23398	. . . . . . . . . . . . . . . . 17 ⊢ ((((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) ∧ (𝑐‘𝑚) ∈ ℝ ∧ (1 / 𝑖) ∈ ℝ*) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
204	133, 203	mp3an1 1450	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐‘𝑚) ∈ ℝ ∧ (1 / 𝑖) ∈ ℝ*) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
205	129, 200, 204	syl2an 599	. . . . . . . . . . . . . . 15 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) ∧ 𝑖 ∈ ℕ) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
206	205	an32s 652	. . . . . . . . . . . . . 14 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
207	157	fvconst2 7019	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑚) = (topGen‘ran (,)))
208	207	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑚) = (topGen‘ran (,)))
209	206, 208	eleqtrrd 2841	. . . . . . . . . . . . 13 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑚))
210		noel 4245	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝑚 ∈ ∅
211		difid 4285	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑁) ∖ (1...𝑁)) = ∅
212	211	eleq2i 2829	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ((1...𝑁) ∖ (1...𝑁)) ↔ 𝑚 ∈ ∅)
213	210, 212	mtbir 326	. . . . . . . . . . . . . . 15 ⊢ ¬ 𝑚 ∈ ((1...𝑁) ∖ (1...𝑁))
214	213	pm2.21i 119	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ((1...𝑁) ∖ (1...𝑁)) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑚))
215	214	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ ((1...𝑁) ∖ (1...𝑁))) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑚))
216	197, 198, 197, 209, 215	ptopn 22480	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (∏t‘((1...𝑁) × {(topGen‘ran (,))})))
217	216, 146	eleqtrrdi 2849	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ 𝑅)
218		ovex 7246	. . . . . . . . . . . . 13 ⊢ ((0[,]1) ↑m (1...𝑁)) ∈ V
219	46, 218	eqeltri 2834	. . . . . . . . . . . 12 ⊢ 𝐼 ∈ V
220		elrestr 16933	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Haus ∧ 𝐼 ∈ V ∧ X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ 𝑅) → (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∈ (𝑅 ↾t 𝐼))
221	177, 219, 220	mp3an12 1453	. . . . . . . . . . 11 ⊢ (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ 𝑅 → (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∈ (𝑅 ↾t 𝐼))
222	217, 221	syl 17	. . . . . . . . . 10 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∈ (𝑅 ↾t 𝐼))
223		difss 4046	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))
224		imassrn 5940	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ⊆ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))
225	223, 224	sstri 3910	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))
226	225, 54	sstrid 3912	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ 𝐼)
227		haust1 22249	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Haus → 𝑅 ∈ Fre)
228	177, 227	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 𝑅 ∈ Fre
229		restt1 22264	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Fre ∧ 𝐼 ∈ V) → (𝑅 ↾t 𝐼) ∈ Fre)
230	228, 219, 229	mp2an 692	. . . . . . . . . . . 12 ⊢ (𝑅 ↾t 𝐼) ∈ Fre
231		funmpt 6418	. . . . . . . . . . . . . 14 ⊢ Fun (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))
232		imafi 8853	. . . . . . . . . . . . . 14 ⊢ ((Fun (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ (1..^𝑖) ∈ Fin) → ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∈ Fin)
233	231, 111, 232	mp2an 692	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∈ Fin
234		diffi 8906	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∈ Fin → (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ Fin)
235	233, 234	ax-mp 5	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ Fin
236	188	t1ficld 22224	. . . . . . . . . . . 12 ⊢ (((𝑅 ↾t 𝐼) ∈ Fre ∧ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ 𝐼 ∧ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ Fin) → (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ (Clsd‘(𝑅 ↾t 𝐼)))
237	230, 235, 236	mp3an13 1454	. . . . . . . . . . 11 ⊢ ((((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ 𝐼 → (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ (Clsd‘(𝑅 ↾t 𝐼)))
238	226, 237	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ (Clsd‘(𝑅 ↾t 𝐼)))
239	188	difopn 21931	. . . . . . . . . 10 ⊢ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∈ (𝑅 ↾t 𝐼) ∧ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ (Clsd‘(𝑅 ↾t 𝐼))) → ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∈ (𝑅 ↾t 𝐼))
240	222, 238, 239	syl2anr 600	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ)) → ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∈ (𝑅 ↾t 𝐼))
241	240	anassrs 471	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑖 ∈ ℕ) → ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∈ (𝑅 ↾t 𝐼))
242		eleq2 2826	. . . . . . . . . . 11 ⊢ (𝑣 = ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → (𝑐 ∈ 𝑣 ↔ 𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}))))
243		ineq1 4120	. . . . . . . . . . . 12 ⊢ (𝑣 = ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) = (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})))
244	243	neeq1d 3000	. . . . . . . . . . 11 ⊢ (𝑣 = ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → ((𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅ ↔ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅))
245	242, 244	imbi12d 348	. . . . . . . . . 10 ⊢ (𝑣 = ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → ((𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅) ↔ (𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)))
246	245	rspcva 3535	. . . . . . . . 9 ⊢ ((((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∈ (𝑅 ↾t 𝐼) ∧ ∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)) → (𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅))
247	127	ffnd 6546	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝐼 → 𝑐 Fn (1...𝑁))
248	247	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 Fn (1...𝑁))
249	137	ralrimiva 3105	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → ∀𝑚 ∈ (1...𝑁)(𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
250	103	elixp 8585	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ (𝑐 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
251	248, 249, 250	sylanbrc 586	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
252		simpl 486	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ 𝐼)
253	251, 252	elind 4108	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼))
254		neldifsnd 4706	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → ¬ 𝑐 ∈ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}))
255	253, 254	eldifd 3877	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})))
256	255	adantll 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})))
257		simplr 769	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) → ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
258	257	anim1i 618	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) → (∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))))
259		simpl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}) → 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))))
260	258, 259	anim12i 616	. . . . . . . . . . . . . . 15 ⊢ (((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})) → ((∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))))
261		elin 3882	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ↔ (𝑗 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∧ 𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})))
262		andir 1009	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∨ (((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐})) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})) ↔ (((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})) ∨ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐}) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}))))
263		eldif 3876	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ↔ (𝑗 ∈ (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∧ ¬ 𝑗 ∈ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})))
264		elin 3882	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ↔ (𝑗 ∈ X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ 𝑗 ∈ 𝐼))
265		vex 3412	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑗 ∈ V
266	265	elixp 8585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ (𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
267	266	anbi1i 627	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑗 ∈ X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ 𝑗 ∈ 𝐼) ↔ ((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼))
268	264, 267	bitri 278	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ↔ ((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼))
269		ianor 982	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ (𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ ¬ 𝑗 ∈ {𝑐}) ↔ (¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∨ ¬ ¬ 𝑗 ∈ {𝑐}))
270		eldif 3876	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ↔ (𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ ¬ 𝑗 ∈ {𝑐}))
271	269, 270	xchnxbir 336	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑗 ∈ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ↔ (¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∨ ¬ ¬ 𝑗 ∈ {𝑐}))
272	268, 271	anbi12i 630	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∧ ¬ 𝑗 ∈ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ↔ (((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ (¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∨ ¬ ¬ 𝑗 ∈ {𝑐})))
273		andi 1008	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ (¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∨ ¬ ¬ 𝑗 ∈ {𝑐})) ↔ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∨ (((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐})))
274	263, 272, 273	3bitri 300	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ↔ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∨ (((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐})))
275		eldif 3876	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐}) ↔ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}))
276	274, 275	anbi12i 630	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∧ 𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ↔ (((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∨ (((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐})) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})))
277		pm3.24 406	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ (¬ 𝑗 ∈ {𝑐} ∧ ¬ ¬ 𝑗 ∈ {𝑐})
278		simpr 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐}) → ¬ ¬ 𝑗 ∈ {𝑐})
279		simpr 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}) → ¬ 𝑗 ∈ {𝑐})
280	278, 279	anim12ci 617	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐}) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})) → (¬ 𝑗 ∈ {𝑐} ∧ ¬ ¬ 𝑗 ∈ {𝑐}))
281	277, 280	mto 200	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐}) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}))
282	281	biorfi 939	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})) ↔ (((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})) ∨ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐}) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}))))
283	262, 276, 282	3bitr4i 306	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∧ 𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ↔ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})))
284	261, 283	bitri 278	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ↔ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})))
285		ancom 464	. . . . . . . . . . . . . . . 16 ⊢ (((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ↔ (∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ (¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))))))
286		anass 472	. . . . . . . . . . . . . . . 16 ⊢ (((∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) ↔ (∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ (¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))))))
287	285, 286	bitr4i 281	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ↔ ((∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))))
288	260, 284, 287	3imtr4i 295	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) → ((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
289		ancom 464	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) ↔ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))))
290		eldif 3876	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ↔ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))))
291	289, 290	bitr4i 281	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) ↔ 𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))))
292		imadmrn 5939	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ dom (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) = ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))
293	62, 63	dmmpti 6522	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) = ℕ
294	293	imaeq2i 5927	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ dom (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) = ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ ℕ)
295	292, 294	eqtr3i 2767	. . . . . . . . . . . . . . . . . . . 20 ⊢ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) = ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ ℕ)
296	295	difeq1i 4033	. . . . . . . . . . . . . . . . . . 19 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) = (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ ℕ) ∖ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)))
297		imadifss 35489	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ ℕ) ∖ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ⊆ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (ℕ ∖ (1..^𝑖)))
298	296, 297	eqsstri 3935	. . . . . . . . . . . . . . . . . 18 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ⊆ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (ℕ ∖ (1..^𝑖)))
299		imass2 5970	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℕ ∖ (1..^𝑖)) ⊆ (ℤ≥‘𝑖) → ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (ℕ ∖ (1..^𝑖))) ⊆ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (ℤ≥‘𝑖)))
300	92, 299	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (ℕ ∖ (1..^𝑖))) ⊆ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (ℤ≥‘𝑖)))
301		df-ima 5564	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (ℤ≥‘𝑖)) = ran ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ↾ (ℤ≥‘𝑖))
302		uznnssnn 12491	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ ℕ → (ℤ≥‘𝑖) ⊆ ℕ)
303	302	resmptd 5908	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ↾ (ℤ≥‘𝑖)) = (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))))
304	303	rneqd 5807	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ ℕ → ran ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ↾ (ℤ≥‘𝑖)) = ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))))
305	301, 304	syl5eq 2790	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (ℤ≥‘𝑖)) = ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))))
306	300, 305	sseqtrd 3941	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (ℕ ∖ (1..^𝑖))) ⊆ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))))
307	298, 306	sstrid 3912	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ → (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ⊆ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))))
308	307	sseld 3900	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℕ → (𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) → 𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))))
309	291, 308	syl5bi 245	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℕ → ((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) → 𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))))
310	309	anim1d 614	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ → (((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) → (𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))))
311	288, 310	syl5 34	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ℕ → (𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) → (𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))))
312	311	eximdv 1925	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ℕ → (∃𝑗 𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) → ∃𝑗(𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))))
313		n0 4261	. . . . . . . . . . . 12 ⊢ ((((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅ ↔ ∃𝑗 𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})))
314	62	rgenw 3073	. . . . . . . . . . . . . 14 ⊢ ∀𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ V
315		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) = (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))
316		fveq1 6716	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) → (𝑗‘𝑚) = (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚))
317	316	eleq1d 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) → ((𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
318	317	ralbidv 3118	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) → (∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ ∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
319	315, 318	rexrnmptw 6914	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ V → (∃𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
320	314, 319	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (∃𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
321		df-rex 3067	. . . . . . . . . . . . 13 ⊢ (∃𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ ∃𝑗(𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
322	320, 321	bitr3i 280	. . . . . . . . . . . 12 ⊢ (∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ ∃𝑗(𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
323	312, 313, 322	3imtr4g 299	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ → ((((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅ → ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
324	323	adantl 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑖 ∈ ℕ) → ((((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅ → ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
325	256, 324	embantd 59	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑖 ∈ ℕ) → ((𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅) → ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
326	246, 325	syl5 34	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑖 ∈ ℕ) → ((((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∈ (𝑅 ↾t 𝐼) ∧ ∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)) → ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
327	241, 326	mpand 695	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑖 ∈ ℕ) → (∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅) → ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
328	327	ralrimdva 3110	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅) → ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
329	196, 328	sylbid 243	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑐 ∈ ((limPt‘(𝑅 ↾t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) → ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
330	329	reximdva 3193	. . . 4 ⊢ (𝜑 → (∃𝑐 ∈ 𝐼 𝑐 ∈ ((limPt‘(𝑅 ↾t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))) → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
331	191, 330	syld 47	. . 3 ⊢ (𝜑 → (¬ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
332	145, 331	pm2.61d 182	. 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
333		poimir.0	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
334		poimir.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn 𝑅))
335		poimirlem30.x	. . . 4 ⊢ 𝑋 = ((𝐹‘(((1st ‘(𝐺‘𝑘)) ∘f + ((((2nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘f / ((1...𝑁) × {𝑘})))‘𝑛)
336		poimirlem30.4	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ◡ ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟𝑋)
337	333, 46, 146, 334, 335, 32, 37, 336	poimirlem29 35543	. . 3 ⊢ (𝜑 → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(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))))
338	337	reximdv 3192	. 2 ⊢ (𝜑 → (∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))))
339	332, 338	mpd 15	1 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   ∧ w3a 1089  ∀wal 1541   = wceq 1543  ⊤wtru 1544  ∃wex 1787   ∈ wcel 2110  {cab 2714   ≠ wne 2940  ∀wral 3061  ∃wrex 3062  {crab 3065  Vcvv 3408   ∖ cdif 3863   ∪ cun 3864   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237  {csn 4541  {cpr 4543  ∪ cuni 4819  ∪ ciun 4904   class class class wbr 5053   ↦ cmpt 5135   × cxp 5549  ^◡ccnv 5550  dom cdm 5551  ran crn 5552   ↾ cres 5553   “ cima 5554   ∘ ccom 5555  Fun wfun 6374   Fn wfn 6375  ⟶wf 6376  –1-1-onto→wf1o 6379  ‘cfv 6380  (class class class)co 7213   ∘_f cof 7467  ωcom 7644  1^st c1st 7759  2^nd c2nd 7760   ↑_m cmap 8508  Xcixp 8578   ≈ cen 8623  Fincfn 8626  ℝcr 10728  0cc0 10729  1c1 10730   + caddc 10732   · cmul 10734  ℝ^*cxr 10866   < clt 10867   ≤ cle 10868   − cmin 11062   / cdiv 11489  ℕcn 11830  ℕ₀cn0 12090  ℤ_≥cuz 12438  ℝ⁺crp 12586  (,)cioo 12935  [,]cicc 12938  ...cfz 13095  ..^cfzo 13238  abscabs 14797   ↾_t crest 16925  topGenctg 16942  ∏_tcpt 16943  ∞Metcxmet 20348  ballcbl 20350  MetOpencmopn 20353  Topctop 21790  Clsdccld 21913  limPtclp 22031   Cn ccn 22121  Frect1 22204  Hauscha 22205  Compccmp 22283  IIcii 23772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fi 9027  df-sup 9058  df-inf 9059  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-z 12177  df-uz 12439  df-q 12545  df-rp 12587  df-xneg 12704  df-xadd 12705  df-xmul 12706  df-ioo 12939  df-icc 12942  df-fz 13096  df-fzo 13239  df-fl 13367  df-seq 13575  df-exp 13636  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-rest 16927  df-topgen 16948  df-pt 16949  df-psmet 20355  df-xmet 20356  df-met 20357  df-bl 20358  df-mopn 20359  df-top 21791  df-topon 21808  df-bases 21843  df-cld 21916  df-ntr 21917  df-cls 21918  df-lp 22033  df-cn 22124  df-cnp 22125  df-t1 22211  df-haus 22212  df-cmp 22284  df-tx 22459  df-hmeo 22652  df-hmph 22653  df-ii 23774

This theorem is referenced by:  poimirlem32  35546