Theorem poimirlem30 36108

Description: Lemma for poimir 36111 combining poimirlem29 36107 with bwth 22761. (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 13617	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑘) → 𝑖 ∈ ℕ0)
2	1	nn0red 12474	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑘) → 𝑖 ∈ ℝ)
3		nndivre 12194	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ)
4	2, 3	sylan 580	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ)
5		elfzole1 13580	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑘) → 0 ≤ 𝑖)
6	2, 5	jca 512	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑘) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
7		nnrp 12926	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
8	7	rpregt0d 12963	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
9		divge0 12024	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ℝ ∧ 0 ≤ 𝑖) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝑖 / 𝑘))
10	6, 8, 9	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑖 / 𝑘))
11		elfzo0le 13616	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑘) → 𝑖 ≤ 𝑘)
12	11	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 𝑖 ≤ 𝑘)
13	2	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 𝑖 ∈ ℝ)
14		1red 11156	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ)
15	7	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ+)
16	13, 14, 15	ledivmuld 13010	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖 ≤ (𝑘 · 1)))
17		nncn 12161	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
18	17	mulid1d 11172	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → (𝑘 · 1) = 𝑘)
19	18	breq2d 5117	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → (𝑖 ≤ (𝑘 · 1) ↔ 𝑖 ≤ 𝑘))
20	19	adantl 482	. . . . . . . . . . . . . . . 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 13383	. . . . . . . . . . . . . 14 ⊢ ((𝑖 / 𝑘) ∈ (0[,]1) ↔ ((𝑖 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝑖 / 𝑘) ∧ (𝑖 / 𝑘) ≤ 1))
24	4, 10, 22, 23	syl3anbrc 1343	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ (0[,]1))
25	24	ancoms 459	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0..^𝑘)) → (𝑖 / 𝑘) ∈ (0[,]1))
26		elsni 4603	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ {𝑘} → 𝑗 = 𝑘)
27	26	oveq2d 7373	. . . . . . . . . . . . 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 246	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0..^𝑘)) → (𝑗 ∈ {𝑘} → (𝑖 / 𝑗) ∈ (0[,]1)))
30	29	impr 455	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ (𝑖 ∈ (0..^𝑘) ∧ 𝑗 ∈ {𝑘})) → (𝑖 / 𝑗) ∈ (0[,]1))
31	30	adantll 712	. . . . . . . . 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 7035	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ((ℕ0 ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
34		xp1st 7953	. . . . . . . . . . 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 8803	. . . . . . . . . . 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 6500	. . . . . . . . . 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 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐺‘𝑘)):(1...𝑁)⟶(0..^𝑘))
40		vex 3449	. . . . . . . . . . 11 ⊢ 𝑘 ∈ V
41	40	fconst 6728	. . . . . . . . . 10 ⊢ ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘}
42	41	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘})
43		fzfid 13878	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑁) ∈ Fin)
44		inidm 4178	. . . . . . . . 9 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
45	31, 39, 42, 43, 43, 44	off 7635	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
46		poimir.i	. . . . . . . . . 10 ⊢ 𝐼 = ((0[,]1) ↑m (1...𝑁))
47	46	eleq2i 2829	. . . . . . . . 9 ⊢ (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑m (1...𝑁)))
48		ovex 7390	. . . . . . . . . 10 ⊢ (0[,]1) ∈ V
49		ovex 7390	. . . . . . . . . 10 ⊢ (1...𝑁) ∈ V
50	48, 49	elmap 8809	. . . . . . . . 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 7063	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))):ℕ⟶𝐼)
54	53	frnd 6676	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ⊆ 𝐼)
55		ominf 9202	. . . . . . 7 ⊢ ¬ ω ∈ Fin
56		nnenom 13885	. . . . . . . . 9 ⊢ ℕ ≈ ω
57		enfi 9134	. . . . . . . . 9 ⊢ (ℕ ≈ ω → (ℕ ∈ Fin ↔ ω ∈ Fin))
58	56, 57	ax-mp 5	. . . . . . . 8 ⊢ (ℕ ∈ Fin ↔ ω ∈ Fin)
59		iunid 5020	. . . . . . . . . . 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 6011	. . . . . . . . . 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 7192	. . . . . . . . . 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 7390	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ V
63		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) = (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))
64	62, 63	fnmpti 6644	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) Fn ℕ
65		dffn3 6681	. . . . . . . . . . . 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 6690	. . . . . . . . . . 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 2773	. . . . . . . . 9 ⊢ ℕ = ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐})
70	69	eleq1i 2828	. . . . . . . 8 ⊢ (ℕ ∈ Fin ↔ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
71	58, 70	bitr3i 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 3075	. . . . . . . . . . . 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 3097	. . . . . . . . . . 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 3103	. . . . . . . . . . 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 3096	. . . . . . . . 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 3075	. . . . . . . . 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 12806	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ≥‘1)
81		elnnuz 12807	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ ↔ 𝑖 ∈ (ℤ≥‘1))
82		fzouzsplit 13607	. . . . . . . . . . . . . . . . 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	eqtrid 2788	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℕ → ℕ = ((1..^𝑖) ∪ (ℤ≥‘𝑖)))
85	84	difeq1d 4081	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ → (ℕ ∖ (1..^𝑖)) = (((1..^𝑖) ∪ (ℤ≥‘𝑖)) ∖ (1..^𝑖)))
86		uncom 4113	. . . . . . . . . . . . . . . 16 ⊢ ((1..^𝑖) ∪ (ℤ≥‘𝑖)) = ((ℤ≥‘𝑖) ∪ (1..^𝑖))
87	86	difeq1i 4078	. . . . . . . . . . . . . . 15 ⊢ (((1..^𝑖) ∪ (ℤ≥‘𝑖)) ∖ (1..^𝑖)) = (((ℤ≥‘𝑖) ∪ (1..^𝑖)) ∖ (1..^𝑖))
88		difun2 4440	. . . . . . . . . . . . . . 15 ⊢ (((ℤ≥‘𝑖) ∪ (1..^𝑖)) ∖ (1..^𝑖)) = ((ℤ≥‘𝑖) ∖ (1..^𝑖))
89	87, 88	eqtri 2764	. . . . . . . . . . . . . 14 ⊢ (((1..^𝑖) ∪ (ℤ≥‘𝑖)) ∖ (1..^𝑖)) = ((ℤ≥‘𝑖) ∖ (1..^𝑖))
90	85, 89	eqtrdi 2792	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ℕ → (ℕ ∖ (1..^𝑖)) = ((ℤ≥‘𝑖) ∖ (1..^𝑖)))
91		difss 4091	. . . . . . . . . . . . 13 ⊢ ((ℤ≥‘𝑖) ∖ (1..^𝑖)) ⊆ (ℤ≥‘𝑖)
92	90, 91	eqsstrdi 3998	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ℕ → (ℕ ∖ (1..^𝑖)) ⊆ (ℤ≥‘𝑖))
93		ssralv 4010	. . . . . . . . . . . 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 451	. . . . . . . . . . . . . . 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 3920	. . . . . . . . . . . . . . . 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 1821	. . . . . . . . . . . . 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 3065	. . . . . . . . . . . . 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 3449	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
104	63	mptiniseg 6191	. . . . . . . . . . . . . . . 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 3972	. . . . . . . . . . . . . 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 4029	. . . . . . . . . . . . . 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 3065	. . . . . . . . . . . . . 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 13879	. . . . . . . . . . . . 13 ⊢ (1..^𝑖) ∈ Fin
112		ssfi 9117	. . . . . . . . . . . . 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 688	. . . . . . . . . . . 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 3145	. . . . . . . . 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 9284	. . . . . . . 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 413	. . . . . . 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 4011	. . . . 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 13416	. . . . . . . . . . . 12 ⊢ (0[,]1) ⊆ ℝ
126		elmapi 8787	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ((0[,]1) ↑m (1...𝑁)) → 𝑐:(1...𝑁)⟶(0[,]1))
127	126, 46	eleq2s 2856	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝐼 → 𝑐:(1...𝑁)⟶(0[,]1))
128	127	ffvelcdmda 7035	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) → (𝑐‘𝑚) ∈ (0[,]1))
129	125, 128	sselid 3942	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) → (𝑐‘𝑚) ∈ ℝ)
130		nnrp 12926	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℝ+)
131	130	rpreccld 12967	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ → (1 / 𝑖) ∈ ℝ+)
132		eqid 2736	. . . . . . . . . . . . 13 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
133	132	rexmet 24154	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
134		blcntr 23766	. . . . . . . . . . . 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 1448	. . . . . . . . . . 11 ⊢ (((𝑐‘𝑚) ∈ ℝ ∧ (1 / 𝑖) ∈ ℝ+) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
136	129, 131, 135	syl2an 596	. . . . . . . . . 10 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) ∧ 𝑖 ∈ ℕ) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
137	136	an32s 650	. . . . . . . . 9 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
138		fveq1 6841	. . . . . . . . . 10 ⊢ (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) = (𝑐‘𝑚))
139	138	eleq1d 2822	. . . . . . . . 9 ⊢ (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) = 𝑐 → ((((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
140	137, 139	syl5ibrcom 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 3151	. . . . . . 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 3167	. . . . . 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 3164	. . . . 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 3084	. . . 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 8845	. . . . . . . . 9 ⊢ X𝑛 ∈ (1...𝑁)(0[,]1) = ((0[,]1) ↑m (1...𝑁))
148	46, 147	eqtr4i 2767	. . . . . . . 8 ⊢ 𝐼 = X𝑛 ∈ (1...𝑁)(0[,]1)
149	146, 148	oveq12i 7369	. . . . . . 7 ⊢ (𝑅 ↾t 𝐼) = ((∏t‘((1...𝑁) × {(topGen‘ran (,))})) ↾t X𝑛 ∈ (1...𝑁)(0[,]1))
150		fzfid 13878	. . . . . . . . 9 ⊢ (⊤ → (1...𝑁) ∈ Fin)
151		retop 24125	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) ∈ Top
152	151	fconst6 6732	. . . . . . . . . 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 36077	. . . . . . . 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 1548	. . . . . . 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 6855	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) ∈ V
158	157	fvconst2 7153	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
159	158	oveq1d 7372	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑁) → ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1)) = ((topGen‘ran (,)) ↾t (0[,]1)))
160	159	mpteq2ia 5208	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...𝑁) ↦ ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1))) = (𝑛 ∈ (1...𝑁) ↦ ((topGen‘ran (,)) ↾t (0[,]1)))
161		fconstmpt 5694	. . . . . . . . 9 ⊢ ((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))}) = (𝑛 ∈ (1...𝑁) ↦ ((topGen‘ran (,)) ↾t (0[,]1)))
162	160, 161	eqtr4i 2767	. . . . . . . 8 ⊢ (𝑛 ∈ (1...𝑁) ↦ ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1))) = ((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))})
163	162	fveq2i 6845	. . . . . . 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 2768	. . . . . 6 ⊢ (𝑅 ↾t 𝐼) = (∏t‘((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))}))
165		fzfi 13877	. . . . . . 7 ⊢ (1...𝑁) ∈ Fin
166		dfii2 24245	. . . . . . . . 9 ⊢ II = ((topGen‘ran (,)) ↾t (0[,]1))
167		iicmp 24249	. . . . . . . . 9 ⊢ II ∈ Comp
168	166, 167	eqeltrri 2835	. . . . . . . 8 ⊢ ((topGen‘ran (,)) ↾t (0[,]1)) ∈ Comp
169	168	fconst6 6732	. . . . . . 7 ⊢ ((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))}):(1...𝑁)⟶Comp
170		ptcmpfi 23164	. . . . . . 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 690	. . . . . 6 ⊢ (∏t‘((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))})) ∈ Comp
172	164, 171	eqeltri 2834	. . . . 5 ⊢ (𝑅 ↾t 𝐼) ∈ Comp
173		rehaus 24162	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) ∈ Haus
174	173	fconst6 6732	. . . . . . . . . . 11 ⊢ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Haus
175		pthaus 22989	. . . . . . . . . . 11 ⊢ (((1...𝑁) ∈ Fin ∧ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Haus) → (∏t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Haus)
176	165, 174, 175	mp2an 690	. . . . . . . . . 10 ⊢ (∏t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Haus
177	146, 176	eqeltri 2834	. . . . . . . . 9 ⊢ 𝑅 ∈ Haus
178		haustop 22682	. . . . . . . . 9 ⊢ (𝑅 ∈ Haus → 𝑅 ∈ Top)
179	177, 178	ax-mp 5	. . . . . . . 8 ⊢ 𝑅 ∈ Top
180		reex 11142	. . . . . . . . . 10 ⊢ ℝ ∈ V
181		mapss 8827	. . . . . . . . . 10 ⊢ ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑m (1...𝑁)) ⊆ (ℝ ↑m (1...𝑁)))
182	180, 125, 181	mp2an 690	. . . . . . . . 9 ⊢ ((0[,]1) ↑m (1...𝑁)) ⊆ (ℝ ↑m (1...𝑁))
183	46, 182	eqsstri 3978	. . . . . . . 8 ⊢ 𝐼 ⊆ (ℝ ↑m (1...𝑁))
184		uniretop 24126	. . . . . . . . . . 11 ⊢ ℝ = ∪ (topGen‘ran (,))
185	146, 184	ptuniconst 22949	. . . . . . . . . 10 ⊢ (((1...𝑁) ∈ Fin ∧ (topGen‘ran (,)) ∈ Top) → (ℝ ↑m (1...𝑁)) = ∪ 𝑅)
186	165, 151, 185	mp2an 690	. . . . . . . . 9 ⊢ (ℝ ↑m (1...𝑁)) = ∪ 𝑅
187	186	restuni 22513	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝐼 ⊆ (ℝ ↑m (1...𝑁))) → 𝐼 = ∪ (𝑅 ↾t 𝐼))
188	179, 183, 187	mp2an 690	. . . . . . 7 ⊢ 𝐼 = ∪ (𝑅 ↾t 𝐼)
189	188	bwth 22761	. . . . . 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 1121	. . . . 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 587	. . . 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 22746	. . . . . . . . 9 ⊢ ((𝑅 ↾t 𝐼) ∈ Comp → (𝑅 ↾t 𝐼) ∈ Top)
193	172, 192	ax-mp 5	. . . . . . . 8 ⊢ (𝑅 ↾t 𝐼) ∈ Top
194	188	islp3 22497	. . . . . . . 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 1448	. . . . . . 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 580	. . . . . 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 13878	. . . . . . . . . . . . 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 12195	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ → (1 / 𝑖) ∈ ℝ)
200	199	rexrd 11205	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℕ → (1 / 𝑖) ∈ ℝ*)
201		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
202	132, 201	tgioo 24159	. . . . . . . . . . . . . . . . . 18 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
203	202	blopn 23856	. . . . . . . . . . . . . . . . 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 1448	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐‘𝑚) ∈ ℝ ∧ (1 / 𝑖) ∈ ℝ*) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
205	129, 200, 204	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) ∧ 𝑖 ∈ ℕ) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
206	205	an32s 650	. . . . . . . . . . . . . 14 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
207	157	fvconst2 7153	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑚) = (topGen‘ran (,)))
208	207	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑚) = (topGen‘ran (,)))
209	206, 208	eleqtrrd 2841	. . . . . . . . . . . . 13 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑚))
210		noel 4290	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝑚 ∈ ∅
211		difid 4330	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑁) ∖ (1...𝑁)) = ∅
212	211	eleq2i 2829	. . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 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 22934	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (∏t‘((1...𝑁) × {(topGen‘ran (,))})))
217	216, 146	eleqtrrdi 2849	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ 𝑅)
218		ovex 7390	. . . . . . . . . . . . 13 ⊢ ((0[,]1) ↑m (1...𝑁)) ∈ V
219	46, 218	eqeltri 2834	. . . . . . . . . . . 12 ⊢ 𝐼 ∈ V
220		elrestr 17310	. . . . . . . . . . . 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 1451	. . . . . . . . . . 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 4091	. . . . . . . . . . . . 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 6024	. . . . . . . . . . . . 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 3953	. . . . . . . . . . . 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 3955	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ 𝐼)
227		haust1 22703	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Haus → 𝑅 ∈ Fre)
228	177, 227	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 𝑅 ∈ Fre
229		restt1 22718	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Fre ∧ 𝐼 ∈ V) → (𝑅 ↾t 𝐼) ∈ Fre)
230	228, 219, 229	mp2an 690	. . . . . . . . . . . 12 ⊢ (𝑅 ↾t 𝐼) ∈ Fre
231		funmpt 6539	. . . . . . . . . . . . . 14 ⊢ Fun (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))
232		imafi 9119	. . . . . . . . . . . . . 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 690	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∈ Fin
234		diffi 9123	. . . . . . . . . . . . 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 22678	. . . . . . . . . . . 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 1452	. . . . . . . . . . 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 22385	. . . . . . . . . 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 597	. . . . . . . . 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 468	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑖 ∈ ℕ) → ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∈ (𝑅 ↾t 𝐼))
242		eleq2 2826	. . . . . . . . . . 11 ⊢ (𝑣 = ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → (𝑐 ∈ 𝑣 ↔ 𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}))))
243		ineq1 4165	. . . . . . . . . . . 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 3003	. . . . . . . . . . 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 3579	. . . . . . . . 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 6669	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝐼 → 𝑐 Fn (1...𝑁))
248	247	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 Fn (1...𝑁))
249	137	ralrimiva 3143	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → ∀𝑚 ∈ (1...𝑁)(𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
250	103	elixp 8842	. . . . . . . . . . . . . 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 583	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
252		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ 𝐼)
253	251, 252	elind 4154	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼))
254		neldifsnd 4753	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → ¬ 𝑐 ∈ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}))
255	253, 254	eldifd 3921	. . . . . . . . . . 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 712	. . . . . . . . . 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 767	. . . . . . . . . . . . . . . . 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 615	. . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . 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 613	. . . . . . . . . . . . . . 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 3926	. . . . . . . . . . . . . . . 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 1007	. . . . . . . . . . . . . . . . 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 3920	. . . . . . . . . . . . . . . . . . 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 3926	. . . . . . . . . . . . . . . . . . . . 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 3449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑗 ∈ V
266	265	elixp 8842	. . . . . . . . . . . . . . . . . . . . . 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 624	. . . . . . . . . . . . . . . . . . . . 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 980	. . . . . . . . . . . . . . . . . . . . 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 3920	. . . . . . . . . . . . . . . . . . . . 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 627	. . . . . . . . . . . . . . . . . . 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 1006	. . . . . . . . . . . . . . . . . . 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 3920	. . . . . . . . . . . . . . . . . 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 627	. . . . . . . . . . . . . . . . 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		pm3.24 403	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ (¬ 𝑗 ∈ {𝑐} ∧ ¬ ¬ 𝑗 ∈ {𝑐})
278		simpr 485	. . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}) → ¬ 𝑗 ∈ {𝑐})
280	278, 279	anim12ci 614	. . . . . . . . . . . . . . . . . . 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 937	. . . . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . . 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 469	. . . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . . . 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 3920	. . . . . . . . . . . . . . . . 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 6023	. . . . . . . . . . . . . . . . . . . . 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 6645	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) = ℕ
294	293	imaeq2i 6011	. . . . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) = ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) “ ℕ)
296	295	difeq1i 4078	. . . . . . . . . . . . . . . . . . 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 36053	. . . . . . . . . . . . . . . . . . 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 3978	. . . . . . . . . . . . . . . . . 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 6054	. . . . . . . . . . . . . . . . . . . 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 5646	. . . . . . . . . . . . . . . . . . . 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 12820	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ ℕ → (ℤ≥‘𝑖) ⊆ ℕ)
303	302	resmptd 5994	. . . . . . . . . . . . . . . . . . . . 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 5893	. . . . . . . . . . . . . . . . . . . 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	eqtrid 2788	. . . . . . . . . . . . . . . . . . 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 3984	. . . . . . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . . . . . 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 3943	. . . . . . . . . . . . . . . 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	biimtrid 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 611	. . . . . . . . . . . . . 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 1920	. . . . . . . . . . . 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 4306	. . . . . . . . . . . 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 3068	. . . . . . . . . . . . . 14 ⊢ ∀𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) ∈ V
315		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))) = (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})))
316		fveq1 6841	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) → (𝑗‘𝑚) = (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚))
317	316	eleq1d 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = ((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘})) → ((𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ (((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
318	317	ralbidv 3174	. . . . . . . . . . . . . . 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 7045	. . . . . . . . . . . . . 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 3074	. . . . . . . . . . . . 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 482	. . . . . . . . . 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 693	. . . . . . 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 3151	. . . . . 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 3165	. . . 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 36107	. . 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 3167	. 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 396   ∨ wo 845   ∧ w3a 1087  ∀wal 1539   = wceq 1541  ⊤wtru 1542  ∃wex 1781   ∈ wcel 2106  {cab 2713   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  {crab 3407  Vcvv 3445   ∖ cdif 3907   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  {csn 4586  {cpr 4588  ∪ cuni 4865  ∪ ciun 4954   class class class wbr 5105   ↦ cmpt 5188   × cxp 5631  ^◡ccnv 5632  dom cdm 5633  ran crn 5634   ↾ cres 5635   “ cima 5636   ∘ ccom 5637  Fun wfun 6490   Fn wfn 6491  ⟶wf 6492  –1-1-onto→wf1o 6495  ‘cfv 6496  (class class class)co 7357   ∘_f cof 7615  ωcom 7802  1^st c1st 7919  2^nd c2nd 7920   ↑_m cmap 8765  Xcixp 8835   ≈ cen 8880  Fincfn 8883  ℝcr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  ℝ^*cxr 11188   < clt 11189   ≤ cle 11190   − cmin 11385   / cdiv 11812  ℕcn 12153  ℕ₀cn0 12413  ℤ_≥cuz 12763  ℝ⁺crp 12915  (,)cioo 13264  [,]cicc 13267  ...cfz 13424  ..^cfzo 13567  abscabs 15119   ↾_t crest 17302  topGenctg 17319  ∏_tcpt 17320  ∞Metcxmet 20781  ballcbl 20783  MetOpencmopn 20786  Topctop 22242  Clsdccld 22367  limPtclp 22485   Cn ccn 22575  Frect1 22658  Hauscha 22659  Compccmp 22737  IIcii 24238

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-rest 17304  df-topgen 17325  df-pt 17326  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-lp 22487  df-cn 22578  df-cnp 22579  df-t1 22665  df-haus 22666  df-cmp 22738  df-tx 22913  df-hmeo 23106  df-hmph 23107  df-ii 24240

This theorem is referenced by:  poimirlem32  36110