poimirlem30 - Mathbox for Brendan Leahy

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Theorem poimirlem30 37993

Description: Lemma for poimir 37996 combining poimirlem29 37992 with bwth 23391. (Contributed by Brendan Leahy, 21-Aug-2020.)

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

**Assertion**
Ref	Expression
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 13659	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑘) → 𝑖 ∈ ℕ₀)
2	1	nn0red 12496	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑘) → 𝑖 ∈ ℝ)
3		nndivre 12215	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ)
4	2, 3	sylan 581	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ)
5		elfzole1 13619	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑘) → 0 ≤ 𝑖)
6	2, 5	jca 511	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑘) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
7		nnrp 12951	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ⁺)
8	7	rpregt0d 12989	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
9		divge0 12022	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ℝ ∧ 0 ≤ 𝑖) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝑖 / 𝑘))
10	6, 8, 9	syl2an 597	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑖 / 𝑘))
11		elfzo0le 13655	. . . . . . . . . . . . . . . 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 11142	. . . . . . . . . . . . . . . . 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 13036	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖 ≤ (𝑘 · 1)))
17		nncn 12179	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
18	17	mulridd 11159	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → (𝑘 · 1) = 𝑘)
19	18	breq2d 5098	. . . . . . . . . . . . . . . . 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 279	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖 ≤ 𝑘))
22	12, 21	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ≤ 1)
23		elicc01 13416	. . . . . . . . . . . . . 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 458	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0..^𝑘)) → (𝑖 / 𝑘) ∈ (0[,]1))
26		elsni 4585	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ {𝑘} → 𝑗 = 𝑘)
27	26	oveq2d 7380	. . . . . . . . . . . . 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 247	. . . . . . . . . . 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 715	. . . . . . . . 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	ffvelcdmda 7034	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
34		xp1st 7971	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑘) ∈ ((ℕ₀ ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^st ‘(𝐺‘𝑘)) ∈ (ℕ₀ ↑_m (1...𝑁)))
35		elmapfn 8809	. . . . . . . . . . 11 ⊢ ((1^st ‘(𝐺‘𝑘)) ∈ (ℕ₀ ↑_m (1...𝑁)) → (1^st ‘(𝐺‘𝑘)) Fn (1...𝑁))
36	33, 34, 35	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐺‘𝑘)) Fn (1...𝑁))
37		poimirlem30.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran (1^st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘))
38		df-f 6500	. . . . . . . . . 10 ⊢ ((1^st ‘(𝐺‘𝑘)):(1...𝑁)⟶(0..^𝑘) ↔ ((1^st ‘(𝐺‘𝑘)) Fn (1...𝑁) ∧ ran (1^st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘)))
39	36, 37, 38	sylanbrc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐺‘𝑘)):(1...𝑁)⟶(0..^𝑘))
40		vex 3434	. . . . . . . . . . 11 ⊢ 𝑘 ∈ V
41	40	fconst 6724	. . . . . . . . . 10 ⊢ ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘}
42	41	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘})
43		fzfid 13932	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑁) ∈ Fin)
44		inidm 4168	. . . . . . . . 9 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
45	31, 39, 42, 43, 43, 44	off 7646	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1^st ‘(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 ⊢ (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑_m (1...𝑁)))
48		ovex 7397	. . . . . . . . . 10 ⊢ (0[,]1) ∈ V
49		ovex 7397	. . . . . . . . . 10 ⊢ (1...𝑁) ∈ V
50	48, 49	elmap 8816	. . . . . . . . 9 ⊢ (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑_m (1...𝑁)) ↔ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
51	47, 50	bitri 275	. . . . . . . 8 ⊢ (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
52	45, 51	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) ∈ 𝐼)
53	52	fmpttd 7065	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))):ℕ⟶𝐼)
54	53	frnd 6674	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ⊆ 𝐼)
55		ominf 9171	. . . . . . 7 ⊢ ¬ ω ∈ Fin
56		nnenom 13939	. . . . . . . . 9 ⊢ ℕ ≈ ω
57		enfi 9118	. . . . . . . . 9 ⊢ (ℕ ≈ ω → (ℕ ∈ Fin ↔ ω ∈ Fin))
58	56, 57	ax-mp 5	. . . . . . . 8 ⊢ (ℕ ∈ Fin ↔ ω ∈ Fin)
59		iunid 5004	. . . . . . . . . . 11 ⊢ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))){𝑐} = ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))
60	59	imaeq2i 6021	. . . . . . . . . 10 ⊢ (^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))){𝑐}) = (^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))))
61		imaiun 7197	. . . . . . . . . 10 ⊢ (^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))){𝑐}) = ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))(^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐})
62		ovex 7397	. . . . . . . . . . . . 13 ⊢ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) ∈ V
63		eqid 2737	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) = (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))
64	62, 63	fnmpti 6639	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) Fn ℕ
65		dffn3 6678	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) Fn ℕ ↔ (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))):ℕ⟶ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))))
66	64, 65	mpbi 230	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))):ℕ⟶ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))
67		fimacnv 6688	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))):ℕ⟶ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) → (^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))) = ℕ)
68	66, 67	ax-mp 5	. . . . . . . . . 10 ⊢ (^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))) = ℕ
69	60, 61, 68	3eqtr3ri 2769	. . . . . . . . 9 ⊢ ℕ = ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))(^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐})
70	69	eleq1i 2828	. . . . . . . 8 ⊢ (ℕ ∈ Fin ↔ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))(^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
71	58, 70	bitr3i 277	. . . . . . 7 ⊢ (ω ∈ Fin ↔ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))(^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
72	55, 71	mtbi 322	. . . . . 6 ⊢ ¬ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))(^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin
73		ralnex 3064	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑖) ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ¬ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐)
74	73	rexbii 3085	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑖) ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ∃𝑖 ∈ ℕ ¬ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐)
75		rexnal 3090	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ ℕ ¬ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ¬ ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐)
76	74, 75	bitri 275	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑖) ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ¬ ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐)
77	76	ralbii 3084	. . . . . . . . 9 ⊢ (∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑖) ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ¬ ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐)
78		ralnex 3064	. . . . . . . . 9 ⊢ (∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ¬ ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ¬ ∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐)
79	77, 78	bitri 275	. . . . . . . 8 ⊢ (∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑖) ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ¬ ∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐)
80		nnuz 12824	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
81		elnnuz 12825	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ ↔ 𝑖 ∈ (ℤ_≥‘1))
82		fzouzsplit 13646	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (ℤ_≥‘1) → (ℤ_≥‘1) = ((1..^𝑖) ∪ (ℤ_≥‘𝑖)))
83	81, 82	sylbi 217	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℕ → (ℤ_≥‘1) = ((1..^𝑖) ∪ (ℤ_≥‘𝑖)))
84	80, 83	eqtrid 2784	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℕ → ℕ = ((1..^𝑖) ∪ (ℤ_≥‘𝑖)))
85	84	difeq1d 4066	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ → (ℕ ∖ (1..^𝑖)) = (((1..^𝑖) ∪ (ℤ_≥‘𝑖)) ∖ (1..^𝑖)))
86		uncom 4099	. . . . . . . . . . . . . . . 16 ⊢ ((1..^𝑖) ∪ (ℤ_≥‘𝑖)) = ((ℤ_≥‘𝑖) ∪ (1..^𝑖))
87	86	difeq1i 4063	. . . . . . . . . . . . . . 15 ⊢ (((1..^𝑖) ∪ (ℤ_≥‘𝑖)) ∖ (1..^𝑖)) = (((ℤ_≥‘𝑖) ∪ (1..^𝑖)) ∖ (1..^𝑖))
88		difun2 4422	. . . . . . . . . . . . . . 15 ⊢ (((ℤ_≥‘𝑖) ∪ (1..^𝑖)) ∖ (1..^𝑖)) = ((ℤ_≥‘𝑖) ∖ (1..^𝑖))
89	87, 88	eqtri 2760	. . . . . . . . . . . . . 14 ⊢ (((1..^𝑖) ∪ (ℤ_≥‘𝑖)) ∖ (1..^𝑖)) = ((ℤ_≥‘𝑖) ∖ (1..^𝑖))
90	85, 89	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ℕ → (ℕ ∖ (1..^𝑖)) = ((ℤ_≥‘𝑖) ∖ (1..^𝑖)))
91		difss 4077	. . . . . . . . . . . . 13 ⊢ ((ℤ_≥‘𝑖) ∖ (1..^𝑖)) ⊆ (ℤ_≥‘𝑖)
92	90, 91	eqsstrdi 3967	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ℕ → (ℕ ∖ (1..^𝑖)) ⊆ (ℤ_≥‘𝑖))
93		ssralv 3991	. . . . . . . . . . . 12 ⊢ ((ℕ ∖ (1..^𝑖)) ⊆ (ℤ_≥‘𝑖) → (∀𝑘 ∈ (ℤ_≥‘𝑖) ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑘 ∈ (ℕ ∖ (1..^𝑖)) ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐))
94	92, 93	syl 17	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ → (∀𝑘 ∈ (ℤ_≥‘𝑖) ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑘 ∈ (ℕ ∖ (1..^𝑖)) ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐))
95		impexp 450	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ℕ ∧ ¬ 𝑘 ∈ (1..^𝑖)) → ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐) ↔ (𝑘 ∈ ℕ → (¬ 𝑘 ∈ (1..^𝑖) → ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐)))
96		eldif 3900	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℕ ∖ (1..^𝑖)) ↔ (𝑘 ∈ ℕ ∧ ¬ 𝑘 ∈ (1..^𝑖)))
97	96	imbi1i 349	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (ℕ ∖ (1..^𝑖)) → ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐) ↔ ((𝑘 ∈ ℕ ∧ ¬ 𝑘 ∈ (1..^𝑖)) → ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐))
98		con34b 316	. . . . . . . . . . . . . . . 16 ⊢ ((((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖)) ↔ (¬ 𝑘 ∈ (1..^𝑖) → ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐))
99	98	imbi2i 336	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ → (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖))) ↔ (𝑘 ∈ ℕ → (¬ 𝑘 ∈ (1..^𝑖) → ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐)))
100	95, 97, 99	3bitr4i 303	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (ℕ ∖ (1..^𝑖)) → ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐) ↔ (𝑘 ∈ ℕ → (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖))))
101	100	albii 1821	. . . . . . . . . . . . 13 ⊢ (∀𝑘(𝑘 ∈ (ℕ ∖ (1..^𝑖)) → ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐) ↔ ∀𝑘(𝑘 ∈ ℕ → (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖))))
102		df-ral 3053	. . . . . . . . . . . . 13 ⊢ (∀𝑘 ∈ (ℕ ∖ (1..^𝑖)) ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ∀𝑘(𝑘 ∈ (ℕ ∖ (1..^𝑖)) → ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐))
103		vex 3434	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
104	63	mptiniseg 6201	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ V → (^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) = {𝑘 ∈ ℕ ∣ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐})
105	103, 104	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) = {𝑘 ∈ ℕ ∣ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐}
106	105	sseq1i 3951	. . . . . . . . . . . . . 14 ⊢ ((^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ⊆ (1..^𝑖) ↔ {𝑘 ∈ ℕ ∣ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐} ⊆ (1..^𝑖))
107		rabss 4011	. . . . . . . . . . . . . 14 ⊢ ({𝑘 ∈ ℕ ∣ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐} ⊆ (1..^𝑖) ↔ ∀𝑘 ∈ ℕ (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖)))
108		df-ral 3053	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ ℕ (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖)) ↔ ∀𝑘(𝑘 ∈ ℕ → (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖))))
109	106, 107, 108	3bitri 297	. . . . . . . . . . . . 13 ⊢ ((^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ⊆ (1..^𝑖) ↔ ∀𝑘(𝑘 ∈ ℕ → (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖))))
110	101, 102, 109	3bitr4i 303	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ (ℕ ∖ (1..^𝑖)) ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 ↔ (^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ⊆ (1..^𝑖))
111		fzofi 13933	. . . . . . . . . . . . 13 ⊢ (1..^𝑖) ∈ Fin
112		ssfi 9104	. . . . . . . . . . . . 13 ⊢ (((1..^𝑖) ∈ Fin ∧ (^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ⊆ (1..^𝑖)) → (^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
113	111, 112	mpan 691	. . . . . . . . . . . 12 ⊢ ((^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ⊆ (1..^𝑖) → (^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
114	110, 113	sylbi 217	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ (ℕ ∖ (1..^𝑖)) ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → (^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
115	94, 114	syl6 35	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℕ → (∀𝑘 ∈ (ℤ_≥‘𝑖) ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → (^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin))
116	115	rexlimiv 3132	. . . . . . . . 9 ⊢ (∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑖) ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → (^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
117	116	ralimi 3075	. . . . . . . 8 ⊢ (∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑖) ¬ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))(^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
118	79, 117	sylbir 235	. . . . . . 7 ⊢ (¬ ∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))(^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
119		iunfi 9250	. . . . . . . 8 ⊢ ((ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∈ Fin ∧ ∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))(^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin) → ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))(^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
120	119	ex 412	. . . . . . 7 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∈ Fin → (∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))(^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin → ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))(^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin))
121	118, 120	syl5 34	. . . . . 6 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∈ Fin → (¬ ∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))(^◡(𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin))
122	72, 121	mt3i 149	. . . . 5 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐)
123		ssrexv 3992	. . . . 5 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ⊆ 𝐼 → (∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐))
124	54, 122, 123	syl2im 40	. . . 4 ⊢ (𝜑 → (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐))
125		unitssre 13449	. . . . . . . . . . . 12 ⊢ (0[,]1) ⊆ ℝ
126		elmapi 8793	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ((0[,]1) ↑_m (1...𝑁)) → 𝑐:(1...𝑁)⟶(0[,]1))
127	126, 46	eleq2s 2855	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝐼 → 𝑐:(1...𝑁)⟶(0[,]1))
128	127	ffvelcdmda 7034	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) → (𝑐‘𝑚) ∈ (0[,]1))
129	125, 128	sselid 3920	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) → (𝑐‘𝑚) ∈ ℝ)
130		nnrp 12951	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℝ⁺)
131	130	rpreccld 12993	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ → (1 / 𝑖) ∈ ℝ⁺)
132		eqid 2737	. . . . . . . . . . . . 13 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
133	132	rexmet 24772	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
134		blcntr 24394	. . . . . . . . . . . 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 1451	. . . . . . . . . . 11 ⊢ (((𝑐‘𝑚) ∈ ℝ ∧ (1 / 𝑖) ∈ ℝ⁺) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
136	129, 131, 135	syl2an 597	. . . . . . . . . 10 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) ∧ 𝑖 ∈ ℕ) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
137	136	an32s 653	. . . . . . . . 9 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
138		fveq1 6837	. . . . . . . . . 10 ⊢ (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) = (𝑐‘𝑚))
139	138	eleq1d 2822	. . . . . . . . 9 ⊢ (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → ((((1^st ‘(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 247	. . . . . . . 8 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
141	140	ralrimdva 3138	. . . . . . 7 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
142	141	reximdv 3153	. . . . . 6 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → (∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
143	142	ralimdva 3150	. . . . 5 ⊢ (𝑐 ∈ 𝐼 → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
144	143	reximia 3073	. . . 4 ⊢ (∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) = 𝑐 → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
145	124, 144	syl6 35	. . 3 ⊢ (𝜑 → (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
146		poimir.r	. . . . . . . 8 ⊢ 𝑅 = (∏_t‘((1...𝑁) × {(topGen‘ran (,))}))
147	49, 48	ixpconst 8852	. . . . . . . . 9 ⊢ X𝑛 ∈ (1...𝑁)(0[,]1) = ((0[,]1) ↑_m (1...𝑁))
148	46, 147	eqtr4i 2763	. . . . . . . 8 ⊢ 𝐼 = X𝑛 ∈ (1...𝑁)(0[,]1)
149	146, 148	oveq12i 7376	. . . . . . 7 ⊢ (𝑅 ↾_t 𝐼) = ((∏_t‘((1...𝑁) × {(topGen‘ran (,))})) ↾_t X𝑛 ∈ (1...𝑁)(0[,]1))
150		fzfid 13932	. . . . . . . . 9 ⊢ (⊤ → (1...𝑁) ∈ Fin)
151		retop 24742	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) ∈ Top
152	151	fconst6 6728	. . . . . . . . . 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 37962	. . . . . . . 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 1549	. . . . . . 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 6851	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) ∈ V
158	157	fvconst2 7156	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
159	158	oveq1d 7379	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑁) → ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾_t (0[,]1)) = ((topGen‘ran (,)) ↾_t (0[,]1)))
160	159	mpteq2ia 5181	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...𝑁) ↦ ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾_t (0[,]1))) = (𝑛 ∈ (1...𝑁) ↦ ((topGen‘ran (,)) ↾_t (0[,]1)))
161		fconstmpt 5690	. . . . . . . . 9 ⊢ ((1...𝑁) × {((topGen‘ran (,)) ↾_t (0[,]1))}) = (𝑛 ∈ (1...𝑁) ↦ ((topGen‘ran (,)) ↾_t (0[,]1)))
162	160, 161	eqtr4i 2763	. . . . . . . 8 ⊢ (𝑛 ∈ (1...𝑁) ↦ ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾_t (0[,]1))) = ((1...𝑁) × {((topGen‘ran (,)) ↾_t (0[,]1))})
163	162	fveq2i 6841	. . . . . . 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 2764	. . . . . 6 ⊢ (𝑅 ↾_t 𝐼) = (∏_t‘((1...𝑁) × {((topGen‘ran (,)) ↾_t (0[,]1))}))
165		fzfi 13931	. . . . . . 7 ⊢ (1...𝑁) ∈ Fin
166		dfii2 24865	. . . . . . . . 9 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
167		iicmp 24869	. . . . . . . . 9 ⊢ II ∈ Comp
168	166, 167	eqeltrri 2834	. . . . . . . 8 ⊢ ((topGen‘ran (,)) ↾_t (0[,]1)) ∈ Comp
169	168	fconst6 6728	. . . . . . 7 ⊢ ((1...𝑁) × {((topGen‘ran (,)) ↾_t (0[,]1))}):(1...𝑁)⟶Comp
170		ptcmpfi 23794	. . . . . . 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 693	. . . . . 6 ⊢ (∏_t‘((1...𝑁) × {((topGen‘ran (,)) ↾_t (0[,]1))})) ∈ Comp
172	164, 171	eqeltri 2833	. . . . 5 ⊢ (𝑅 ↾_t 𝐼) ∈ Comp
173		rehaus 24780	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) ∈ Haus
174	173	fconst6 6728	. . . . . . . . . . 11 ⊢ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Haus
175		pthaus 23619	. . . . . . . . . . 11 ⊢ (((1...𝑁) ∈ Fin ∧ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Haus) → (∏_t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Haus)
176	165, 174, 175	mp2an 693	. . . . . . . . . 10 ⊢ (∏_t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Haus
177	146, 176	eqeltri 2833	. . . . . . . . 9 ⊢ 𝑅 ∈ Haus
178		haustop 23312	. . . . . . . . 9 ⊢ (𝑅 ∈ Haus → 𝑅 ∈ Top)
179	177, 178	ax-mp 5	. . . . . . . 8 ⊢ 𝑅 ∈ Top
180		reex 11126	. . . . . . . . . 10 ⊢ ℝ ∈ V
181		mapss 8834	. . . . . . . . . 10 ⊢ ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁)))
182	180, 125, 181	mp2an 693	. . . . . . . . 9 ⊢ ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁))
183	46, 182	eqsstri 3969	. . . . . . . 8 ⊢ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))
184		uniretop 24743	. . . . . . . . . . 11 ⊢ ℝ = ∪ (topGen‘ran (,))
185	146, 184	ptuniconst 23579	. . . . . . . . . 10 ⊢ (((1...𝑁) ∈ Fin ∧ (topGen‘ran (,)) ∈ Top) → (ℝ ↑_m (1...𝑁)) = ∪ 𝑅)
186	165, 151, 185	mp2an 693	. . . . . . . . 9 ⊢ (ℝ ↑_m (1...𝑁)) = ∪ 𝑅
187	186	restuni 23143	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))) → 𝐼 = ∪ (𝑅 ↾_t 𝐼))
188	179, 183, 187	mp2an 693	. . . . . . 7 ⊢ 𝐼 = ∪ (𝑅 ↾_t 𝐼)
189	188	bwth 23391	. . . . . 6 ⊢ (((𝑅 ↾_t 𝐼) ∈ Comp ∧ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ⊆ 𝐼 ∧ ¬ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∈ Fin) → ∃𝑐 ∈ 𝐼 𝑐 ∈ ((limPt‘(𝑅 ↾_t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))))
190	189	3expia 1122	. . . . 5 ⊢ (((𝑅 ↾_t 𝐼) ∈ Comp ∧ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ⊆ 𝐼) → (¬ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ 𝐼 𝑐 ∈ ((limPt‘(𝑅 ↾_t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))))))
191	172, 54, 190	sylancr 588	. . . 4 ⊢ (𝜑 → (¬ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ 𝐼 𝑐 ∈ ((limPt‘(𝑅 ↾_t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))))))
192		cmptop 23376	. . . . . . . . 9 ⊢ ((𝑅 ↾_t 𝐼) ∈ Comp → (𝑅 ↾_t 𝐼) ∈ Top)
193	172, 192	ax-mp 5	. . . . . . . 8 ⊢ (𝑅 ↾_t 𝐼) ∈ Top
194	188	islp3 23127	. . . . . . . 8 ⊢ (((𝑅 ↾_t 𝐼) ∈ Top ∧ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ⊆ 𝐼 ∧ 𝑐 ∈ 𝐼) → (𝑐 ∈ ((limPt‘(𝑅 ↾_t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))) ↔ ∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)))
195	193, 194	mp3an1 1451	. . . . . . 7 ⊢ ((ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ⊆ 𝐼 ∧ 𝑐 ∈ 𝐼) → (𝑐 ∈ ((limPt‘(𝑅 ↾_t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))) ↔ ∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)))
196	54, 195	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑐 ∈ ((limPt‘(𝑅 ↾_t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))) ↔ ∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)))
197		fzfid 13932	. . . . . . . . . . . . 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 12216	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ → (1 / 𝑖) ∈ ℝ)
200	199	rexrd 11192	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℕ → (1 / 𝑖) ∈ ℝ^*)
201		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
202	132, 201	tgioo 24777	. . . . . . . . . . . . . . . . . 18 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
203	202	blopn 24481	. . . . . . . . . . . . . . . . 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 1451	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐‘𝑚) ∈ ℝ ∧ (1 / 𝑖) ∈ ℝ^*) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
205	129, 200, 204	syl2an 597	. . . . . . . . . . . . . . 15 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) ∧ 𝑖 ∈ ℕ) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
206	205	an32s 653	. . . . . . . . . . . . . 14 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
207	157	fvconst2 7156	. . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . 13 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑚))
210		noel 4279	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝑚 ∈ ∅
211		difid 4317	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑁) ∖ (1...𝑁)) = ∅
212	211	eleq2i 2829	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ((1...𝑁) ∖ (1...𝑁)) ↔ 𝑚 ∈ ∅)
213	210, 212	mtbir 323	. . . . . . . . . . . . . . 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 23564	. . . . . . . . . . . 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 2848	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ 𝑅)
218		ovex 7397	. . . . . . . . . . . . 13 ⊢ ((0[,]1) ↑_m (1...𝑁)) ∈ V
219	46, 218	eqeltri 2833	. . . . . . . . . . . 12 ⊢ 𝐼 ∈ V
220		elrestr 17388	. . . . . . . . . . . 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 1454	. . . . . . . . . . 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 4077	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))
224		imassrn 6034	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ⊆ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))
225	223, 224	sstri 3932	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))
226	225, 54	sstrid 3934	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ 𝐼)
227		haust1 23333	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Haus → 𝑅 ∈ Fre)
228	177, 227	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 𝑅 ∈ Fre
229		restt1 23348	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Fre ∧ 𝐼 ∈ V) → (𝑅 ↾_t 𝐼) ∈ Fre)
230	228, 219, 229	mp2an 693	. . . . . . . . . . . 12 ⊢ (𝑅 ↾_t 𝐼) ∈ Fre
231		funmpt 6534	. . . . . . . . . . . . . 14 ⊢ Fun (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))
232		imafi 9222	. . . . . . . . . . . . . 14 ⊢ ((Fun (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ (1..^𝑖) ∈ Fin) → ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∈ Fin)
233	231, 111, 232	mp2an 693	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∈ Fin
234		diffi 9106	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∈ Fin → (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ Fin)
235	233, 234	ax-mp 5	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ Fin
236	188	t1ficld 23308	. . . . . . . . . . . 12 ⊢ (((𝑅 ↾_t 𝐼) ∈ Fre ∧ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ 𝐼 ∧ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ Fin) → (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ (Clsd‘(𝑅 ↾_t 𝐼)))
237	230, 235, 236	mp3an13 1455	. . . . . . . . . . 11 ⊢ ((((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ 𝐼 → (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ (Clsd‘(𝑅 ↾_t 𝐼)))
238	226, 237	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ (Clsd‘(𝑅 ↾_t 𝐼)))
239	188	difopn 23015	. . . . . . . . . 10 ⊢ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∈ (𝑅 ↾_t 𝐼) ∧ (((𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∈ (𝑅 ↾_t 𝐼))
240	222, 238, 239	syl2anr 598	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ)) → ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(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 ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}))))
243		ineq1 4154	. . . . . . . . . . . 12 ⊢ (𝑣 = ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) = (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})))
244	243	neeq1d 2992	. . . . . . . . . . 11 ⊢ (𝑣 = ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → ((𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅ ↔ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → ((𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅) ↔ (𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)))
246	245	rspcva 3563	. . . . . . . . 9 ⊢ ((((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∈ (𝑅 ↾_t 𝐼) ∧ ∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)) → (𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅))
247	127	ffnd 6667	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝐼 → 𝑐 Fn (1...𝑁))
248	247	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 Fn (1...𝑁))
249	137	ralrimiva 3130	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → ∀𝑚 ∈ (1...𝑁)(𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
250	103	elixp 8849	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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 4141	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼))
254		neldifsnd 4737	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → ¬ 𝑐 ∈ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}))
255	253, 254	eldifd 3901	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})))
256	255	adantll 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(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 616	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) → (∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))))
259		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}) → 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))))
260	258, 259	anim12i 614	. . . . . . . . . . . . . . 15 ⊢ (((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})) → ((∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))))
261		elin 3906	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ↔ (𝑗 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∧ 𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})))
262		andir 1011	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}))))
263		eldif 3900	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})))
264		elin 3906	. . . . . . . . . . . . . . . . . . . . 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 3434	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑗 ∈ V
266	265	elixp 8849	. . . . . . . . . . . . . . . . . . . . . 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 625	. . . . . . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . . . . . 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 984	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ (𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ ¬ 𝑗 ∈ {𝑐}) ↔ (¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∨ ¬ ¬ 𝑗 ∈ {𝑐}))
270		eldif 3900	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ↔ (𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ ¬ 𝑗 ∈ {𝑐}))
271	269, 270	xchnxbir 333	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑗 ∈ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ↔ (¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∨ ¬ ¬ 𝑗 ∈ {𝑐}))
272	268, 271	anbi12i 629	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∧ ¬ 𝑗 ∈ (((𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∨ ¬ ¬ 𝑗 ∈ {𝑐})))
273		andi 1010	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ (¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(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 297	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(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 3900	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐}) ↔ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}))
276	274, 275	anbi12i 629	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∧ 𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(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 ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}) → ¬ 𝑗 ∈ {𝑐})
280	278, 279	anim12ci 615	. . . . . . . . . . . . . . . . . . 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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})) → (¬ 𝑗 ∈ {𝑐} ∧ ¬ ¬ 𝑗 ∈ {𝑐}))
281	277, 280	mto 197	. . . . . . . . . . . . . . . . . 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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}))
282	281	biorfri 940	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}))))
283	262, 276, 282	3bitr4i 303	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∧ 𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})))
284	261, 283	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})))
285		ancom 460	. . . . . . . . . . . . . . . 16 ⊢ (((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))) ↔ (∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ (¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))))))
287	285, 286	bitr4i 278	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))))
288	260, 284, 287	3imtr4i 292	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) → ((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))) ↔ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))))
290		eldif 3900	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ↔ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))))
291	289, 290	bitr4i 278	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))) ↔ 𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))))
292		imadmrn 6033	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ dom (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))) = ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))
293	62, 63	dmmpti 6640	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) = ℕ
294	293	imaeq2i 6021	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ dom (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))) = ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ ℕ)
295	292, 294	eqtr3i 2762	. . . . . . . . . . . . . . . . . . . 20 ⊢ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) = ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ ℕ)
296	295	difeq1i 4063	. . . . . . . . . . . . . . . . . . 19 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) = (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ ℕ) ∖ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)))
297		imadifss 37938	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ ℕ) ∖ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ⊆ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (ℕ ∖ (1..^𝑖)))
298	296, 297	eqsstri 3969	. . . . . . . . . . . . . . . . . 18 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ⊆ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (ℕ ∖ (1..^𝑖)))
299		imass2 6065	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℕ ∖ (1..^𝑖)) ⊆ (ℤ_≥‘𝑖) → ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (ℕ ∖ (1..^𝑖))) ⊆ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (ℤ_≥‘𝑖)))
300	92, 299	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (ℕ ∖ (1..^𝑖))) ⊆ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (ℤ_≥‘𝑖)))
301		df-ima 5641	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (ℤ_≥‘𝑖)) = ran ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ↾ (ℤ_≥‘𝑖))
302		uznnssnn 12842	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ ℕ → (ℤ_≥‘𝑖) ⊆ ℕ)
303	302	resmptd 6003	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ↾ (ℤ_≥‘𝑖)) = (𝑘 ∈ (ℤ_≥‘𝑖) ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))))
304	303	rneqd 5891	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ ℕ → ran ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ↾ (ℤ_≥‘𝑖)) = ran (𝑘 ∈ (ℤ_≥‘𝑖) ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))))
305	301, 304	eqtrid 2784	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (ℤ_≥‘𝑖)) = ran (𝑘 ∈ (ℤ_≥‘𝑖) ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))))
306	300, 305	sseqtrd 3959	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (ℕ ∖ (1..^𝑖))) ⊆ ran (𝑘 ∈ (ℤ_≥‘𝑖) ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))))
307	298, 306	sstrid 3934	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ → (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ⊆ ran (𝑘 ∈ (ℤ_≥‘𝑖) ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))))
308	307	sseld 3921	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℕ → (𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) → 𝑗 ∈ ran (𝑘 ∈ (ℤ_≥‘𝑖) ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))))
309	291, 308	biimtrid 242	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℕ → ((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))) → 𝑗 ∈ ran (𝑘 ∈ (ℤ_≥‘𝑖) ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))))
310	309	anim1d 612	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ → (((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) → (𝑗 ∈ ran (𝑘 ∈ (ℤ_≥‘𝑖) ↦ ((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) → (𝑗 ∈ ran (𝑘 ∈ (ℤ_≥‘𝑖) ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))))
312	311	eximdv 1919	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ℕ → (∃𝑗 𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) → ∃𝑗(𝑗 ∈ ran (𝑘 ∈ (ℤ_≥‘𝑖) ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))))
313		n0 4294	. . . . . . . . . . . 12 ⊢ ((((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅ ↔ ∃𝑗 𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})))
314	62	rgenw 3056	. . . . . . . . . . . . . 14 ⊢ ∀𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) ∈ V
315		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘𝑖) ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) = (𝑘 ∈ (ℤ_≥‘𝑖) ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))
316		fveq1 6837	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) → (𝑗‘𝑚) = (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚))
317	316	eleq1d 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) → ((𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ (((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
318	317	ralbidv 3161	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) → (∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ ∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
319	315, 318	rexrnmptw 7045	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑖)((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})) ∈ V → (∃𝑗 ∈ ran (𝑘 ∈ (ℤ_≥‘𝑖) ↦ ((1^st ‘(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)(((1^st ‘(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) ↦ ((1^st ‘(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)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
321		df-rex 3063	. . . . . . . . . . . . 13 ⊢ (∃𝑗 ∈ ran (𝑘 ∈ (ℤ_≥‘𝑖) ↦ ((1^st ‘(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) ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
322	320, 321	bitr3i 277	. . . . . . . . . . . 12 ⊢ (∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ ∃𝑗(𝑗 ∈ ran (𝑘 ∈ (ℤ_≥‘𝑖) ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
323	312, 313, 322	3imtr4g 296	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ → ((((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅ → ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅ → ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅) → ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(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 ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∈ (𝑅 ↾_t 𝐼) ∧ ∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)) → ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
327	241, 326	mpand 696	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑖 ∈ ℕ) → (∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅) → ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
328	327	ralrimdva 3138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅) → ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
329	196, 328	sylbid 240	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑐 ∈ ((limPt‘(𝑅 ↾_t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))) → ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
330	329	reximdva 3151	. . . 4 ⊢ (𝜑 → (∃𝑐 ∈ 𝐼 𝑐 ∈ ((limPt‘(𝑅 ↾_t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘})))) → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
331	191, 330	syld 47	. . 3 ⊢ (𝜑 → (¬ ran (𝑘 ∈ ℕ ↦ ((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
332	145, 331	pm2.61d 179	. 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
333		poimir.0	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
334		poimir.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
335		poimirlem30.x	. . . 4 ⊢ 𝑋 = ((𝐹‘(((1^st ‘(𝐺‘𝑘)) ∘_f + ((((2^nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘_f / ((1...𝑁) × {𝑘})))‘𝑛)
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 37992	. . 3 ⊢ (𝜑 → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))))
338	337	reximdv 3153	. 2 ⊢ (𝜑 → (∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1^st ‘(𝐺‘𝑘)) ∘_f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∃𝑐 ∈ 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)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 ∀wal 1540 = wceq 1542 ⊤wtru 1543 ∃wex 1781 ∈ wcel 2114 {cab 2715 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 {crab 3390 Vcvv 3430 ∖ cdif 3887 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 ∅c0 4274 {csn 4568 {cpr 4570 ∪ cuni 4851 ∪ ciun 4934 class class class wbr 5086 ↦ cmpt 5167 × cxp 5626 ^◡ccnv 5627 dom cdm 5628 ran crn 5629 ↾ cres 5630 “ cima 5631 ∘ ccom 5632 Fun wfun 6490 Fn wfn 6491 ⟶wf 6492 –1-1-onto→wf1o 6495 ‘cfv 6496 (class class class)co 7364 ∘_f cof 7626 ωcom 7814 1^st c1st 7937 2^nd c2nd 7938 ↑_m cmap 8770 Xcixp 8842 ≈ cen 8887 Fincfn 8890 ℝcr 11034 0cc0 11035 1c1 11036 + caddc 11038 · cmul 11040 ℝ^*cxr 11175 < clt 11176 ≤ cle 11177 − cmin 11374 / cdiv 11804 ℕcn 12171 ℕ₀cn0 12434 ℤ_≥cuz 12785 ℝ⁺crp 12939 (,)cioo 13295 [,]cicc 13298 ...cfz 13458 ..^cfzo 13605 abscabs 15193 ↾_t crest 17380 topGenctg 17397 ∏_tcpt 17398 ∞Metcxmet 21335 ballcbl 21337 MetOpencmopn 21340 Topctop 22874 Clsdccld 22997 limPtclp 23115 Cn ccn 23205 Frect1 23288 Hauscha 23289 Compccmp 23367 IIcii 24858

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5306 ax-pr 5374 ax-un 7686 ax-inf2 9559 ax-cnex 11091 ax-resscn 11092 ax-1cn 11093 ax-icn 11094 ax-addcl 11095 ax-addrcl 11096 ax-mulcl 11097 ax-mulrcl 11098 ax-mulcom 11099 ax-addass 11100 ax-mulass 11101 ax-distr 11102 ax-i2m1 11103 ax-1ne0 11104 ax-1rid 11105 ax-rnegex 11106 ax-rrecex 11107 ax-cnre 11108 ax-pre-lttri 11109 ax-pre-lttrn 11110 ax-pre-ltadd 11111 ax-pre-mulgt0 11112 ax-pre-sup 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5523 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5581 df-we 5583 df-xp 5634 df-rel 5635 df-cnv 5636 df-co 5637 df-dm 5638 df-rn 5639 df-res 5640 df-ima 5641 df-pred 6263 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7321 df-ov 7367 df-oprab 7368 df-mpo 7369 df-of 7628 df-om 7815 df-1st 7939 df-2nd 7940 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fi 9321 df-sup 9352 df-inf 9353 df-pnf 11178 df-mnf 11179 df-xr 11180 df-ltxr 11181 df-le 11182 df-sub 11376 df-neg 11377 df-div 11805 df-nn 12172 df-2 12241 df-3 12242 df-n0 12435 df-z 12522 df-uz 12786 df-q 12896 df-rp 12940 df-xneg 13060 df-xadd 13061 df-xmul 13062 df-ioo 13299 df-icc 13302 df-fz 13459 df-fzo 13606 df-fl 13748 df-seq 13961 df-exp 14021 df-cj 15058 df-re 15059 df-im 15060 df-sqrt 15194 df-abs 15195 df-rest 17382 df-topgen 17403 df-pt 17404 df-psmet 21342 df-xmet 21343 df-met 21344 df-bl 21345 df-mopn 21346 df-top 22875 df-topon 22892 df-bases 22927 df-cld 23000 df-ntr 23001 df-cls 23002 df-lp 23117 df-cn 23208 df-cnp 23209 df-t1 23295 df-haus 23296 df-cmp 23368 df-tx 23543 df-hmeo 23736 df-hmph 23737 df-ii 24860

This theorem is referenced by: poimirlem32 37995

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