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Theorem vitalilem3 23379

Description: Lemma for vitali 23382. (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
vitalilem3	⊢ (𝜑 → Disj 𝑚 ∈ ℕ (𝑇‘𝑚))

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 vitalilem3 Dummy variables 𝑘 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprlr 803	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑤 ∈ (𝑇‘𝑚))
2		simprll 802	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑚 ∈ ℕ)
3		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝐺‘𝑛) = (𝐺‘𝑚))
4	3	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (𝑠 − (𝐺‘𝑛)) = (𝑠 − (𝐺‘𝑚)))
5	4	eleq1d 2686	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ((𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹))
6	5	rabbidv 3189	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
7		vitali.6	. . . . . . . . . . . . . 14 ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹})
8		reex 10027	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
9	8	rabex 4813	. . . . . . . . . . . . . 14 ⊢ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} ∈ V
10	6, 7, 9	fvmpt 6282	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → (𝑇‘𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
11	2, 10	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑇‘𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
12	1, 11	eleqtrd 2703	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
13		oveq1 6657	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑤 → (𝑠 − (𝐺‘𝑚)) = (𝑤 − (𝐺‘𝑚)))
14	13	eleq1d 2686	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑤 → ((𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹 ↔ (𝑤 − (𝐺‘𝑚)) ∈ ran 𝐹))
15	14	elrab 3363	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} ↔ (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺‘𝑚)) ∈ ran 𝐹))
16	12, 15	sylib 208	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺‘𝑚)) ∈ ran 𝐹))
17	16	simpld 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑤 ∈ ℝ)
18	17	recnd 10068	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑤 ∈ ℂ)
19		vitali.5	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
20		f1of 6137	. . . . . . . . . . . . 13 ⊢ (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
21	19, 20	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
22		inss1 3833	. . . . . . . . . . . 12 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℚ
23		fss 6056	. . . . . . . . . . . 12 ⊢ ((𝐺:ℕ⟶(ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ⊆ ℚ) → 𝐺:ℕ⟶ℚ)
24	21, 22, 23	sylancl 694	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ⟶ℚ)
25	24	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝐺:ℕ⟶ℚ)
26	25, 2	ffvelrnd 6360	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝐺‘𝑚) ∈ ℚ)
27		qcn 11802	. . . . . . . . 9 ⊢ ((𝐺‘𝑚) ∈ ℚ → (𝐺‘𝑚) ∈ ℂ)
28	26, 27	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝐺‘𝑚) ∈ ℂ)
29		simprrl 804	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑘 ∈ ℕ)
30	25, 29	ffvelrnd 6360	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝐺‘𝑘) ∈ ℚ)
31		qcn 11802	. . . . . . . . 9 ⊢ ((𝐺‘𝑘) ∈ ℚ → (𝐺‘𝑘) ∈ ℂ)
32	30, 31	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝐺‘𝑘) ∈ ℂ)
33		vitali.1	. . . . . . . . . . . . 13 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)}
34	33	vitalilem1 23376	. . . . . . . . . . . 12 ⊢ ∼ Er (0[,]1)
35	34	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → ∼ Er (0[,]1))
36		vitali.2	. . . . . . . . . . . . . . . . 17 ⊢ 𝑆 = ((0[,]1) / ∼ )
37		vitali.3	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 Fn 𝑆)
38		vitali.4	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
39		vitali.7	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
40	33, 36, 37, 38, 19, 7, 39	vitalilem2 23378	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∧ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2)))
41	40	simp1d 1073	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ⊆ (0[,]1))
42	41	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → ran 𝐹 ⊆ (0[,]1))
43	16	simprd 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑤 − (𝐺‘𝑚)) ∈ ran 𝐹)
44	42, 43	sseldd 3604	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑤 − (𝐺‘𝑚)) ∈ (0[,]1))
45		simprrr 805	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑤 ∈ (𝑇‘𝑘))
46		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
47	46	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → (𝑠 − (𝐺‘𝑛)) = (𝑠 − (𝐺‘𝑘)))
48	47	eleq1d 2686	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑘 → ((𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺‘𝑘)) ∈ ran 𝐹))
49	48	rabbidv 3189	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑘)) ∈ ran 𝐹})
50	8	rabex 4813	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑘)) ∈ ran 𝐹} ∈ V
51	49, 7, 50	fvmpt 6282	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → (𝑇‘𝑘) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑘)) ∈ ran 𝐹})
52	29, 51	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑇‘𝑘) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑘)) ∈ ran 𝐹})
53	45, 52	eleqtrd 2703	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑘)) ∈ ran 𝐹})
54		oveq1 6657	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = 𝑤 → (𝑠 − (𝐺‘𝑘)) = (𝑤 − (𝐺‘𝑘)))
55	54	eleq1d 2686	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = 𝑤 → ((𝑠 − (𝐺‘𝑘)) ∈ ran 𝐹 ↔ (𝑤 − (𝐺‘𝑘)) ∈ ran 𝐹))
56	55	elrab 3363	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑘)) ∈ ran 𝐹} ↔ (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺‘𝑘)) ∈ ran 𝐹))
57	53, 56	sylib 208	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺‘𝑘)) ∈ ran 𝐹))
58	57	simprd 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑤 − (𝐺‘𝑘)) ∈ ran 𝐹)
59	42, 58	sseldd 3604	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑤 − (𝐺‘𝑘)) ∈ (0[,]1))
60	44, 59	jca 554	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → ((𝑤 − (𝐺‘𝑚)) ∈ (0[,]1) ∧ (𝑤 − (𝐺‘𝑘)) ∈ (0[,]1)))
61	18, 28, 32	nnncan1d 10426	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → ((𝑤 − (𝐺‘𝑚)) − (𝑤 − (𝐺‘𝑘))) = ((𝐺‘𝑘) − (𝐺‘𝑚)))
62		qsubcl 11807	. . . . . . . . . . . . . 14 ⊢ (((𝐺‘𝑘) ∈ ℚ ∧ (𝐺‘𝑚) ∈ ℚ) → ((𝐺‘𝑘) − (𝐺‘𝑚)) ∈ ℚ)
63	30, 26, 62	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → ((𝐺‘𝑘) − (𝐺‘𝑚)) ∈ ℚ)
64	61, 63	eqeltrd 2701	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → ((𝑤 − (𝐺‘𝑚)) − (𝑤 − (𝐺‘𝑘))) ∈ ℚ)
65		oveq12 6659	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = (𝑤 − (𝐺‘𝑚)) ∧ 𝑦 = (𝑤 − (𝐺‘𝑘))) → (𝑥 − 𝑦) = ((𝑤 − (𝐺‘𝑚)) − (𝑤 − (𝐺‘𝑘))))
66	65	eleq1d 2686	. . . . . . . . . . . . 13 ⊢ ((𝑥 = (𝑤 − (𝐺‘𝑚)) ∧ 𝑦 = (𝑤 − (𝐺‘𝑘))) → ((𝑥 − 𝑦) ∈ ℚ ↔ ((𝑤 − (𝐺‘𝑚)) − (𝑤 − (𝐺‘𝑘))) ∈ ℚ))
67	66, 33	brab2a 5194	. . . . . . . . . . . 12 ⊢ ((𝑤 − (𝐺‘𝑚)) ∼ (𝑤 − (𝐺‘𝑘)) ↔ (((𝑤 − (𝐺‘𝑚)) ∈ (0[,]1) ∧ (𝑤 − (𝐺‘𝑘)) ∈ (0[,]1)) ∧ ((𝑤 − (𝐺‘𝑚)) − (𝑤 − (𝐺‘𝑘))) ∈ ℚ))
68	60, 64, 67	sylanbrc 698	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑤 − (𝐺‘𝑚)) ∼ (𝑤 − (𝐺‘𝑘)))
69	35, 68	erthi 7793	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → [(𝑤 − (𝐺‘𝑚))] ∼ = [(𝑤 − (𝐺‘𝑘))] ∼ )
70	69	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝐹‘[(𝑤 − (𝐺‘𝑚))] ∼ ) = (𝐹‘[(𝑤 − (𝐺‘𝑘))] ∼ ))
71		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ ([𝑣] ∼ = 𝑤 → (𝐹‘[𝑣] ∼ ) = (𝐹‘𝑤))
72	71	eceq1d 7783	. . . . . . . . . . . . . . . 16 ⊢ ([𝑣] ∼ = 𝑤 → [(𝐹‘[𝑣] ∼ )] ∼ = [(𝐹‘𝑤)] ∼ )
73	72	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ ([𝑣] ∼ = 𝑤 → (𝐹‘[(𝐹‘[𝑣] ∼ )] ∼ ) = (𝐹‘[(𝐹‘𝑤)] ∼ ))
74	73, 71	eqeq12d 2637	. . . . . . . . . . . . . 14 ⊢ ([𝑣] ∼ = 𝑤 → ((𝐹‘[(𝐹‘[𝑣] ∼ )] ∼ ) = (𝐹‘[𝑣] ∼ ) ↔ (𝐹‘[(𝐹‘𝑤)] ∼ ) = (𝐹‘𝑤)))
75	34	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ∼ Er (0[,]1))
76		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0[,]1) ∈ V
77		erex 7766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ( ∼ Er (0[,]1) → ((0[,]1) ∈ V → ∼ ∈ V))
78	34, 76, 77	mp2 9	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ∼ ∈ V
79	78	ecelqsi 7803	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ (0[,]1) → [𝑣] ∼ ∈ ((0[,]1) / ∼ ))
80	79, 36	syl6eleqr 2712	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ (0[,]1) → [𝑣] ∼ ∈ 𝑆)
81	80	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [𝑣] ∼ ∈ 𝑆)
82	38	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
83		simpr 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ (0[,]1))
84		erdm 7752	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ( ∼ Er (0[,]1) → dom ∼ = (0[,]1))
85	34, 84	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom ∼ = (0[,]1)
86	85	eleq2i 2693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ dom ∼ ↔ 𝑣 ∈ (0[,]1))
87		ecdmn0 7789	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ dom ∼ ↔ [𝑣] ∼ ≠ ∅)
88	86, 87	bitr3i 266	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ (0[,]1) ↔ [𝑣] ∼ ≠ ∅)
89	83, 88	sylib 208	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [𝑣] ∼ ≠ ∅)
90		neeq1 2856	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = [𝑣] ∼ → (𝑧 ≠ ∅ ↔ [𝑣] ∼ ≠ ∅))
91		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = [𝑣] ∼ → (𝐹‘𝑧) = (𝐹‘[𝑣] ∼ ))
92		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = [𝑣] ∼ → 𝑧 = [𝑣] ∼ )
93	91, 92	eleq12d 2695	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = [𝑣] ∼ → ((𝐹‘𝑧) ∈ 𝑧 ↔ (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ))
94	90, 93	imbi12d 334	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = [𝑣] ∼ → ((𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧) ↔ ([𝑣] ∼ ≠ ∅ → (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ )))
95	94	rspcv 3305	. . . . . . . . . . . . . . . . . . 19 ⊢ ([𝑣] ∼ ∈ 𝑆 → (∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧) → ([𝑣] ∼ ≠ ∅ → (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ )))
96	81, 82, 89, 95	syl3c 66	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ )
97		fvex 6201	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹‘[𝑣] ∼ ) ∈ V
98		vex 3203	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑣 ∈ V
99	97, 98	elec 7786	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ↔ 𝑣 ∼ (𝐹‘[𝑣] ∼ ))
100	96, 99	sylib 208	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∼ (𝐹‘[𝑣] ∼ ))
101	75, 100	erthi 7793	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [𝑣] ∼ = [(𝐹‘[𝑣] ∼ )] ∼ )
102	101	eqcomd 2628	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [(𝐹‘[𝑣] ∼ )] ∼ = [𝑣] ∼ )
103	102	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[(𝐹‘[𝑣] ∼ )] ∼ ) = (𝐹‘[𝑣] ∼ ))
104	36, 74, 103	ectocld 7814	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑆) → (𝐹‘[(𝐹‘𝑤)] ∼ ) = (𝐹‘𝑤))
105	104	ralrimiva 2966	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑤 ∈ 𝑆 (𝐹‘[(𝐹‘𝑤)] ∼ ) = (𝐹‘𝑤))
106		eceq1 7782	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐹‘𝑤) → [𝑧] ∼ = [(𝐹‘𝑤)] ∼ )
107	106	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑤) → (𝐹‘[𝑧] ∼ ) = (𝐹‘[(𝐹‘𝑤)] ∼ ))
108		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑤) → 𝑧 = (𝐹‘𝑤))
109	107, 108	eqeq12d 2637	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐹‘𝑤) → ((𝐹‘[𝑧] ∼ ) = 𝑧 ↔ (𝐹‘[(𝐹‘𝑤)] ∼ ) = (𝐹‘𝑤)))
110	109	ralrn 6362	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝑆 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ∼ ) = 𝑧 ↔ ∀𝑤 ∈ 𝑆 (𝐹‘[(𝐹‘𝑤)] ∼ ) = (𝐹‘𝑤)))
111	37, 110	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ∼ ) = 𝑧 ↔ ∀𝑤 ∈ 𝑆 (𝐹‘[(𝐹‘𝑤)] ∼ ) = (𝐹‘𝑤)))
112	105, 111	mpbird 247	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ∼ ) = 𝑧)
113	112	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → ∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ∼ ) = 𝑧)
114		eceq1 7782	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑤 − (𝐺‘𝑚)) → [𝑧] ∼ = [(𝑤 − (𝐺‘𝑚))] ∼ )
115	114	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑤 − (𝐺‘𝑚)) → (𝐹‘[𝑧] ∼ ) = (𝐹‘[(𝑤 − (𝐺‘𝑚))] ∼ ))
116		id 22	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑤 − (𝐺‘𝑚)) → 𝑧 = (𝑤 − (𝐺‘𝑚)))
117	115, 116	eqeq12d 2637	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑤 − (𝐺‘𝑚)) → ((𝐹‘[𝑧] ∼ ) = 𝑧 ↔ (𝐹‘[(𝑤 − (𝐺‘𝑚))] ∼ ) = (𝑤 − (𝐺‘𝑚))))
118	117	rspcv 3305	. . . . . . . . . 10 ⊢ ((𝑤 − (𝐺‘𝑚)) ∈ ran 𝐹 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ∼ ) = 𝑧 → (𝐹‘[(𝑤 − (𝐺‘𝑚))] ∼ ) = (𝑤 − (𝐺‘𝑚))))
119	43, 113, 118	sylc 65	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝐹‘[(𝑤 − (𝐺‘𝑚))] ∼ ) = (𝑤 − (𝐺‘𝑚)))
120		eceq1 7782	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑤 − (𝐺‘𝑘)) → [𝑧] ∼ = [(𝑤 − (𝐺‘𝑘))] ∼ )
121	120	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑤 − (𝐺‘𝑘)) → (𝐹‘[𝑧] ∼ ) = (𝐹‘[(𝑤 − (𝐺‘𝑘))] ∼ ))
122		id 22	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑤 − (𝐺‘𝑘)) → 𝑧 = (𝑤 − (𝐺‘𝑘)))
123	121, 122	eqeq12d 2637	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑤 − (𝐺‘𝑘)) → ((𝐹‘[𝑧] ∼ ) = 𝑧 ↔ (𝐹‘[(𝑤 − (𝐺‘𝑘))] ∼ ) = (𝑤 − (𝐺‘𝑘))))
124	123	rspcv 3305	. . . . . . . . . 10 ⊢ ((𝑤 − (𝐺‘𝑘)) ∈ ran 𝐹 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ∼ ) = 𝑧 → (𝐹‘[(𝑤 − (𝐺‘𝑘))] ∼ ) = (𝑤 − (𝐺‘𝑘))))
125	58, 113, 124	sylc 65	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝐹‘[(𝑤 − (𝐺‘𝑘))] ∼ ) = (𝑤 − (𝐺‘𝑘)))
126	70, 119, 125	3eqtr3d 2664	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑤 − (𝐺‘𝑚)) = (𝑤 − (𝐺‘𝑘)))
127	18, 28, 32, 126	subcand 10433	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝐺‘𝑚) = (𝐺‘𝑘))
128	19	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
129		f1of1 6136	. . . . . . . . 9 ⊢ (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)))
130	128, 129	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)))
131		f1fveq 6519	. . . . . . . 8 ⊢ ((𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)) ∧ (𝑚 ∈ ℕ ∧ 𝑘 ∈ ℕ)) → ((𝐺‘𝑚) = (𝐺‘𝑘) ↔ 𝑚 = 𝑘))
132	130, 2, 29, 131	syl12anc 1324	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → ((𝐺‘𝑚) = (𝐺‘𝑘) ↔ 𝑚 = 𝑘))
133	127, 132	mpbid 222	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑚 = 𝑘)
134	133	ex 450	. . . . 5 ⊢ (𝜑 → (((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘))) → 𝑚 = 𝑘))
135	134	alrimivv 1856	. . . 4 ⊢ (𝜑 → ∀𝑚∀𝑘(((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘))) → 𝑚 = 𝑘))
136		eleq1 2689	. . . . . 6 ⊢ (𝑚 = 𝑘 → (𝑚 ∈ ℕ ↔ 𝑘 ∈ ℕ))
137		fveq2 6191	. . . . . . 7 ⊢ (𝑚 = 𝑘 → (𝑇‘𝑚) = (𝑇‘𝑘))
138	137	eleq2d 2687	. . . . . 6 ⊢ (𝑚 = 𝑘 → (𝑤 ∈ (𝑇‘𝑚) ↔ 𝑤 ∈ (𝑇‘𝑘)))
139	136, 138	anbi12d 747	. . . . 5 ⊢ (𝑚 = 𝑘 → ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ↔ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘))))
140	139	mo4 2517	. . . 4 ⊢ (∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ↔ ∀𝑚∀𝑘(((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘))) → 𝑚 = 𝑘))
141	135, 140	sylibr 224	. . 3 ⊢ (𝜑 → ∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)))
142	141	alrimiv 1855	. 2 ⊢ (𝜑 → ∀𝑤∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)))
143		dfdisj2 4622	. 2 ⊢ (Disj 𝑚 ∈ ℕ (𝑇‘𝑚) ↔ ∀𝑤∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)))
144	142, 143	sylibr 224	1 ⊢ (𝜑 → Disj 𝑚 ∈ ℕ (𝑇‘𝑚))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 = wceq 1483 ∈ wcel 1990 ∃*wmo 2471 ≠ wne 2794 ∀wral 2912 {crab 2916 Vcvv 3200 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 ∪ ciun 4520 Disj wdisj 4620 class class class wbr 4653 {copab 4712 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 Fn wfn 5883 ⟶wf 5884 –1-1→wf1 5885 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 Er wer 7739 [cec 7740 / cqs 7741 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 − cmin 10266 -cneg 10267 ℕcn 11020 2c2 11070 ℚcq 11788 [,]cicc 12178 volcvol 23232

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-ec 7744 df-qs 7748 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-q 11789 df-icc 12182

This theorem is referenced by: vitalilem4 23380

W3C validator