Step |
Hyp |
Ref |
Expression |
1 |
|
vitali.1 |
⊢ ∼ = { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ∧ ( 𝑥 − 𝑦 ) ∈ ℚ ) } |
2 |
|
vitali.2 |
⊢ 𝑆 = ( ( 0 [,] 1 ) / ∼ ) |
3 |
|
vitali.3 |
⊢ ( 𝜑 → 𝐹 Fn 𝑆 ) |
4 |
|
vitali.4 |
⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝑆 ( 𝑧 ≠ ∅ → ( 𝐹 ‘ 𝑧 ) ∈ 𝑧 ) ) |
5 |
|
vitali.5 |
⊢ ( 𝜑 → 𝐺 : ℕ –1-1-onto→ ( ℚ ∩ ( - 1 [,] 1 ) ) ) |
6 |
|
vitali.6 |
⊢ 𝑇 = ( 𝑛 ∈ ℕ ↦ { 𝑠 ∈ ℝ ∣ ( 𝑠 − ( 𝐺 ‘ 𝑛 ) ) ∈ ran 𝐹 } ) |
7 |
|
vitali.7 |
⊢ ( 𝜑 → ¬ ran 𝐹 ∈ ( 𝒫 ℝ ∖ dom vol ) ) |
8 |
|
simprlr |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) |
9 |
|
simprll |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → 𝑚 ∈ ℕ ) |
10 |
|
fveq2 |
⊢ ( 𝑛 = 𝑚 → ( 𝐺 ‘ 𝑛 ) = ( 𝐺 ‘ 𝑚 ) ) |
11 |
10
|
oveq2d |
⊢ ( 𝑛 = 𝑚 → ( 𝑠 − ( 𝐺 ‘ 𝑛 ) ) = ( 𝑠 − ( 𝐺 ‘ 𝑚 ) ) ) |
12 |
11
|
eleq1d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝑠 − ( 𝐺 ‘ 𝑛 ) ) ∈ ran 𝐹 ↔ ( 𝑠 − ( 𝐺 ‘ 𝑚 ) ) ∈ ran 𝐹 ) ) |
13 |
12
|
rabbidv |
⊢ ( 𝑛 = 𝑚 → { 𝑠 ∈ ℝ ∣ ( 𝑠 − ( 𝐺 ‘ 𝑛 ) ) ∈ ran 𝐹 } = { 𝑠 ∈ ℝ ∣ ( 𝑠 − ( 𝐺 ‘ 𝑚 ) ) ∈ ran 𝐹 } ) |
14 |
|
reex |
⊢ ℝ ∈ V |
15 |
14
|
rabex |
⊢ { 𝑠 ∈ ℝ ∣ ( 𝑠 − ( 𝐺 ‘ 𝑚 ) ) ∈ ran 𝐹 } ∈ V |
16 |
13 6 15
|
fvmpt |
⊢ ( 𝑚 ∈ ℕ → ( 𝑇 ‘ 𝑚 ) = { 𝑠 ∈ ℝ ∣ ( 𝑠 − ( 𝐺 ‘ 𝑚 ) ) ∈ ran 𝐹 } ) |
17 |
9 16
|
syl |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝑇 ‘ 𝑚 ) = { 𝑠 ∈ ℝ ∣ ( 𝑠 − ( 𝐺 ‘ 𝑚 ) ) ∈ ran 𝐹 } ) |
18 |
8 17
|
eleqtrd |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → 𝑤 ∈ { 𝑠 ∈ ℝ ∣ ( 𝑠 − ( 𝐺 ‘ 𝑚 ) ) ∈ ran 𝐹 } ) |
19 |
|
oveq1 |
⊢ ( 𝑠 = 𝑤 → ( 𝑠 − ( 𝐺 ‘ 𝑚 ) ) = ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ) |
20 |
19
|
eleq1d |
⊢ ( 𝑠 = 𝑤 → ( ( 𝑠 − ( 𝐺 ‘ 𝑚 ) ) ∈ ran 𝐹 ↔ ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ∈ ran 𝐹 ) ) |
21 |
20
|
elrab |
⊢ ( 𝑤 ∈ { 𝑠 ∈ ℝ ∣ ( 𝑠 − ( 𝐺 ‘ 𝑚 ) ) ∈ ran 𝐹 } ↔ ( 𝑤 ∈ ℝ ∧ ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ∈ ran 𝐹 ) ) |
22 |
18 21
|
sylib |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝑤 ∈ ℝ ∧ ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ∈ ran 𝐹 ) ) |
23 |
22
|
simpld |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → 𝑤 ∈ ℝ ) |
24 |
23
|
recnd |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → 𝑤 ∈ ℂ ) |
25 |
|
f1of |
⊢ ( 𝐺 : ℕ –1-1-onto→ ( ℚ ∩ ( - 1 [,] 1 ) ) → 𝐺 : ℕ ⟶ ( ℚ ∩ ( - 1 [,] 1 ) ) ) |
26 |
5 25
|
syl |
⊢ ( 𝜑 → 𝐺 : ℕ ⟶ ( ℚ ∩ ( - 1 [,] 1 ) ) ) |
27 |
|
inss1 |
⊢ ( ℚ ∩ ( - 1 [,] 1 ) ) ⊆ ℚ |
28 |
|
fss |
⊢ ( ( 𝐺 : ℕ ⟶ ( ℚ ∩ ( - 1 [,] 1 ) ) ∧ ( ℚ ∩ ( - 1 [,] 1 ) ) ⊆ ℚ ) → 𝐺 : ℕ ⟶ ℚ ) |
29 |
26 27 28
|
sylancl |
⊢ ( 𝜑 → 𝐺 : ℕ ⟶ ℚ ) |
30 |
29
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → 𝐺 : ℕ ⟶ ℚ ) |
31 |
30 9
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝐺 ‘ 𝑚 ) ∈ ℚ ) |
32 |
|
qcn |
⊢ ( ( 𝐺 ‘ 𝑚 ) ∈ ℚ → ( 𝐺 ‘ 𝑚 ) ∈ ℂ ) |
33 |
31 32
|
syl |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝐺 ‘ 𝑚 ) ∈ ℂ ) |
34 |
|
simprrl |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → 𝑘 ∈ ℕ ) |
35 |
30 34
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝐺 ‘ 𝑘 ) ∈ ℚ ) |
36 |
|
qcn |
⊢ ( ( 𝐺 ‘ 𝑘 ) ∈ ℚ → ( 𝐺 ‘ 𝑘 ) ∈ ℂ ) |
37 |
35 36
|
syl |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝐺 ‘ 𝑘 ) ∈ ℂ ) |
38 |
1
|
vitalilem1 |
⊢ ∼ Er ( 0 [,] 1 ) |
39 |
38
|
a1i |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ∼ Er ( 0 [,] 1 ) ) |
40 |
1 2 3 4 5 6 7
|
vitalilem2 |
⊢ ( 𝜑 → ( ran 𝐹 ⊆ ( 0 [,] 1 ) ∧ ( 0 [,] 1 ) ⊆ ∪ 𝑚 ∈ ℕ ( 𝑇 ‘ 𝑚 ) ∧ ∪ 𝑚 ∈ ℕ ( 𝑇 ‘ 𝑚 ) ⊆ ( - 1 [,] 2 ) ) ) |
41 |
40
|
simp1d |
⊢ ( 𝜑 → ran 𝐹 ⊆ ( 0 [,] 1 ) ) |
42 |
41
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ran 𝐹 ⊆ ( 0 [,] 1 ) ) |
43 |
22
|
simprd |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ∈ ran 𝐹 ) |
44 |
42 43
|
sseldd |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ∈ ( 0 [,] 1 ) ) |
45 |
|
simprrr |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) |
46 |
|
fveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝐺 ‘ 𝑛 ) = ( 𝐺 ‘ 𝑘 ) ) |
47 |
46
|
oveq2d |
⊢ ( 𝑛 = 𝑘 → ( 𝑠 − ( 𝐺 ‘ 𝑛 ) ) = ( 𝑠 − ( 𝐺 ‘ 𝑘 ) ) ) |
48 |
47
|
eleq1d |
⊢ ( 𝑛 = 𝑘 → ( ( 𝑠 − ( 𝐺 ‘ 𝑛 ) ) ∈ ran 𝐹 ↔ ( 𝑠 − ( 𝐺 ‘ 𝑘 ) ) ∈ ran 𝐹 ) ) |
49 |
48
|
rabbidv |
⊢ ( 𝑛 = 𝑘 → { 𝑠 ∈ ℝ ∣ ( 𝑠 − ( 𝐺 ‘ 𝑛 ) ) ∈ ran 𝐹 } = { 𝑠 ∈ ℝ ∣ ( 𝑠 − ( 𝐺 ‘ 𝑘 ) ) ∈ ran 𝐹 } ) |
50 |
14
|
rabex |
⊢ { 𝑠 ∈ ℝ ∣ ( 𝑠 − ( 𝐺 ‘ 𝑘 ) ) ∈ ran 𝐹 } ∈ V |
51 |
49 6 50
|
fvmpt |
⊢ ( 𝑘 ∈ ℕ → ( 𝑇 ‘ 𝑘 ) = { 𝑠 ∈ ℝ ∣ ( 𝑠 − ( 𝐺 ‘ 𝑘 ) ) ∈ ran 𝐹 } ) |
52 |
34 51
|
syl |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝑇 ‘ 𝑘 ) = { 𝑠 ∈ ℝ ∣ ( 𝑠 − ( 𝐺 ‘ 𝑘 ) ) ∈ ran 𝐹 } ) |
53 |
45 52
|
eleqtrd |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → 𝑤 ∈ { 𝑠 ∈ ℝ ∣ ( 𝑠 − ( 𝐺 ‘ 𝑘 ) ) ∈ ran 𝐹 } ) |
54 |
|
oveq1 |
⊢ ( 𝑠 = 𝑤 → ( 𝑠 − ( 𝐺 ‘ 𝑘 ) ) = ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ) |
55 |
54
|
eleq1d |
⊢ ( 𝑠 = 𝑤 → ( ( 𝑠 − ( 𝐺 ‘ 𝑘 ) ) ∈ ran 𝐹 ↔ ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ∈ ran 𝐹 ) ) |
56 |
55
|
elrab |
⊢ ( 𝑤 ∈ { 𝑠 ∈ ℝ ∣ ( 𝑠 − ( 𝐺 ‘ 𝑘 ) ) ∈ ran 𝐹 } ↔ ( 𝑤 ∈ ℝ ∧ ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ∈ ran 𝐹 ) ) |
57 |
53 56
|
sylib |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝑤 ∈ ℝ ∧ ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ∈ ran 𝐹 ) ) |
58 |
57
|
simprd |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ∈ ran 𝐹 ) |
59 |
42 58
|
sseldd |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ∈ ( 0 [,] 1 ) ) |
60 |
24 33 37
|
nnncan1d |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) − ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ) = ( ( 𝐺 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑚 ) ) ) |
61 |
|
qsubcl |
⊢ ( ( ( 𝐺 ‘ 𝑘 ) ∈ ℚ ∧ ( 𝐺 ‘ 𝑚 ) ∈ ℚ ) → ( ( 𝐺 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑚 ) ) ∈ ℚ ) |
62 |
35 31 61
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( ( 𝐺 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑚 ) ) ∈ ℚ ) |
63 |
60 62
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) − ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ) ∈ ℚ ) |
64 |
|
oveq12 |
⊢ ( ( 𝑥 = ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ∧ 𝑦 = ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ) → ( 𝑥 − 𝑦 ) = ( ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) − ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ) ) |
65 |
64
|
eleq1d |
⊢ ( ( 𝑥 = ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ∧ 𝑦 = ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ) → ( ( 𝑥 − 𝑦 ) ∈ ℚ ↔ ( ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) − ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ) ∈ ℚ ) ) |
66 |
65 1
|
brab2a |
⊢ ( ( 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 |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ∼ ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ) |
68 |
39 67
|
erthi |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → [ ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ] ∼ = [ ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ] ∼ ) |
69 |
68
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝐹 ‘ [ ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ] ∼ ) = ( 𝐹 ‘ [ ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ] ∼ ) ) |
70 |
|
eceq1 |
⊢ ( 𝑧 = ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) → [ 𝑧 ] ∼ = [ ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ] ∼ ) |
71 |
70
|
fveq2d |
⊢ ( 𝑧 = ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) → ( 𝐹 ‘ [ 𝑧 ] ∼ ) = ( 𝐹 ‘ [ ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ] ∼ ) ) |
72 |
|
id |
⊢ ( 𝑧 = ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) → 𝑧 = ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ) |
73 |
71 72
|
eqeq12d |
⊢ ( 𝑧 = ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) → ( ( 𝐹 ‘ [ 𝑧 ] ∼ ) = 𝑧 ↔ ( 𝐹 ‘ [ ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ] ∼ ) = ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ) ) |
74 |
|
fveq2 |
⊢ ( [ 𝑣 ] ∼ = 𝑤 → ( 𝐹 ‘ [ 𝑣 ] ∼ ) = ( 𝐹 ‘ 𝑤 ) ) |
75 |
74
|
eceq1d |
⊢ ( [ 𝑣 ] ∼ = 𝑤 → [ ( 𝐹 ‘ [ 𝑣 ] ∼ ) ] ∼ = [ ( 𝐹 ‘ 𝑤 ) ] ∼ ) |
76 |
75
|
fveq2d |
⊢ ( [ 𝑣 ] ∼ = 𝑤 → ( 𝐹 ‘ [ ( 𝐹 ‘ [ 𝑣 ] ∼ ) ] ∼ ) = ( 𝐹 ‘ [ ( 𝐹 ‘ 𝑤 ) ] ∼ ) ) |
77 |
76 74
|
eqeq12d |
⊢ ( [ 𝑣 ] ∼ = 𝑤 → ( ( 𝐹 ‘ [ ( 𝐹 ‘ [ 𝑣 ] ∼ ) ] ∼ ) = ( 𝐹 ‘ [ 𝑣 ] ∼ ) ↔ ( 𝐹 ‘ [ ( 𝐹 ‘ 𝑤 ) ] ∼ ) = ( 𝐹 ‘ 𝑤 ) ) ) |
78 |
38
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( 0 [,] 1 ) ) → ∼ Er ( 0 [,] 1 ) ) |
79 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( 0 [,] 1 ) ) → 𝑣 ∈ ( 0 [,] 1 ) ) |
80 |
|
erdm |
⊢ ( ∼ Er ( 0 [,] 1 ) → dom ∼ = ( 0 [,] 1 ) ) |
81 |
38 80
|
ax-mp |
⊢ dom ∼ = ( 0 [,] 1 ) |
82 |
81
|
eleq2i |
⊢ ( 𝑣 ∈ dom ∼ ↔ 𝑣 ∈ ( 0 [,] 1 ) ) |
83 |
|
ecdmn0 |
⊢ ( 𝑣 ∈ dom ∼ ↔ [ 𝑣 ] ∼ ≠ ∅ ) |
84 |
82 83
|
bitr3i |
⊢ ( 𝑣 ∈ ( 0 [,] 1 ) ↔ [ 𝑣 ] ∼ ≠ ∅ ) |
85 |
79 84
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( 0 [,] 1 ) ) → [ 𝑣 ] ∼ ≠ ∅ ) |
86 |
|
neeq1 |
⊢ ( 𝑧 = [ 𝑣 ] ∼ → ( 𝑧 ≠ ∅ ↔ [ 𝑣 ] ∼ ≠ ∅ ) ) |
87 |
|
fveq2 |
⊢ ( 𝑧 = [ 𝑣 ] ∼ → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ [ 𝑣 ] ∼ ) ) |
88 |
|
id |
⊢ ( 𝑧 = [ 𝑣 ] ∼ → 𝑧 = [ 𝑣 ] ∼ ) |
89 |
87 88
|
eleq12d |
⊢ ( 𝑧 = [ 𝑣 ] ∼ → ( ( 𝐹 ‘ 𝑧 ) ∈ 𝑧 ↔ ( 𝐹 ‘ [ 𝑣 ] ∼ ) ∈ [ 𝑣 ] ∼ ) ) |
90 |
86 89
|
imbi12d |
⊢ ( 𝑧 = [ 𝑣 ] ∼ → ( ( 𝑧 ≠ ∅ → ( 𝐹 ‘ 𝑧 ) ∈ 𝑧 ) ↔ ( [ 𝑣 ] ∼ ≠ ∅ → ( 𝐹 ‘ [ 𝑣 ] ∼ ) ∈ [ 𝑣 ] ∼ ) ) ) |
91 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( 0 [,] 1 ) ) → ∀ 𝑧 ∈ 𝑆 ( 𝑧 ≠ ∅ → ( 𝐹 ‘ 𝑧 ) ∈ 𝑧 ) ) |
92 |
|
ovex |
⊢ ( 0 [,] 1 ) ∈ V |
93 |
|
erex |
⊢ ( ∼ Er ( 0 [,] 1 ) → ( ( 0 [,] 1 ) ∈ V → ∼ ∈ V ) ) |
94 |
38 92 93
|
mp2 |
⊢ ∼ ∈ V |
95 |
94
|
ecelqsi |
⊢ ( 𝑣 ∈ ( 0 [,] 1 ) → [ 𝑣 ] ∼ ∈ ( ( 0 [,] 1 ) / ∼ ) ) |
96 |
95 2
|
eleqtrrdi |
⊢ ( 𝑣 ∈ ( 0 [,] 1 ) → [ 𝑣 ] ∼ ∈ 𝑆 ) |
97 |
96
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( 0 [,] 1 ) ) → [ 𝑣 ] ∼ ∈ 𝑆 ) |
98 |
90 91 97
|
rspcdva |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( 0 [,] 1 ) ) → ( [ 𝑣 ] ∼ ≠ ∅ → ( 𝐹 ‘ [ 𝑣 ] ∼ ) ∈ [ 𝑣 ] ∼ ) ) |
99 |
85 98
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( 0 [,] 1 ) ) → ( 𝐹 ‘ [ 𝑣 ] ∼ ) ∈ [ 𝑣 ] ∼ ) |
100 |
|
fvex |
⊢ ( 𝐹 ‘ [ 𝑣 ] ∼ ) ∈ V |
101 |
|
vex |
⊢ 𝑣 ∈ V |
102 |
100 101
|
elec |
⊢ ( ( 𝐹 ‘ [ 𝑣 ] ∼ ) ∈ [ 𝑣 ] ∼ ↔ 𝑣 ∼ ( 𝐹 ‘ [ 𝑣 ] ∼ ) ) |
103 |
99 102
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( 0 [,] 1 ) ) → 𝑣 ∼ ( 𝐹 ‘ [ 𝑣 ] ∼ ) ) |
104 |
78 103
|
erthi |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( 0 [,] 1 ) ) → [ 𝑣 ] ∼ = [ ( 𝐹 ‘ [ 𝑣 ] ∼ ) ] ∼ ) |
105 |
104
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( 0 [,] 1 ) ) → [ ( 𝐹 ‘ [ 𝑣 ] ∼ ) ] ∼ = [ 𝑣 ] ∼ ) |
106 |
105
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( 0 [,] 1 ) ) → ( 𝐹 ‘ [ ( 𝐹 ‘ [ 𝑣 ] ∼ ) ] ∼ ) = ( 𝐹 ‘ [ 𝑣 ] ∼ ) ) |
107 |
2 77 106
|
ectocld |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑆 ) → ( 𝐹 ‘ [ ( 𝐹 ‘ 𝑤 ) ] ∼ ) = ( 𝐹 ‘ 𝑤 ) ) |
108 |
107
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑤 ∈ 𝑆 ( 𝐹 ‘ [ ( 𝐹 ‘ 𝑤 ) ] ∼ ) = ( 𝐹 ‘ 𝑤 ) ) |
109 |
|
eceq1 |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑤 ) → [ 𝑧 ] ∼ = [ ( 𝐹 ‘ 𝑤 ) ] ∼ ) |
110 |
109
|
fveq2d |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑤 ) → ( 𝐹 ‘ [ 𝑧 ] ∼ ) = ( 𝐹 ‘ [ ( 𝐹 ‘ 𝑤 ) ] ∼ ) ) |
111 |
|
id |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑤 ) → 𝑧 = ( 𝐹 ‘ 𝑤 ) ) |
112 |
110 111
|
eqeq12d |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑤 ) → ( ( 𝐹 ‘ [ 𝑧 ] ∼ ) = 𝑧 ↔ ( 𝐹 ‘ [ ( 𝐹 ‘ 𝑤 ) ] ∼ ) = ( 𝐹 ‘ 𝑤 ) ) ) |
113 |
112
|
ralrn |
⊢ ( 𝐹 Fn 𝑆 → ( ∀ 𝑧 ∈ ran 𝐹 ( 𝐹 ‘ [ 𝑧 ] ∼ ) = 𝑧 ↔ ∀ 𝑤 ∈ 𝑆 ( 𝐹 ‘ [ ( 𝐹 ‘ 𝑤 ) ] ∼ ) = ( 𝐹 ‘ 𝑤 ) ) ) |
114 |
3 113
|
syl |
⊢ ( 𝜑 → ( ∀ 𝑧 ∈ ran 𝐹 ( 𝐹 ‘ [ 𝑧 ] ∼ ) = 𝑧 ↔ ∀ 𝑤 ∈ 𝑆 ( 𝐹 ‘ [ ( 𝐹 ‘ 𝑤 ) ] ∼ ) = ( 𝐹 ‘ 𝑤 ) ) ) |
115 |
108 114
|
mpbird |
⊢ ( 𝜑 → ∀ 𝑧 ∈ ran 𝐹 ( 𝐹 ‘ [ 𝑧 ] ∼ ) = 𝑧 ) |
116 |
115
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ∀ 𝑧 ∈ ran 𝐹 ( 𝐹 ‘ [ 𝑧 ] ∼ ) = 𝑧 ) |
117 |
73 116 43
|
rspcdva |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝐹 ‘ [ ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ] ∼ ) = ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) ) |
118 |
|
eceq1 |
⊢ ( 𝑧 = ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) → [ 𝑧 ] ∼ = [ ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ] ∼ ) |
119 |
118
|
fveq2d |
⊢ ( 𝑧 = ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) → ( 𝐹 ‘ [ 𝑧 ] ∼ ) = ( 𝐹 ‘ [ ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ] ∼ ) ) |
120 |
|
id |
⊢ ( 𝑧 = ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) → 𝑧 = ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ) |
121 |
119 120
|
eqeq12d |
⊢ ( 𝑧 = ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) → ( ( 𝐹 ‘ [ 𝑧 ] ∼ ) = 𝑧 ↔ ( 𝐹 ‘ [ ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ] ∼ ) = ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ) ) |
122 |
121 116 58
|
rspcdva |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝐹 ‘ [ ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ] ∼ ) = ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ) |
123 |
69 117 122
|
3eqtr3d |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝑤 − ( 𝐺 ‘ 𝑚 ) ) = ( 𝑤 − ( 𝐺 ‘ 𝑘 ) ) ) |
124 |
24 33 37 123
|
subcand |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( 𝐺 ‘ 𝑚 ) = ( 𝐺 ‘ 𝑘 ) ) |
125 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → 𝐺 : ℕ –1-1-onto→ ( ℚ ∩ ( - 1 [,] 1 ) ) ) |
126 |
|
f1of1 |
⊢ ( 𝐺 : ℕ –1-1-onto→ ( ℚ ∩ ( - 1 [,] 1 ) ) → 𝐺 : ℕ –1-1→ ( ℚ ∩ ( - 1 [,] 1 ) ) ) |
127 |
125 126
|
syl |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → 𝐺 : ℕ –1-1→ ( ℚ ∩ ( - 1 [,] 1 ) ) ) |
128 |
|
f1fveq |
⊢ ( ( 𝐺 : ℕ –1-1→ ( ℚ ∩ ( - 1 [,] 1 ) ) ∧ ( 𝑚 ∈ ℕ ∧ 𝑘 ∈ ℕ ) ) → ( ( 𝐺 ‘ 𝑚 ) = ( 𝐺 ‘ 𝑘 ) ↔ 𝑚 = 𝑘 ) ) |
129 |
127 9 34 128
|
syl12anc |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → ( ( 𝐺 ‘ 𝑚 ) = ( 𝐺 ‘ 𝑘 ) ↔ 𝑚 = 𝑘 ) ) |
130 |
124 129
|
mpbid |
⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) → 𝑚 = 𝑘 ) |
131 |
130
|
ex |
⊢ ( 𝜑 → ( ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) → 𝑚 = 𝑘 ) ) |
132 |
131
|
alrimivv |
⊢ ( 𝜑 → ∀ 𝑚 ∀ 𝑘 ( ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) → 𝑚 = 𝑘 ) ) |
133 |
|
eleq1w |
⊢ ( 𝑚 = 𝑘 → ( 𝑚 ∈ ℕ ↔ 𝑘 ∈ ℕ ) ) |
134 |
|
fveq2 |
⊢ ( 𝑚 = 𝑘 → ( 𝑇 ‘ 𝑚 ) = ( 𝑇 ‘ 𝑘 ) ) |
135 |
134
|
eleq2d |
⊢ ( 𝑚 = 𝑘 → ( 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ↔ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) |
136 |
133 135
|
anbi12d |
⊢ ( 𝑚 = 𝑘 → ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ↔ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) ) |
137 |
136
|
mo4 |
⊢ ( ∃* 𝑚 ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ↔ ∀ 𝑚 ∀ 𝑘 ( ( ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ∧ ( 𝑘 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑘 ) ) ) → 𝑚 = 𝑘 ) ) |
138 |
132 137
|
sylibr |
⊢ ( 𝜑 → ∃* 𝑚 ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ) |
139 |
138
|
alrimiv |
⊢ ( 𝜑 → ∀ 𝑤 ∃* 𝑚 ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ) |
140 |
|
dfdisj2 |
⊢ ( Disj 𝑚 ∈ ℕ ( 𝑇 ‘ 𝑚 ) ↔ ∀ 𝑤 ∃* 𝑚 ( 𝑚 ∈ ℕ ∧ 𝑤 ∈ ( 𝑇 ‘ 𝑚 ) ) ) |
141 |
139 140
|
sylibr |
⊢ ( 𝜑 → Disj 𝑚 ∈ ℕ ( 𝑇 ‘ 𝑚 ) ) |