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Theorem vitalilem3 25563

Description: Lemma for vitali 25566. (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 779	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑤 ∈ (𝑇‘𝑚))
2		simprll 778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑚 ∈ ℕ)
3		fveq2 6876	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝐺‘𝑛) = (𝐺‘𝑚))
4	3	oveq2d 7421	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (𝑠 − (𝐺‘𝑛)) = (𝑠 − (𝐺‘𝑚)))
5	4	eleq1d 2819	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ((𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹))
6	5	rabbidv 3423	. . . . . . . . . . . . . 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 11220	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
9	8	rabex 5309	. . . . . . . . . . . . . 14 ⊢ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} ∈ V
10	6, 7, 9	fvmpt 6986	. . . . . . . . . . . . 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 2836	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
13		oveq1 7412	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑤 → (𝑠 − (𝐺‘𝑚)) = (𝑤 − (𝐺‘𝑚)))
14	13	eleq1d 2819	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑤 → ((𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹 ↔ (𝑤 − (𝐺‘𝑚)) ∈ ran 𝐹))
15	14	elrab 3671	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} ↔ (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺‘𝑚)) ∈ ran 𝐹))
16	12, 15	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺‘𝑚)) ∈ ran 𝐹))
17	16	simpld 494	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑤 ∈ ℝ)
18	17	recnd 11263	. . . . . . . 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 6818	. . . . . . . . . . . . 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 4212	. . . . . . . . . . . 12 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℚ
23		fss 6722	. . . . . . . . . . . 12 ⊢ ((𝐺:ℕ⟶(ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ⊆ ℚ) → 𝐺:ℕ⟶ℚ)
24	21, 22, 23	sylancl 586	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ⟶ℚ)
25	24	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝐺:ℕ⟶ℚ)
26	25, 2	ffvelcdmd 7075	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝐺‘𝑚) ∈ ℚ)
27		qcn 12979	. . . . . . . . 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 780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑘 ∈ ℕ)
30	25, 29	ffvelcdmd 7075	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝐺‘𝑘) ∈ ℚ)
31		qcn 12979	. . . . . . . . 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 25561	. . . . . . . . . . . 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	. . . . . . . . . . . . . . . 16 ⊢ 𝑆 = ((0[,]1) / ∼ )
37		vitali.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 Fn 𝑆)
38		vitali.4	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
39		vitali.7	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
40	33, 36, 37, 38, 19, 7, 39	vitalilem2 25562	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∧ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2)))
41	40	simp1d 1142	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐹 ⊆ (0[,]1))
42	41	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → ran 𝐹 ⊆ (0[,]1))
43	16	simprd 495	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑤 − (𝐺‘𝑚)) ∈ ran 𝐹)
44	42, 43	sseldd 3959	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑤 − (𝐺‘𝑚)) ∈ (0[,]1))
45		simprrr 781	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑤 ∈ (𝑇‘𝑘))
46		fveq2 6876	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
47	46	oveq2d 7421	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑘 → (𝑠 − (𝐺‘𝑛)) = (𝑠 − (𝐺‘𝑘)))
48	47	eleq1d 2819	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → ((𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺‘𝑘)) ∈ ran 𝐹))
49	48	rabbidv 3423	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑘)) ∈ ran 𝐹})
50	8	rabex 5309	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑘)) ∈ ran 𝐹} ∈ V
51	49, 7, 50	fvmpt 6986	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → (𝑇‘𝑘) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑘)) ∈ ran 𝐹})
52	29, 51	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑇‘𝑘) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑘)) ∈ ran 𝐹})
53	45, 52	eleqtrd 2836	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑘)) ∈ ran 𝐹})
54		oveq1 7412	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = 𝑤 → (𝑠 − (𝐺‘𝑘)) = (𝑤 − (𝐺‘𝑘)))
55	54	eleq1d 2819	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑤 → ((𝑠 − (𝐺‘𝑘)) ∈ ran 𝐹 ↔ (𝑤 − (𝐺‘𝑘)) ∈ ran 𝐹))
56	55	elrab 3671	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑘)) ∈ ran 𝐹} ↔ (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺‘𝑘)) ∈ ran 𝐹))
57	53, 56	sylib 218	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑤 ∈ ℝ ∧ (𝑤 − (𝐺‘𝑘)) ∈ ran 𝐹))
58	57	simprd 495	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑤 − (𝐺‘𝑘)) ∈ ran 𝐹)
59	42, 58	sseldd 3959	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑤 − (𝐺‘𝑘)) ∈ (0[,]1))
60	18, 28, 32	nnncan1d 11628	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → ((𝑤 − (𝐺‘𝑚)) − (𝑤 − (𝐺‘𝑘))) = ((𝐺‘𝑘) − (𝐺‘𝑚)))
61		qsubcl 12984	. . . . . . . . . . . . . 14 ⊢ (((𝐺‘𝑘) ∈ ℚ ∧ (𝐺‘𝑚) ∈ ℚ) → ((𝐺‘𝑘) − (𝐺‘𝑚)) ∈ ℚ)
62	30, 26, 61	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → ((𝐺‘𝑘) − (𝐺‘𝑚)) ∈ ℚ)
63	60, 62	eqeltrd 2834	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → ((𝑤 − (𝐺‘𝑚)) − (𝑤 − (𝐺‘𝑘))) ∈ ℚ)
64		oveq12 7414	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = (𝑤 − (𝐺‘𝑚)) ∧ 𝑦 = (𝑤 − (𝐺‘𝑘))) → (𝑥 − 𝑦) = ((𝑤 − (𝐺‘𝑚)) − (𝑤 − (𝐺‘𝑘))))
65	64	eleq1d 2819	. . . . . . . . . . . . 13 ⊢ ((𝑥 = (𝑤 − (𝐺‘𝑚)) ∧ 𝑦 = (𝑤 − (𝐺‘𝑘))) → ((𝑥 − 𝑦) ∈ ℚ ↔ ((𝑤 − (𝐺‘𝑚)) − (𝑤 − (𝐺‘𝑘))) ∈ ℚ))
66	65, 33	brab2a 5748	. . . . . . . . . . . 12 ⊢ ((𝑤 − (𝐺‘𝑚)) ∼ (𝑤 − (𝐺‘𝑘)) ↔ (((𝑤 − (𝐺‘𝑚)) ∈ (0[,]1) ∧ (𝑤 − (𝐺‘𝑘)) ∈ (0[,]1)) ∧ ((𝑤 − (𝐺‘𝑚)) − (𝑤 − (𝐺‘𝑘))) ∈ ℚ))
67	44, 59, 63, 66	syl21anbrc 1345	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑤 − (𝐺‘𝑚)) ∼ (𝑤 − (𝐺‘𝑘)))
68	35, 67	erthi 8772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → [(𝑤 − (𝐺‘𝑚))] ∼ = [(𝑤 − (𝐺‘𝑘))] ∼ )
69	68	fveq2d 6880	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝐹‘[(𝑤 − (𝐺‘𝑚))] ∼ ) = (𝐹‘[(𝑤 − (𝐺‘𝑘))] ∼ ))
70		eceq1 8758	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑤 − (𝐺‘𝑚)) → [𝑧] ∼ = [(𝑤 − (𝐺‘𝑚))] ∼ )
71	70	fveq2d 6880	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑤 − (𝐺‘𝑚)) → (𝐹‘[𝑧] ∼ ) = (𝐹‘[(𝑤 − (𝐺‘𝑚))] ∼ ))
72		id 22	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑤 − (𝐺‘𝑚)) → 𝑧 = (𝑤 − (𝐺‘𝑚)))
73	71, 72	eqeq12d 2751	. . . . . . . . . 10 ⊢ (𝑧 = (𝑤 − (𝐺‘𝑚)) → ((𝐹‘[𝑧] ∼ ) = 𝑧 ↔ (𝐹‘[(𝑤 − (𝐺‘𝑚))] ∼ ) = (𝑤 − (𝐺‘𝑚))))
74		fveq2 6876	. . . . . . . . . . . . . . . . 17 ⊢ ([𝑣] ∼ = 𝑤 → (𝐹‘[𝑣] ∼ ) = (𝐹‘𝑤))
75	74	eceq1d 8759	. . . . . . . . . . . . . . . 16 ⊢ ([𝑣] ∼ = 𝑤 → [(𝐹‘[𝑣] ∼ )] ∼ = [(𝐹‘𝑤)] ∼ )
76	75	fveq2d 6880	. . . . . . . . . . . . . . 15 ⊢ ([𝑣] ∼ = 𝑤 → (𝐹‘[(𝐹‘[𝑣] ∼ )] ∼ ) = (𝐹‘[(𝐹‘𝑤)] ∼ ))
77	76, 74	eqeq12d 2751	. . . . . . . . . . . . . 14 ⊢ ([𝑣] ∼ = 𝑤 → ((𝐹‘[(𝐹‘[𝑣] ∼ )] ∼ ) = (𝐹‘[𝑣] ∼ ) ↔ (𝐹‘[(𝐹‘𝑤)] ∼ ) = (𝐹‘𝑤)))
78	34	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ∼ Er (0[,]1))
79		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∈ (0[,]1))
80		erdm 8729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ( ∼ Er (0[,]1) → dom ∼ = (0[,]1))
81	34, 80	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom ∼ = (0[,]1)
82	81	eleq2i 2826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ dom ∼ ↔ 𝑣 ∈ (0[,]1))
83		ecdmn0 8768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ dom ∼ ↔ [𝑣] ∼ ≠ ∅)
84	82, 83	bitr3i 277	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ (0[,]1) ↔ [𝑣] ∼ ≠ ∅)
85	79, 84	sylib 218	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [𝑣] ∼ ≠ ∅)
86		neeq1 2994	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = [𝑣] ∼ → (𝑧 ≠ ∅ ↔ [𝑣] ∼ ≠ ∅))
87		fveq2 6876	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = [𝑣] ∼ → (𝐹‘𝑧) = (𝐹‘[𝑣] ∼ ))
88		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = [𝑣] ∼ → 𝑧 = [𝑣] ∼ )
89	87, 88	eleq12d 2828	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = [𝑣] ∼ → ((𝐹‘𝑧) ∈ 𝑧 ↔ (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ))
90	86, 89	imbi12d 344	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = [𝑣] ∼ → ((𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧) ↔ ([𝑣] ∼ ≠ ∅ → (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ )))
91	38	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
92		ovex 7438	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0[,]1) ∈ V
93		erex 8743	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ( ∼ Er (0[,]1) → ((0[,]1) ∈ V → ∼ ∈ V))
94	34, 92, 93	mp2 9	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∼ ∈ V
95	94	ecelqsi 8787	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ (0[,]1) → [𝑣] ∼ ∈ ((0[,]1) / ∼ ))
96	95, 36	eleqtrrdi 2845	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ (0[,]1) → [𝑣] ∼ ∈ 𝑆)
97	96	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [𝑣] ∼ ∈ 𝑆)
98	90, 91, 97	rspcdva 3602	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → ([𝑣] ∼ ≠ ∅ → (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ))
99	85, 98	mpd 15	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ )
100		fvex 6889	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹‘[𝑣] ∼ ) ∈ V
101		vex 3463	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑣 ∈ V
102	100, 101	elec 8765	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘[𝑣] ∼ ) ∈ [𝑣] ∼ ↔ 𝑣 ∼ (𝐹‘[𝑣] ∼ ))
103	99, 102	sylib 218	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → 𝑣 ∼ (𝐹‘[𝑣] ∼ ))
104	78, 103	erthi 8772	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [𝑣] ∼ = [(𝐹‘[𝑣] ∼ )] ∼ )
105	104	eqcomd 2741	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → [(𝐹‘[𝑣] ∼ )] ∼ = [𝑣] ∼ )
106	105	fveq2d 6880	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘[(𝐹‘[𝑣] ∼ )] ∼ ) = (𝐹‘[𝑣] ∼ ))
107	36, 77, 106	ectocld 8798	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑆) → (𝐹‘[(𝐹‘𝑤)] ∼ ) = (𝐹‘𝑤))
108	107	ralrimiva 3132	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑤 ∈ 𝑆 (𝐹‘[(𝐹‘𝑤)] ∼ ) = (𝐹‘𝑤))
109		eceq1 8758	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐹‘𝑤) → [𝑧] ∼ = [(𝐹‘𝑤)] ∼ )
110	109	fveq2d 6880	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑤) → (𝐹‘[𝑧] ∼ ) = (𝐹‘[(𝐹‘𝑤)] ∼ ))
111		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑤) → 𝑧 = (𝐹‘𝑤))
112	110, 111	eqeq12d 2751	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐹‘𝑤) → ((𝐹‘[𝑧] ∼ ) = 𝑧 ↔ (𝐹‘[(𝐹‘𝑤)] ∼ ) = (𝐹‘𝑤)))
113	112	ralrn 7078	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝑆 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ∼ ) = 𝑧 ↔ ∀𝑤 ∈ 𝑆 (𝐹‘[(𝐹‘𝑤)] ∼ ) = (𝐹‘𝑤)))
114	37, 113	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ∼ ) = 𝑧 ↔ ∀𝑤 ∈ 𝑆 (𝐹‘[(𝐹‘𝑤)] ∼ ) = (𝐹‘𝑤)))
115	108, 114	mpbird 257	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ∼ ) = 𝑧)
116	115	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → ∀𝑧 ∈ ran 𝐹(𝐹‘[𝑧] ∼ ) = 𝑧)
117	73, 116, 43	rspcdva 3602	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝐹‘[(𝑤 − (𝐺‘𝑚))] ∼ ) = (𝑤 − (𝐺‘𝑚)))
118		eceq1 8758	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑤 − (𝐺‘𝑘)) → [𝑧] ∼ = [(𝑤 − (𝐺‘𝑘))] ∼ )
119	118	fveq2d 6880	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑤 − (𝐺‘𝑘)) → (𝐹‘[𝑧] ∼ ) = (𝐹‘[(𝑤 − (𝐺‘𝑘))] ∼ ))
120		id 22	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑤 − (𝐺‘𝑘)) → 𝑧 = (𝑤 − (𝐺‘𝑘)))
121	119, 120	eqeq12d 2751	. . . . . . . . . 10 ⊢ (𝑧 = (𝑤 − (𝐺‘𝑘)) → ((𝐹‘[𝑧] ∼ ) = 𝑧 ↔ (𝐹‘[(𝑤 − (𝐺‘𝑘))] ∼ ) = (𝑤 − (𝐺‘𝑘))))
122	121, 116, 58	rspcdva 3602	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝐹‘[(𝑤 − (𝐺‘𝑘))] ∼ ) = (𝑤 − (𝐺‘𝑘)))
123	69, 117, 122	3eqtr3d 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝑤 − (𝐺‘𝑚)) = (𝑤 − (𝐺‘𝑘)))
124	18, 28, 32, 123	subcand 11635	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → (𝐺‘𝑚) = (𝐺‘𝑘))
125	19	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
126		f1of1 6817	. . . . . . . . 9 ⊢ (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)))
127	125, 126	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)))
128		f1fveq 7255	. . . . . . . 8 ⊢ ((𝐺:ℕ–1-1→(ℚ ∩ (-1[,]1)) ∧ (𝑚 ∈ ℕ ∧ 𝑘 ∈ ℕ)) → ((𝐺‘𝑚) = (𝐺‘𝑘) ↔ 𝑚 = 𝑘))
129	127, 2, 29, 128	syl12anc 836	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → ((𝐺‘𝑚) = (𝐺‘𝑘) ↔ 𝑚 = 𝑘))
130	124, 129	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘)))) → 𝑚 = 𝑘)
131	130	ex 412	. . . . 5 ⊢ (𝜑 → (((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘))) → 𝑚 = 𝑘))
132	131	alrimivv 1928	. . . 4 ⊢ (𝜑 → ∀𝑚∀𝑘(((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘))) → 𝑚 = 𝑘))
133		eleq1w 2817	. . . . . 6 ⊢ (𝑚 = 𝑘 → (𝑚 ∈ ℕ ↔ 𝑘 ∈ ℕ))
134		fveq2 6876	. . . . . . 7 ⊢ (𝑚 = 𝑘 → (𝑇‘𝑚) = (𝑇‘𝑘))
135	134	eleq2d 2820	. . . . . 6 ⊢ (𝑚 = 𝑘 → (𝑤 ∈ (𝑇‘𝑚) ↔ 𝑤 ∈ (𝑇‘𝑘)))
136	133, 135	anbi12d 632	. . . . 5 ⊢ (𝑚 = 𝑘 → ((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ↔ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘))))
137	136	mo4 2565	. . . 4 ⊢ (∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ↔ ∀𝑚∀𝑘(((𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)) ∧ (𝑘 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑘))) → 𝑚 = 𝑘))
138	132, 137	sylibr 234	. . 3 ⊢ (𝜑 → ∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)))
139	138	alrimiv 1927	. 2 ⊢ (𝜑 → ∀𝑤∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)))
140		dfdisj2 5088	. 2 ⊢ (Disj 𝑚 ∈ ℕ (𝑇‘𝑚) ↔ ∀𝑤∃*𝑚(𝑚 ∈ ℕ ∧ 𝑤 ∈ (𝑇‘𝑚)))
141	139, 140	sylibr 234	1 ⊢ (𝜑 → Disj 𝑚 ∈ ℕ (𝑇‘𝑚))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∃*wmo 2537 ≠ wne 2932 ∀wral 3051 {crab 3415 Vcvv 3459 ∖ cdif 3923 ∩ cin 3925 ⊆ wss 3926 ∅c0 4308 𝒫 cpw 4575 ∪ ciun 4967 Disj wdisj 5086 class class class wbr 5119 {copab 5181 ↦ cmpt 5201 dom cdm 5654 ran crn 5655 Fn wfn 6526 ⟶wf 6527 –1-1→wf1 6528 –1-1-onto→wf1o 6530 ‘cfv 6531 (class class class)co 7405 Er wer 8716 [cec 8717 / cqs 8718 ℂcc 11127 ℝcr 11128 0cc0 11129 1c1 11130 − cmin 11466 -cneg 11467 ℕcn 12240 2c2 12295 ℚcq 12964 [,]cicc 13365 volcvol 25416

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-disj 5087 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8719 df-ec 8721 df-qs 8725 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-n0 12502 df-z 12589 df-q 12965 df-icc 13369

This theorem is referenced by: vitalilem4 25564

W3C validator