Theorem poimirlem30 35734

Description: Lemma for poimir 35737 combining poimirlem29 35733 with bwth 22469. (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 13360	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑘) → 𝑖 ∈ ℕ0)
2	1	nn0red 12224	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑘) → 𝑖 ∈ ℝ)
3		nndivre 11944	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ)
4	2, 3	sylan 579	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ)
5		elfzole1 13324	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑘) → 0 ≤ 𝑖)
6	2, 5	jca 511	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑘) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
7		nnrp 12670	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
8	7	rpregt0d 12707	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
9		divge0 11774	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ℝ ∧ 0 ≤ 𝑖) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝑖 / 𝑘))
10	6, 8, 9	syl2an 595	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑖 / 𝑘))
11		elfzo0le 13359	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑘) → 𝑖 ≤ 𝑘)
12	11	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 𝑖 ≤ 𝑘)
13	2	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 𝑖 ∈ ℝ)
14		1red 10907	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ)
15	7	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ+)
16	13, 14, 15	ledivmuld 12754	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖 ≤ (𝑘 · 1)))
17		nncn 11911	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
18	17	mulid1d 10923	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → (𝑘 · 1) = 𝑘)
19	18	breq2d 5082	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → (𝑖 ≤ (𝑘 · 1) ↔ 𝑖 ≤ 𝑘))
20	19	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 ≤ (𝑘 · 1) ↔ 𝑖 ≤ 𝑘))
21	16, 20	bitrd 278	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖 ≤ 𝑘))
22	12, 21	mpbird 256	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ≤ 1)
23		elicc01 13127	. . . . . . . . . . . . . 14 ⊢ ((𝑖 / 𝑘) ∈ (0[,]1) ↔ ((𝑖 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝑖 / 𝑘) ∧ (𝑖 / 𝑘) ≤ 1))
24	4, 10, 22, 23	syl3anbrc 1341	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ (0[,]1))
25	24	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0..^𝑘)) → (𝑖 / 𝑘) ∈ (0[,]1))
26		elsni 4575	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ {𝑘} → 𝑗 = 𝑘)
27	26	oveq2d 7271	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ {𝑘} → (𝑖 / 𝑗) = (𝑖 / 𝑘))
28	27	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ {𝑘} → ((𝑖 / 𝑗) ∈ (0[,]1) ↔ (𝑖 / 𝑘) ∈ (0[,]1)))
29	25, 28	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0..^𝑘)) → (𝑗 ∈ {𝑘} → (𝑖 / 𝑗) ∈ (0[,]1)))
30	29	impr 454	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ (𝑖 ∈ (0..^𝑘) ∧ 𝑗 ∈ {𝑘})) → (𝑖 / 𝑗) ∈ (0[,]1))
31	30	adantll 710	. . . . . . . . 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 6943	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ((ℕ0 ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
34		xp1st 7836	. . . . . . . . . . 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 8611	. . . . . . . . . . 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 6422	. . . . . . . . . 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 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐺‘𝑘)):(1...𝑁)⟶(0..^𝑘))
40		vex 3426	. . . . . . . . . . 11 ⊢ 𝑘 ∈ V
41	40	fconst 6644	. . . . . . . . . 10 ⊢ ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘}
42	41	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘})
43		fzfid 13621	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑁) ∈ Fin)
44		inidm 4149	. . . . . . . . 9 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
45	31, 39, 42, 43, 43, 44	off 7529	. . . . . . . 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 2830	. . . . . . . . 9 ⊢ (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑m (1...𝑁)))
48		ovex 7288	. . . . . . . . . 10 ⊢ (0[,]1) ∈ V
49		ovex 7288	. . . . . . . . . 10 ⊢ (1...𝑁) ∈ V
50	48, 49	elmap 8617	. . . . . . . . 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 274	. . . . . . . 8 ⊢ (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
52	45, 51	sylibr 233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ 𝐼)
53	52	fmpttd 6971	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))):ℕ⟶𝐼)
54	53	frnd 6592	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ⊆ 𝐼)
55		ominf 8964	. . . . . . 7 ⊢ ¬ ω ∈ Fin
56		nnenom 13628	. . . . . . . . 9 ⊢ ℕ ≈ ω
57		enfi 8933	. . . . . . . . 9 ⊢ (ℕ ≈ ω → (ℕ ∈ Fin ↔ ω ∈ Fin))
58	56, 57	ax-mp 5	. . . . . . . 8 ⊢ (ℕ ∈ Fin ↔ ω ∈ Fin)
59		iunid 4986	. . . . . . . . . . 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 5956	. . . . . . . . . 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 7100	. . . . . . . . . 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 7288	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ V
63		eqid 2738	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) = (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))
64	62, 63	fnmpti 6560	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) Fn ℕ
65		dffn3 6597	. . . . . . . . . . . 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 229	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))):ℕ⟶ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))
67		fimacnv 6606	. . . . . . . . . . 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 2775	. . . . . . . . 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 2829	. . . . . . . 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 276	. . . . . . 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 321	. . . . . 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 3163	. . . . . . . . . . . 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 3177	. . . . . . . . . . 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 3165	. . . . . . . . . . 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 274	. . . . . . . . . 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 3090	. . . . . . . . 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 3163	. . . . . . . . 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 274	. . . . . . . 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 12550	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ≥‘1)
81		elnnuz 12551	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ ↔ 𝑖 ∈ (ℤ≥‘1))
82		fzouzsplit 13350	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (ℤ≥‘1) → (ℤ≥‘1) = ((1..^𝑖) ∪ (ℤ≥‘𝑖)))
83	81, 82	sylbi 216	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℕ → (ℤ≥‘1) = ((1..^𝑖) ∪ (ℤ≥‘𝑖)))
84	80, 83	syl5eq 2791	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℕ → ℕ = ((1..^𝑖) ∪ (ℤ≥‘𝑖)))
85	84	difeq1d 4052	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ → (ℕ ∖ (1..^𝑖)) = (((1..^𝑖) ∪ (ℤ≥‘𝑖)) ∖ (1..^𝑖)))
86		uncom 4083	. . . . . . . . . . . . . . . 16 ⊢ ((1..^𝑖) ∪ (ℤ≥‘𝑖)) = ((ℤ≥‘𝑖) ∪ (1..^𝑖))
87	86	difeq1i 4049	. . . . . . . . . . . . . . 15 ⊢ (((1..^𝑖) ∪ (ℤ≥‘𝑖)) ∖ (1..^𝑖)) = (((ℤ≥‘𝑖) ∪ (1..^𝑖)) ∖ (1..^𝑖))
88		difun2 4411	. . . . . . . . . . . . . . 15 ⊢ (((ℤ≥‘𝑖) ∪ (1..^𝑖)) ∖ (1..^𝑖)) = ((ℤ≥‘𝑖) ∖ (1..^𝑖))
89	87, 88	eqtri 2766	. . . . . . . . . . . . . 14 ⊢ (((1..^𝑖) ∪ (ℤ≥‘𝑖)) ∖ (1..^𝑖)) = ((ℤ≥‘𝑖) ∖ (1..^𝑖))
90	85, 89	eqtrdi 2795	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ℕ → (ℕ ∖ (1..^𝑖)) = ((ℤ≥‘𝑖) ∖ (1..^𝑖)))
91		difss 4062	. . . . . . . . . . . . 13 ⊢ ((ℤ≥‘𝑖) ∖ (1..^𝑖)) ⊆ (ℤ≥‘𝑖)
92	90, 91	eqsstrdi 3971	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ℕ → (ℕ ∖ (1..^𝑖)) ⊆ (ℤ≥‘𝑖))
93		ssralv 3983	. . . . . . . . . . . 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 450	. . . . . . . . . . . . . . 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 3893	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℕ ∖ (1..^𝑖)) ↔ (𝑘 ∈ ℕ ∧ ¬ 𝑘 ∈ (1..^𝑖)))
97	96	imbi1i 349	. . . . . . . . . . . . . . 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 315	. . . . . . . . . . . . . . . 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 335	. . . . . . . . . . . . . . 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 302	. . . . . . . . . . . . . 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 1823	. . . . . . . . . . . . 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 3068	. . . . . . . . . . . . 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 3426	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
104	63	mptiniseg 6131	. . . . . . . . . . . . . . . 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 3945	. . . . . . . . . . . . . 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 4001	. . . . . . . . . . . . . 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 3068	. . . . . . . . . . . . . 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 296	. . . . . . . . . . . . 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 302	. . . . . . . . . . . 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 13622	. . . . . . . . . . . . 13 ⊢ (1..^𝑖) ∈ Fin
112		ssfi 8918	. . . . . . . . . . . . 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 686	. . . . . . . . . . . 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 216	. . . . . . . . . . 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 3208	. . . . . . . . 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 3086	. . . . . . . 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 234	. . . . . . 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 9037	. . . . . . . 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 412	. . . . . . 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 149	. . . . 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 3984	. . . . 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 13160	. . . . . . . . . . . 12 ⊢ (0[,]1) ⊆ ℝ
126		elmapi 8595	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ((0[,]1) ↑m (1...𝑁)) → 𝑐:(1...𝑁)⟶(0[,]1))
127	126, 46	eleq2s 2857	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝐼 → 𝑐:(1...𝑁)⟶(0[,]1))
128	127	ffvelrnda 6943	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) → (𝑐‘𝑚) ∈ (0[,]1))
129	125, 128	sselid 3915	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) → (𝑐‘𝑚) ∈ ℝ)
130		nnrp 12670	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℝ+)
131	130	rpreccld 12711	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ → (1 / 𝑖) ∈ ℝ+)
132		eqid 2738	. . . . . . . . . . . . 13 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
133	132	rexmet 23860	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
134		blcntr 23474	. . . . . . . . . . . 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 1446	. . . . . . . . . . 11 ⊢ (((𝑐‘𝑚) ∈ ℝ ∧ (1 / 𝑖) ∈ ℝ+) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
136	129, 131, 135	syl2an 595	. . . . . . . . . 10 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) ∧ 𝑖 ∈ ℕ) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
137	136	an32s 648	. . . . . . . . 9 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
138		fveq1 6755	. . . . . . . . . 10 ⊢ (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) = (𝑐‘𝑚))
139	138	eleq1d 2823	. . . . . . . . 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 246	. . . . . . . 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 3112	. . . . . . 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 3201	. . . . . 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 3102	. . . . 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 3172	. . . 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 8653	. . . . . . . . 9 ⊢ X𝑛 ∈ (1...𝑁)(0[,]1) = ((0[,]1) ↑m (1...𝑁))
148	46, 147	eqtr4i 2769	. . . . . . . 8 ⊢ 𝐼 = X𝑛 ∈ (1...𝑁)(0[,]1)
149	146, 148	oveq12i 7267	. . . . . . 7 ⊢ (𝑅 ↾t 𝐼) = ((∏t‘((1...𝑁) × {(topGen‘ran (,))})) ↾t X𝑛 ∈ (1...𝑁)(0[,]1))
150		fzfid 13621	. . . . . . . . 9 ⊢ (⊤ → (1...𝑁) ∈ Fin)
151		retop 23831	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) ∈ Top
152	151	fconst6 6648	. . . . . . . . . 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 35703	. . . . . . . 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 1546	. . . . . . 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 6769	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) ∈ V
158	157	fvconst2 7061	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
159	158	oveq1d 7270	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑁) → ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1)) = ((topGen‘ran (,)) ↾t (0[,]1)))
160	159	mpteq2ia 5173	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...𝑁) ↦ ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1))) = (𝑛 ∈ (1...𝑁) ↦ ((topGen‘ran (,)) ↾t (0[,]1)))
161		fconstmpt 5640	. . . . . . . . 9 ⊢ ((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))}) = (𝑛 ∈ (1...𝑁) ↦ ((topGen‘ran (,)) ↾t (0[,]1)))
162	160, 161	eqtr4i 2769	. . . . . . . 8 ⊢ (𝑛 ∈ (1...𝑁) ↦ ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1))) = ((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))})
163	162	fveq2i 6759	. . . . . . 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 2770	. . . . . 6 ⊢ (𝑅 ↾t 𝐼) = (∏t‘((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))}))
165		fzfi 13620	. . . . . . 7 ⊢ (1...𝑁) ∈ Fin
166		dfii2 23951	. . . . . . . . 9 ⊢ II = ((topGen‘ran (,)) ↾t (0[,]1))
167		iicmp 23955	. . . . . . . . 9 ⊢ II ∈ Comp
168	166, 167	eqeltrri 2836	. . . . . . . 8 ⊢ ((topGen‘ran (,)) ↾t (0[,]1)) ∈ Comp
169	168	fconst6 6648	. . . . . . 7 ⊢ ((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))}):(1...𝑁)⟶Comp
170		ptcmpfi 22872	. . . . . . 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 688	. . . . . 6 ⊢ (∏t‘((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))})) ∈ Comp
172	164, 171	eqeltri 2835	. . . . 5 ⊢ (𝑅 ↾t 𝐼) ∈ Comp
173		rehaus 23868	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) ∈ Haus
174	173	fconst6 6648	. . . . . . . . . . 11 ⊢ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Haus
175		pthaus 22697	. . . . . . . . . . 11 ⊢ (((1...𝑁) ∈ Fin ∧ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Haus) → (∏t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Haus)
176	165, 174, 175	mp2an 688	. . . . . . . . . 10 ⊢ (∏t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Haus
177	146, 176	eqeltri 2835	. . . . . . . . 9 ⊢ 𝑅 ∈ Haus
178		haustop 22390	. . . . . . . . 9 ⊢ (𝑅 ∈ Haus → 𝑅 ∈ Top)
179	177, 178	ax-mp 5	. . . . . . . 8 ⊢ 𝑅 ∈ Top
180		reex 10893	. . . . . . . . . 10 ⊢ ℝ ∈ V
181		mapss 8635	. . . . . . . . . 10 ⊢ ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑m (1...𝑁)) ⊆ (ℝ ↑m (1...𝑁)))
182	180, 125, 181	mp2an 688	. . . . . . . . 9 ⊢ ((0[,]1) ↑m (1...𝑁)) ⊆ (ℝ ↑m (1...𝑁))
183	46, 182	eqsstri 3951	. . . . . . . 8 ⊢ 𝐼 ⊆ (ℝ ↑m (1...𝑁))
184		uniretop 23832	. . . . . . . . . . 11 ⊢ ℝ = ∪ (topGen‘ran (,))
185	146, 184	ptuniconst 22657	. . . . . . . . . 10 ⊢ (((1...𝑁) ∈ Fin ∧ (topGen‘ran (,)) ∈ Top) → (ℝ ↑m (1...𝑁)) = ∪ 𝑅)
186	165, 151, 185	mp2an 688	. . . . . . . . 9 ⊢ (ℝ ↑m (1...𝑁)) = ∪ 𝑅
187	186	restuni 22221	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝐼 ⊆ (ℝ ↑m (1...𝑁))) → 𝐼 = ∪ (𝑅 ↾t 𝐼))
188	179, 183, 187	mp2an 688	. . . . . . 7 ⊢ 𝐼 = ∪ (𝑅 ↾t 𝐼)
189	188	bwth 22469	. . . . . 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 1119	. . . . 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 586	. . . 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 22454	. . . . . . . . 9 ⊢ ((𝑅 ↾t 𝐼) ∈ Comp → (𝑅 ↾t 𝐼) ∈ Top)
193	172, 192	ax-mp 5	. . . . . . . 8 ⊢ (𝑅 ↾t 𝐼) ∈ Top
194	188	islp3 22205	. . . . . . . 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 1446	. . . . . . 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 579	. . . . . 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 13621	. . . . . . . . . . . . 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 11945	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ → (1 / 𝑖) ∈ ℝ)
200	199	rexrd 10956	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℕ → (1 / 𝑖) ∈ ℝ*)
201		eqid 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
202	132, 201	tgioo 23865	. . . . . . . . . . . . . . . . . 18 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
203	202	blopn 23562	. . . . . . . . . . . . . . . . 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 1446	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐‘𝑚) ∈ ℝ ∧ (1 / 𝑖) ∈ ℝ*) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
205	129, 200, 204	syl2an 595	. . . . . . . . . . . . . . 15 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) ∧ 𝑖 ∈ ℕ) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
206	205	an32s 648	. . . . . . . . . . . . . 14 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
207	157	fvconst2 7061	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑚) = (topGen‘ran (,)))
208	207	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑚) = (topGen‘ran (,)))
209	206, 208	eleqtrrd 2842	. . . . . . . . . . . . 13 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑚))
210		noel 4261	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝑚 ∈ ∅
211		difid 4301	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑁) ∖ (1...𝑁)) = ∅
212	211	eleq2i 2830	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ((1...𝑁) ∖ (1...𝑁)) ↔ 𝑚 ∈ ∅)
213	210, 212	mtbir 322	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 22642	. . . . . . . . . . . 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 2850	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ 𝑅)
218		ovex 7288	. . . . . . . . . . . . 13 ⊢ ((0[,]1) ↑m (1...𝑁)) ∈ V
219	46, 218	eqeltri 2835	. . . . . . . . . . . 12 ⊢ 𝐼 ∈ V
220		elrestr 17056	. . . . . . . . . . . 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 1449	. . . . . . . . . . 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 4062	. . . . . . . . . . . . 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 5969	. . . . . . . . . . . . 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 3926	. . . . . . . . . . . 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 3928	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ 𝐼)
227		haust1 22411	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Haus → 𝑅 ∈ Fre)
228	177, 227	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 𝑅 ∈ Fre
229		restt1 22426	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Fre ∧ 𝐼 ∈ V) → (𝑅 ↾t 𝐼) ∈ Fre)
230	228, 219, 229	mp2an 688	. . . . . . . . . . . 12 ⊢ (𝑅 ↾t 𝐼) ∈ Fre
231		funmpt 6456	. . . . . . . . . . . . . 14 ⊢ Fun (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))
232		imafi 8920	. . . . . . . . . . . . . 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 688	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∈ Fin
234		diffi 8979	. . . . . . . . . . . . 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 22386	. . . . . . . . . . . 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 1450	. . . . . . . . . . 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 22093	. . . . . . . . . 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 596	. . . . . . . . 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 467	. . . . . . . 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 2827	. . . . . . . . . . 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 4136	. . . . . . . . . . . 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 3002	. . . . . . . . . . 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 344	. . . . . . . . . 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 3550	. . . . . . . . 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 6585	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝐼 → 𝑐 Fn (1...𝑁))
248	247	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 Fn (1...𝑁))
249	137	ralrimiva 3107	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → ∀𝑚 ∈ (1...𝑁)(𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
250	103	elixp 8650	. . . . . . . . . . . . . 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 582	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
252		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ 𝐼)
253	251, 252	elind 4124	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼))
254		neldifsnd 4723	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → ¬ 𝑐 ∈ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}))
255	253, 254	eldifd 3894	. . . . . . . . . . 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 710	. . . . . . . . . 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 765	. . . . . . . . . . . . . . . . 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 614	. . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . 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 612	. . . . . . . . . . . . . . 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 3899	. . . . . . . . . . . . . . . 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 1005	. . . . . . . . . . . . . . . . 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 3893	. . . . . . . . . . . . . . . . . . 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 3899	. . . . . . . . . . . . . . . . . . . . 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 3426	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑗 ∈ V
266	265	elixp 8650	. . . . . . . . . . . . . . . . . . . . . 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 623	. . . . . . . . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . . . . . . 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 978	. . . . . . . . . . . . . . . . . . . . 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 3893	. . . . . . . . . . . . . . . . . . . . 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 332	. . . . . . . . . . . . . . . . . . . 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 626	. . . . . . . . . . . . . . . . . . 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 1004	. . . . . . . . . . . . . . . . . . 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 296	. . . . . . . . . . . . . . . . . 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 3893	. . . . . . . . . . . . . . . . . 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 626	. . . . . . . . . . . . . . . . 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		pm3.24 402	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ (¬ 𝑗 ∈ {𝑐} ∧ ¬ ¬ 𝑗 ∈ {𝑐})
278		simpr 484	. . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}) → ¬ 𝑗 ∈ {𝑐})
280	278, 279	anim12ci 613	. . . . . . . . . . . . . . . . . . 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 196	. . . . . . . . . . . . . . . . . 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 935	. . . . . . . . . . . . . . . . 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 302	. . . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . . . 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 277	. . . . . . . . . . . . . . 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 291	. . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . . . 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 3893	. . . . . . . . . . . . . . . . 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 277	. . . . . . . . . . . . . . . 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 5968	. . . . . . . . . . . . . . . . . . . . 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 6561	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) = ℕ
294	293	imaeq2i 5956	. . . . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . . . . 20 ⊢ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) = ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ ℕ)
296	295	difeq1i 4049	. . . . . . . . . . . . . . . . . . 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 35679	. . . . . . . . . . . . . . . . . . 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 3951	. . . . . . . . . . . . . . . . . 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 5999	. . . . . . . . . . . . . . . . . . . 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 5593	. . . . . . . . . . . . . . . . . . . 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 12564	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ ℕ → (ℤ≥‘𝑖) ⊆ ℕ)
303	302	resmptd 5937	. . . . . . . . . . . . . . . . . . . . 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 5836	. . . . . . . . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . . . . . . 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 3957	. . . . . . . . . . . . . . . . . 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 3928	. . . . . . . . . . . . . . . . 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 3916	. . . . . . . . . . . . . . . 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 241	. . . . . . . . . . . . . . 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 610	. . . . . . . . . . . . . 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 1921	. . . . . . . . . . . 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 4277	. . . . . . . . . . . 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 3075	. . . . . . . . . . . . . 14 ⊢ ∀𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ V
315		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) = (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))
316		fveq1 6755	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) → (𝑗‘𝑚) = (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚))
317	316	eleq1d 2823	. . . . . . . . . . . . . . . 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 3120	. . . . . . . . . . . . . . 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 6953	. . . . . . . . . . . . . 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 3069	. . . . . . . . . . . . 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 276	. . . . . . . . . . . 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 295	. . . . . . . . . . 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 481	. . . . . . . . . 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 691	. . . . . . 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 3112	. . . . . 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 239	. . . . 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 3202	. . . 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 179	. 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 35733	. . 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 3201	. 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 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085  ∀wal 1537   = wceq 1539  ⊤wtru 1540  ∃wex 1783   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558  {cpr 4560  ∪ cuni 4836  ∪ ciun 4921   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   ↾ cres 5582   “ cima 5583   ∘ ccom 5584  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255   ∘_f cof 7509  ωcom 7687  1^st c1st 7802  2^nd c2nd 7803   ↑_m cmap 8573  Xcixp 8643   ≈ cen 8688  Fincfn 8691  ℝcr 10801  0cc0 10802  1c1 10803   + caddc 10805   · cmul 10807  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941   − cmin 11135   / cdiv 11562  ℕcn 11903  ℕ₀cn0 12163  ℤ_≥cuz 12511  ℝ⁺crp 12659  (,)cioo 13008  [,]cicc 13011  ...cfz 13168  ..^cfzo 13311  abscabs 14873   ↾_t crest 17048  topGenctg 17065  ∏_tcpt 17066  ∞Metcxmet 20495  ballcbl 20497  MetOpencmopn 20500  Topctop 21950  Clsdccld 22075  limPtclp 22193   Cn ccn 22283  Frect1 22366  Hauscha 22367  Compccmp 22445  IIcii 23944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fi 9100  df-sup 9131  df-inf 9132  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-q 12618  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-ioo 13012  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-rest 17050  df-topgen 17071  df-pt 17072  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505  df-mopn 20506  df-top 21951  df-topon 21968  df-bases 22004  df-cld 22078  df-ntr 22079  df-cls 22080  df-lp 22195  df-cn 22286  df-cnp 22287  df-t1 22373  df-haus 22374  df-cmp 22446  df-tx 22621  df-hmeo 22814  df-hmph 22815  df-ii 23946

This theorem is referenced by:  poimirlem32  35736