Theorem vitalilem2 25777

Description: Lemma for vitali 25781. (Contributed by Mario Carneiro, 16-Jun-2014.)

Hypotheses

Ref	Expression
vitali.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)}
vitali.2	⊢ 𝑆 = ((0[,]1) / ∼ )
vitali.3	⊢ (𝜑 → 𝐹 Fn 𝑆)
vitali.4	⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
vitali.5	⊢ (𝜑 → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
vitali.6	⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹})
vitali.7	⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))

Assertion

Ref	Expression
vitalilem2	⊢ (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∧ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2)))

Distinct variable groups:   𝑚,𝑛,𝑠,𝑥,𝑦,𝑧,𝐺   𝜑,𝑚,𝑛,𝑥,𝑧   𝑧,𝑆   𝑇,𝑚,𝑥   𝑚,𝐹,𝑛,𝑠,𝑥,𝑦,𝑧   ∼ ,𝑚,𝑛,𝑠,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑦, 𝑠)   𝑆(𝑥, 𝑦, 𝑚, 𝑛, 𝑠)   𝑇(𝑦, 𝑧, 𝑛, 𝑠)

Proof of Theorem vitalilem2

Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vitali.3	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝑆)
2		vitali.4	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
3		vitali.2	. . . . . . . . 9 ⊢ 𝑆 = ((0[,]1) / ∼ )
4		neeq1 3020	. . . . . . . . 9 ⊢ ([𝑣] ∼ = 𝑧 → ([𝑣] ∼ ≠ ∅ ↔ 𝑧 ≠ ∅))
5		vitali.1	. . . . . . . . . . . . . 14 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)}
6	5	vitalilem1 25776	. . . . . . . . . . . . 13 ⊢ ∼ Er (0[,]1)
7		erdm 8701	. . . . . . . . . . . . 13 ⊢ ( ∼ Er (0[,]1) → dom ∼ = (0[,]1))
8	6, 7	ax-mp 5	. . . . . . . . . . . 12 ⊢ dom ∼ = (0[,]1)
9	8	eleq2i 2855	. . . . . . . . . . 11 ⊢ (𝑣 ∈ dom ∼ ↔ 𝑣 ∈ (0[,]1))
10		ecdmn0 8743	. . . . . . . . . . 11 ⊢ (𝑣 ∈ dom ∼ ↔ [𝑣] ∼ ≠ ∅)
11	9, 10	bitr3i 280	. . . . . . . . . 10 ⊢ (𝑣 ∈ (0[,]1) ↔ [𝑣] ∼ ≠ ∅)
12	11	biimpi 219	. . . . . . . . 9 ⊢ (𝑣 ∈ (0[,]1) → [𝑣] ∼ ≠ ∅)
13	3, 4, 12	ectocl 8777	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ≠ ∅)
14	13	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ≠ ∅)
15		sseq1 3962	. . . . . . . . . 10 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ (0[,]1) ↔ 𝑧 ⊆ (0[,]1)))
16	6	a1i 11	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (0[,]1) → ∼ Er (0[,]1))
17	16	ecss 8742	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0[,]1) → [𝑤] ∼ ⊆ (0[,]1))
18	3, 15, 17	ectocl 8777	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ⊆ (0[,]1))
19	18	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ⊆ (0[,]1))
20	19	sseld 3936	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑧) ∈ 𝑧 → (𝐹‘𝑧) ∈ (0[,]1)))
21	14, 20	embantd 60	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧) → (𝐹‘𝑧) ∈ (0[,]1)))
22	21	ralimdva 3177	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧) → ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ (0[,]1)))
23	2, 22	mpd 16	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ (0[,]1))
24		ffnfv 7114	. . . 4 ⊢ (𝐹:𝑆⟶(0[,]1) ↔ (𝐹 Fn 𝑆 ∧ ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ (0[,]1)))
25	1, 23, 24	sylanbrc 594	. . 3 ⊢ (𝜑 → 𝐹:𝑆⟶(0[,]1))
26	25	frnd 6714	. 2 ⊢ (𝜑 → ran 𝐹 ⊆ (0[,]1))
27		vitali.5	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
28	27	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
29		f1ocnv 6833	. . . . . . 7 ⊢ (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → ◡𝐺:(ℚ ∩ (-1[,]1))–1-1-onto→ℕ)
30		f1of 6820	. . . . . . 7 ⊢ (◡𝐺:(ℚ ∩ (-1[,]1))–1-1-onto→ℕ → ◡𝐺:(ℚ ∩ (-1[,]1))⟶ℕ)
31	28, 29, 30	3syl 19	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ◡𝐺:(ℚ ∩ (-1[,]1))⟶ℕ)
32	11	bilani 509	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [𝑣] ∼ ≠ ∅)
33		neeq1 3020	. . . . . . . . . . . 12 ⊢ (𝑧 = [𝑣] ∼ → (𝑧 ≠ ∅ ↔ [𝑣] ∼ ≠ ∅))
34		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑧 = [𝑣] ∼ → (𝐹‘𝑧) = (𝐹‘[𝑣] ∼ ))
35		id 23	. . . . . . . . . . . . 13 ⊢ (𝑧 = [𝑣] ∼ → 𝑧 = [𝑣] ∼ )
36	34, 35	eleq12d 2857	. . . . . . . . . . . 12 ⊢ (𝑧 = [𝑣] ∼ → ((𝐹‘𝑧) ∈ 𝑧 ↔ (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ))
37	33, 36	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑧 = [𝑣] ∼ → ((𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧) ↔ ([𝑣] ∼ ≠ ∅ → (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ )))
38	2	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
39		ovex 7443	. . . . . . . . . . . . . . 15 ⊢ (0[,]1) ∈ V
40		erex 8715	. . . . . . . . . . . . . . 15 ⊢ ( ∼ Er (0[,]1) → ((0[,]1) ∈ V → ∼ ∈ V))
41	6, 39, 40	mp2 9	. . . . . . . . . . . . . 14 ⊢ ∼ ∈ V
42	41	ecelqsi 8763	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (0[,]1) → [𝑣] ∼ ∈ ((0[,]1) / ∼ ))
43	42	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [𝑣] ∼ ∈ ((0[,]1) / ∼ ))
44	43, 3	eleqtrrdi 2874	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [𝑣] ∼ ∈ 𝑆)
45	37, 38, 44	rspcdva 3582	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ([𝑣] ∼ ≠ ∅ → (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ))
46	32, 45	mpd 16	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ )
47		fvex 6894	. . . . . . . . . . 11 ⊢ (𝐹‘[𝑣] ∼ ) ∈ V
48		vex 3459	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
49	47, 48	elec 8737	. . . . . . . . . 10 ⊢ ((𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ↔ 𝑣 ∼ (𝐹‘[𝑣] ∼ ))
50		oveq12 7419	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = (𝐹‘[𝑣] ∼ )) → (𝑥 − 𝑦) = (𝑣 − (𝐹‘[𝑣] ∼ )))
51	50	eleq1d 2848	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = (𝐹‘[𝑣] ∼ )) → ((𝑥 − 𝑦) ∈ ℚ ↔ (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℚ))
52	51, 5	brab2a 5754	. . . . . . . . . 10 ⊢ (𝑣 ∼ (𝐹‘[𝑣] ∼ ) ↔ ((𝑣 ∈ (0[,]1) ∧ (𝐹‘[𝑣] ∼ ) ∈ (0[,]1)) ∧ (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℚ))
53	49, 52	bitri 278	. . . . . . . . 9 ⊢ ((𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ↔ ((𝑣 ∈ (0[,]1) ∧ (𝐹‘[𝑣] ∼ ) ∈ (0[,]1)) ∧ (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℚ))
54	46, 53	sylib 221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ((𝑣 ∈ (0[,]1) ∧ (𝐹‘[𝑣] ∼ ) ∈ (0[,]1)) ∧ (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℚ))
55	54	simprd 500	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℚ)
56		elicc01 13497	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (0[,]1) ↔ (𝑣 ∈ ℝ ∧ 0 ≤ 𝑣 ∧ 𝑣 ≤ 1))
57	56	bilani 509	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 ∈ ℝ ∧ 0 ≤ 𝑣 ∧ 𝑣 ≤ 1))
58	57	simp1d 1160	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ ℝ)
59	54	simpld 499	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 ∈ (0[,]1) ∧ (𝐹‘[𝑣] ∼ ) ∈ (0[,]1)))
60	59	simprd 500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ (0[,]1))
61		elicc01 13497	. . . . . . . . . . 11 ⊢ ((𝐹‘[𝑣] ∼ ) ∈ (0[,]1) ↔ ((𝐹‘[𝑣] ∼ ) ∈ ℝ ∧ 0 ≤ (𝐹‘[𝑣] ∼ ) ∧ (𝐹‘[𝑣] ∼ ) ≤ 1))
62	60, 61	sylib 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘[𝑣] ∼ ) ∈ ℝ ∧ 0 ≤ (𝐹‘[𝑣] ∼ ) ∧ (𝐹‘[𝑣] ∼ ) ≤ 1))
63	62	simp1d 1160	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ ℝ)
64	58, 63	resubcld 11646	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℝ)
65	63, 58	resubcld 11646	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘[𝑣] ∼ ) − 𝑣) ∈ ℝ)
66		1red 11213	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 1 ∈ ℝ)
67	57	simp2d 1161	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 0 ≤ 𝑣)
68	63, 58	subge02d 11810	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (0 ≤ 𝑣 ↔ ((𝐹‘[𝑣] ∼ ) − 𝑣) ≤ (𝐹‘[𝑣] ∼ )))
69	67, 68	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘[𝑣] ∼ ) − 𝑣) ≤ (𝐹‘[𝑣] ∼ ))
70	62	simp3d 1162	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ≤ 1)
71	65, 63, 66, 69, 70	letrd 11371	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘[𝑣] ∼ ) − 𝑣) ≤ 1)
72	65, 66	lenegd 11797	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (((𝐹‘[𝑣] ∼ ) − 𝑣) ≤ 1 ↔ -1 ≤ -((𝐹‘[𝑣] ∼ ) − 𝑣)))
73	71, 72	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → -1 ≤ -((𝐹‘[𝑣] ∼ ) − 𝑣))
74	63	recnd 11241	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ ℂ)
75	58	recnd 11241	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ ℂ)
76	74, 75	negsubdi2d 11589	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → -((𝐹‘[𝑣] ∼ ) − 𝑣) = (𝑣 − (𝐹‘[𝑣] ∼ )))
77	73, 76	breqtrd 5137	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → -1 ≤ (𝑣 − (𝐹‘[𝑣] ∼ )))
78	62	simp2d 1161	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 0 ≤ (𝐹‘[𝑣] ∼ ))
79	58, 63	subge02d 11810	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (0 ≤ (𝐹‘[𝑣] ∼ ) ↔ (𝑣 − (𝐹‘[𝑣] ∼ )) ≤ 𝑣))
80	78, 79	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ≤ 𝑣)
81	57	simp3d 1162	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ≤ 1)
82	64, 58, 66, 80, 81	letrd 11371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ≤ 1)
83		neg1rr 12208	. . . . . . . . 9 ⊢ -1 ∈ ℝ
84		1re 11212	. . . . . . . . 9 ⊢ 1 ∈ ℝ
85	83, 84	elicc2i 13443	. . . . . . . 8 ⊢ ((𝑣 − (𝐹‘[𝑣] ∼ )) ∈ (-1[,]1) ↔ ((𝑣 − (𝐹‘[𝑣] ∼ )) ∈ ℝ ∧ -1 ≤ (𝑣 − (𝐹‘[𝑣] ∼ )) ∧ (𝑣 − (𝐹‘[𝑣] ∼ )) ≤ 1))
86	64, 77, 82, 85	syl3anbrc 1362	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ (-1[,]1))
87	55, 86	elind 4153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ (ℚ ∩ (-1[,]1)))
88	31, 87	ffvelcdmd 7080	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) ∈ ℕ)
89		oveq1 7417	. . . . . . . 8 ⊢ (𝑠 = 𝑣 → (𝑠 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) = (𝑣 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))))
90	89	eleq1d 2848	. . . . . . 7 ⊢ (𝑠 = 𝑣 → ((𝑠 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹 ↔ (𝑣 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹))
91		f1ocnvfv2 7275	. . . . . . . . . . 11 ⊢ ((𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) ∧ (𝑣 − (𝐹‘[𝑣] ∼ )) ∈ (ℚ ∩ (-1[,]1))) → (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))) = (𝑣 − (𝐹‘[𝑣] ∼ )))
92	27, 87, 91	syl2an2r 697	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))) = (𝑣 − (𝐹‘[𝑣] ∼ )))
93	92	oveq2d 7426	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) = (𝑣 − (𝑣 − (𝐹‘[𝑣] ∼ ))))
94	75, 74	nncand 11578	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝑣 − (𝐹‘[𝑣] ∼ ))) = (𝐹‘[𝑣] ∼ ))
95	93, 94	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) = (𝐹‘[𝑣] ∼ ))
96		fnfvelrn 7075	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝑆 ∧ [𝑣] ∼ ∈ 𝑆) → (𝐹‘[𝑣] ∼ ) ∈ ran 𝐹)
97	1, 44, 96	syl2an2r 697	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ ran 𝐹)
98	95, 97	eqeltrd 2863	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑣 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹)
99	90, 58, 98	elrabd 3652	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹})
100		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑛 = (◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) → (𝐺‘𝑛) = (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))))
101	100	oveq2d 7426	. . . . . . . . . 10 ⊢ (𝑛 = (◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) → (𝑠 − (𝐺‘𝑛)) = (𝑠 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))))
102	101	eleq1d 2848	. . . . . . . . 9 ⊢ (𝑛 = (◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) → ((𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹))
103	102	rabbidv 3423	. . . . . . . 8 ⊢ (𝑛 = (◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹})
104		vitali.6	. . . . . . . 8 ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹})
105		reex 11195	. . . . . . . . 9 ⊢ ℝ ∈ V
106	105	rabex 5309	. . . . . . . 8 ⊢ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹} ∈ V
107	103, 104, 106	fvmpt 6989	. . . . . . 7 ⊢ ((◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) ∈ ℕ → (𝑇‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹})
108	88, 107	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝑇‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) ∈ ran 𝐹})
109	99, 108	eleqtrrd 2866	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ (𝑇‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))))
110		fveq2 6881	. . . . . 6 ⊢ (𝑚 = (◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) → (𝑇‘𝑚) = (𝑇‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ )))))
111	110	eliuni 4962	. . . . 5 ⊢ (((◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))) ∈ ℕ ∧ 𝑣 ∈ (𝑇‘(◡𝐺‘(𝑣 − (𝐹‘[𝑣] ∼ ))))) → 𝑣 ∈ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚))
112	88, 109, 111	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚))
113	112	ex 417	. . 3 ⊢ (𝜑 → (𝑣 ∈ (0[,]1) → 𝑣 ∈ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
114	113	ssrdv 3943	. 2 ⊢ (𝜑 → (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚))
115		eliun 4960	. . . 4 ⊢ (𝑥 ∈ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑇‘𝑚))
116		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐺‘𝑛) = (𝐺‘𝑚))
117	116	oveq2d 7426	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → (𝑠 − (𝐺‘𝑛)) = (𝑠 − (𝐺‘𝑚)))
118	117	eleq1d 2848	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → ((𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹))
119	118	rabbidv 3423	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
120	105	rabex 5309	. . . . . . . . . . . 12 ⊢ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} ∈ V
121	119, 104, 120	fvmpt 6989	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → (𝑇‘𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
122	121	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
123	122	eleq2d 2849	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑥 ∈ (𝑇‘𝑚) ↔ 𝑥 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹}))
124	123	biimpa 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 𝑥 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
125		oveq1 7417	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → (𝑠 − (𝐺‘𝑚)) = (𝑥 − (𝐺‘𝑚)))
126	125	eleq1d 2848	. . . . . . . . 9 ⊢ (𝑠 = 𝑥 → ((𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹 ↔ (𝑥 − (𝐺‘𝑚)) ∈ ran 𝐹))
127	126	elrab 3650	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} ↔ (𝑥 ∈ ℝ ∧ (𝑥 − (𝐺‘𝑚)) ∈ ran 𝐹))
128	124, 127	sylib 221	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝑥 ∈ ℝ ∧ (𝑥 − (𝐺‘𝑚)) ∈ ran 𝐹))
129	128	simpld 499	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 𝑥 ∈ ℝ)
130	83	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → -1 ∈ ℝ)
131		iccssre 13460	. . . . . . . . . 10 ⊢ ((-1 ∈ ℝ ∧ 1 ∈ ℝ) → (-1[,]1) ⊆ ℝ)
132	83, 84, 131	mp2an 704	. . . . . . . . 9 ⊢ (-1[,]1) ⊆ ℝ
133		f1of 6820	. . . . . . . . . . . 12 ⊢ (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
134	27, 133	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
135	134	ffvelcdmda 7079	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ (ℚ ∩ (-1[,]1)))
136	135	elin2d 4158	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ (-1[,]1))
137	132, 136	sselid 3935	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℝ)
138	137	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝐺‘𝑚) ∈ ℝ)
139	136	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝐺‘𝑚) ∈ (-1[,]1))
140	83, 84	elicc2i 13443	. . . . . . . . 9 ⊢ ((𝐺‘𝑚) ∈ (-1[,]1) ↔ ((𝐺‘𝑚) ∈ ℝ ∧ -1 ≤ (𝐺‘𝑚) ∧ (𝐺‘𝑚) ≤ 1))
141	139, 140	sylib 221	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝐺‘𝑚) ∈ ℝ ∧ -1 ≤ (𝐺‘𝑚) ∧ (𝐺‘𝑚) ≤ 1))
142	141	simp2d 1161	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → -1 ≤ (𝐺‘𝑚))
143	26	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ran 𝐹 ⊆ (0[,]1))
144	128	simprd 500	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝑥 − (𝐺‘𝑚)) ∈ ran 𝐹)
145	143, 144	sseldd 3938	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝑥 − (𝐺‘𝑚)) ∈ (0[,]1))
146		elicc01 13497	. . . . . . . . . 10 ⊢ ((𝑥 − (𝐺‘𝑚)) ∈ (0[,]1) ↔ ((𝑥 − (𝐺‘𝑚)) ∈ ℝ ∧ 0 ≤ (𝑥 − (𝐺‘𝑚)) ∧ (𝑥 − (𝐺‘𝑚)) ≤ 1))
147	145, 146	sylib 221	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝑥 − (𝐺‘𝑚)) ∈ ℝ ∧ 0 ≤ (𝑥 − (𝐺‘𝑚)) ∧ (𝑥 − (𝐺‘𝑚)) ≤ 1))
148	147	simp2d 1161	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 0 ≤ (𝑥 − (𝐺‘𝑚)))
149	129, 138	subge0d 11808	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (0 ≤ (𝑥 − (𝐺‘𝑚)) ↔ (𝐺‘𝑚) ≤ 𝑥))
150	148, 149	mpbid 235	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝐺‘𝑚) ≤ 𝑥)
151	130, 138, 129, 142, 150	letrd 11371	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → -1 ≤ 𝑥)
152		peano2re 11387	. . . . . . . 8 ⊢ ((𝐺‘𝑚) ∈ ℝ → ((𝐺‘𝑚) + 1) ∈ ℝ)
153	138, 152	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝐺‘𝑚) + 1) ∈ ℝ)
154		2re 12319	. . . . . . . 8 ⊢ 2 ∈ ℝ
155	154	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 2 ∈ ℝ)
156	147	simp3d 1162	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝑥 − (𝐺‘𝑚)) ≤ 1)
157		1red 11213	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 1 ∈ ℝ)
158	129, 138, 157	lesubadd2d 11817	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝑥 − (𝐺‘𝑚)) ≤ 1 ↔ 𝑥 ≤ ((𝐺‘𝑚) + 1)))
159	156, 158	mpbid 235	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 𝑥 ≤ ((𝐺‘𝑚) + 1))
160	141	simp3d 1162	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → (𝐺‘𝑚) ≤ 1)
161	138, 157, 157, 160	leadd1dd 11832	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝐺‘𝑚) + 1) ≤ (1 + 1))
162		df-2 12307	. . . . . . . 8 ⊢ 2 = (1 + 1)
163	161, 162	breqtrrdi 5153	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → ((𝐺‘𝑚) + 1) ≤ 2)
164	129, 153, 155, 159, 163	letrd 11371	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 𝑥 ≤ 2)
165	83, 154	elicc2i 13443	. . . . . 6 ⊢ (𝑥 ∈ (-1[,]2) ↔ (𝑥 ∈ ℝ ∧ -1 ≤ 𝑥 ∧ 𝑥 ≤ 2))
166	129, 151, 164, 165	syl3anbrc 1362	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ (𝑇‘𝑚)) → 𝑥 ∈ (-1[,]2))
167	166	rexlimdva2 3168	. . . 4 ⊢ (𝜑 → (∃𝑚 ∈ ℕ 𝑥 ∈ (𝑇‘𝑚) → 𝑥 ∈ (-1[,]2)))
168	115, 167	biimtrid 245	. . 3 ⊢ (𝜑 → (𝑥 ∈ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) → 𝑥 ∈ (-1[,]2)))
169	168	ssrdv 3943	. 2 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2))
170	26, 114, 169	3jca 1146	1 ⊢ (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∧ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  {crab 3416  Vcvv 3455   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4562  ∪ ciun 4956   class class class wbr 5109  {copab 5173   ↦ cmpt 5192  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   Fn wfn 6531  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410   Er wer 8687  [cec 8688   / cqs 8689  ℝcr 11103  0cc0 11104  1c1 11105   + caddc 11107   ≤ cle 11248   − cmin 11445  -cneg 11446  ℕcn 12237  2c2 12299  ℚcq 12976  [,]cicc 13379  volcvol 25631

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-ec 8692  df-qs 8696  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448  df-div 11876  df-nn 12238  df-2 12307  df-n0 12509  df-z 12596  df-q 12977  df-icc 13383

This theorem is used by:  vitalilem3  25778  vitalilem4  25779  vitalilem5  25780