Step |
Hyp |
Ref |
Expression |
1 |
|
cvmliftlem.1 |
⊢ 𝑆 = ( 𝑘 ∈ 𝐽 ↦ { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) } ) |
2 |
|
cvmliftlem.b |
⊢ 𝐵 = ∪ 𝐶 |
3 |
|
cvmliftlem.x |
⊢ 𝑋 = ∪ 𝐽 |
4 |
|
cvmliftlem.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ) |
5 |
|
cvmliftlem.g |
⊢ ( 𝜑 → 𝐺 ∈ ( II Cn 𝐽 ) ) |
6 |
|
cvmliftlem.p |
⊢ ( 𝜑 → 𝑃 ∈ 𝐵 ) |
7 |
|
cvmliftlem.e |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 0 ) ) |
8 |
|
cvmliftlem.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
9 |
|
cvmliftlem.t |
⊢ ( 𝜑 → 𝑇 : ( 1 ... 𝑁 ) ⟶ ∪ 𝑗 ∈ 𝐽 ( { 𝑗 } × ( 𝑆 ‘ 𝑗 ) ) ) |
10 |
|
cvmliftlem.a |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 1 ... 𝑁 ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑁 ) [,] ( 𝑘 / 𝑁 ) ) ) ⊆ ( 1st ‘ ( 𝑇 ‘ 𝑘 ) ) ) |
11 |
|
cvmliftlem.l |
⊢ 𝐿 = ( topGen ‘ ran (,) ) |
12 |
|
cvmliftlem.q |
⊢ 𝑄 = seq 0 ( ( 𝑥 ∈ V , 𝑚 ∈ ℕ ↦ ( 𝑧 ∈ ( ( ( 𝑚 − 1 ) / 𝑁 ) [,] ( 𝑚 / 𝑁 ) ) ↦ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑚 ) ) ( 𝑥 ‘ ( ( 𝑚 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ) ) , ( ( I ↾ ℕ ) ∪ { 〈 0 , { 〈 0 , 𝑃 〉 } 〉 } ) ) |
13 |
|
cvmliftlem5.3 |
⊢ 𝑊 = ( ( ( 𝑀 − 1 ) / 𝑁 ) [,] ( 𝑀 / 𝑁 ) ) |
14 |
|
cvmliftlem6.1 |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑀 ∈ ( 1 ... 𝑁 ) ) |
15 |
|
cvmliftlem6.2 |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ ( ◡ 𝐹 “ { ( 𝐺 ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) } ) ) |
16 |
14
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → 𝑀 ∈ ( 1 ... 𝑁 ) ) |
17 |
1 2 3 4 5 6 7 8 9 10 11 16
|
cvmliftlem1 |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ∈ ( 𝑆 ‘ ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) ) ) |
18 |
1
|
cvmsss |
⊢ ( ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ∈ ( 𝑆 ‘ ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) ) → ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ⊆ 𝐶 ) |
19 |
17 18
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ⊆ 𝐶 ) |
20 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ) |
21 |
15
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ ( ◡ 𝐹 “ { ( 𝐺 ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) } ) ) |
22 |
|
cvmcn |
⊢ ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) → 𝐹 ∈ ( 𝐶 Cn 𝐽 ) ) |
23 |
2 3
|
cnf |
⊢ ( 𝐹 ∈ ( 𝐶 Cn 𝐽 ) → 𝐹 : 𝐵 ⟶ 𝑋 ) |
24 |
20 22 23
|
3syl |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → 𝐹 : 𝐵 ⟶ 𝑋 ) |
25 |
|
ffn |
⊢ ( 𝐹 : 𝐵 ⟶ 𝑋 → 𝐹 Fn 𝐵 ) |
26 |
|
fniniseg |
⊢ ( 𝐹 Fn 𝐵 → ( ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ ( ◡ 𝐹 “ { ( 𝐺 ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) } ) ↔ ( ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝐵 ∧ ( 𝐹 ‘ ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ) = ( 𝐺 ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ) ) ) |
27 |
24 25 26
|
3syl |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ ( ◡ 𝐹 “ { ( 𝐺 ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) } ) ↔ ( ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝐵 ∧ ( 𝐹 ‘ ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ) = ( 𝐺 ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ) ) ) |
28 |
21 27
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝐵 ∧ ( 𝐹 ‘ ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ) = ( 𝐺 ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ) ) |
29 |
28
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝐵 ) |
30 |
28
|
simprd |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( 𝐹 ‘ ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ) = ( 𝐺 ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ) |
31 |
|
elfznn |
⊢ ( 𝑀 ∈ ( 1 ... 𝑁 ) → 𝑀 ∈ ℕ ) |
32 |
16 31
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → 𝑀 ∈ ℕ ) |
33 |
32
|
nnred |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → 𝑀 ∈ ℝ ) |
34 |
|
peano2rem |
⊢ ( 𝑀 ∈ ℝ → ( 𝑀 − 1 ) ∈ ℝ ) |
35 |
33 34
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( 𝑀 − 1 ) ∈ ℝ ) |
36 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → 𝑁 ∈ ℕ ) |
37 |
35 36
|
nndivred |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ( 𝑀 − 1 ) / 𝑁 ) ∈ ℝ ) |
38 |
37
|
rexrd |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ( 𝑀 − 1 ) / 𝑁 ) ∈ ℝ* ) |
39 |
33 36
|
nndivred |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( 𝑀 / 𝑁 ) ∈ ℝ ) |
40 |
39
|
rexrd |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( 𝑀 / 𝑁 ) ∈ ℝ* ) |
41 |
33
|
ltm1d |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( 𝑀 − 1 ) < 𝑀 ) |
42 |
36
|
nnred |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → 𝑁 ∈ ℝ ) |
43 |
36
|
nngt0d |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → 0 < 𝑁 ) |
44 |
|
ltdiv1 |
⊢ ( ( ( 𝑀 − 1 ) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ( 𝑁 ∈ ℝ ∧ 0 < 𝑁 ) ) → ( ( 𝑀 − 1 ) < 𝑀 ↔ ( ( 𝑀 − 1 ) / 𝑁 ) < ( 𝑀 / 𝑁 ) ) ) |
45 |
35 33 42 43 44
|
syl112anc |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ( 𝑀 − 1 ) < 𝑀 ↔ ( ( 𝑀 − 1 ) / 𝑁 ) < ( 𝑀 / 𝑁 ) ) ) |
46 |
41 45
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ( 𝑀 − 1 ) / 𝑁 ) < ( 𝑀 / 𝑁 ) ) |
47 |
37 39 46
|
ltled |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ( 𝑀 − 1 ) / 𝑁 ) ≤ ( 𝑀 / 𝑁 ) ) |
48 |
|
lbicc2 |
⊢ ( ( ( ( 𝑀 − 1 ) / 𝑁 ) ∈ ℝ* ∧ ( 𝑀 / 𝑁 ) ∈ ℝ* ∧ ( ( 𝑀 − 1 ) / 𝑁 ) ≤ ( 𝑀 / 𝑁 ) ) → ( ( 𝑀 − 1 ) / 𝑁 ) ∈ ( ( ( 𝑀 − 1 ) / 𝑁 ) [,] ( 𝑀 / 𝑁 ) ) ) |
49 |
38 40 47 48
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ( 𝑀 − 1 ) / 𝑁 ) ∈ ( ( ( 𝑀 − 1 ) / 𝑁 ) [,] ( 𝑀 / 𝑁 ) ) ) |
50 |
49 13
|
eleqtrrdi |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ( 𝑀 − 1 ) / 𝑁 ) ∈ 𝑊 ) |
51 |
1 2 3 4 5 6 7 8 9 10 11 16 13 50
|
cvmliftlem3 |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( 𝐺 ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) ) |
52 |
30 51
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( 𝐹 ‘ ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ) ∈ ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) ) |
53 |
|
eqid |
⊢ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) = ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) |
54 |
1 2 53
|
cvmsiota |
⊢ ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ ( ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ∈ ( 𝑆 ‘ ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) ) ∧ ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝐵 ∧ ( 𝐹 ‘ ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ) ∈ ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) ) ) → ( ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ∧ ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ) |
55 |
20 17 29 52 54
|
syl13anc |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ∧ ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ) |
56 |
55
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ) |
57 |
19 56
|
sseldd |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ∈ 𝐶 ) |
58 |
|
elssuni |
⊢ ( ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ∈ 𝐶 → ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ⊆ ∪ 𝐶 ) |
59 |
57 58
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ⊆ ∪ 𝐶 ) |
60 |
59 2
|
sseqtrrdi |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ⊆ 𝐵 ) |
61 |
1
|
cvmsf1o |
⊢ ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ∈ ( 𝑆 ‘ ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) ) ∧ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ) → ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) : ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) –1-1-onto→ ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) ) |
62 |
20 17 56 61
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) : ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) –1-1-onto→ ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) ) |
63 |
|
f1ocnv |
⊢ ( ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) : ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) –1-1-onto→ ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) → ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) : ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) –1-1-onto→ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) |
64 |
|
f1of |
⊢ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) : ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) –1-1-onto→ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) → ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) : ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) ⟶ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) |
65 |
62 63 64
|
3syl |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) : ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) ⟶ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) |
66 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → 𝑧 ∈ 𝑊 ) |
67 |
1 2 3 4 5 6 7 8 9 10 11 16 13 66
|
cvmliftlem3 |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) ) |
68 |
65 67
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ∈ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) |
69 |
60 68
|
sseldd |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ∈ 𝐵 ) |
70 |
69
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝑧 ∈ 𝑊 ) → ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ∈ 𝐵 ) |
71 |
70
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝑊 ↦ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ) : 𝑊 ⟶ 𝐵 ) |
72 |
14 31
|
syl |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑀 ∈ ℕ ) |
73 |
1 2 3 4 5 6 7 8 9 10 11 12 13
|
cvmliftlem5 |
⊢ ( ( 𝜑 ∧ 𝑀 ∈ ℕ ) → ( 𝑄 ‘ 𝑀 ) = ( 𝑧 ∈ 𝑊 ↦ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ) ) |
74 |
72 73
|
syldan |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑄 ‘ 𝑀 ) = ( 𝑧 ∈ 𝑊 ↦ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ) ) |
75 |
74
|
feq1d |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝑄 ‘ 𝑀 ) : 𝑊 ⟶ 𝐵 ↔ ( 𝑧 ∈ 𝑊 ↦ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ) : 𝑊 ⟶ 𝐵 ) ) |
76 |
71 75
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑄 ‘ 𝑀 ) : 𝑊 ⟶ 𝐵 ) |
77 |
|
fvres |
⊢ ( 𝑧 ∈ 𝑊 → ( ( 𝐺 ↾ 𝑊 ) ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) |
78 |
66 77
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ( 𝐺 ↾ 𝑊 ) ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) |
79 |
|
f1ocnvfv2 |
⊢ ( ( ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) : ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) –1-1-onto→ ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( 1st ‘ ( 𝑇 ‘ 𝑀 ) ) ) → ( ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ) = ( 𝐺 ‘ 𝑧 ) ) |
80 |
62 67 79
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ) = ( 𝐺 ‘ 𝑧 ) ) |
81 |
|
fvres |
⊢ ( ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ∈ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) → ( ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ) = ( 𝐹 ‘ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ) ) |
82 |
68 81
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ) = ( 𝐹 ‘ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ) ) |
83 |
78 80 82
|
3eqtr2rd |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝑧 ∈ 𝑊 ) ) → ( 𝐹 ‘ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ) = ( ( 𝐺 ↾ 𝑊 ) ‘ 𝑧 ) ) |
84 |
83
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝑧 ∈ 𝑊 ) → ( 𝐹 ‘ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ) = ( ( 𝐺 ↾ 𝑊 ) ‘ 𝑧 ) ) |
85 |
84
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝑊 ↦ ( 𝐹 ‘ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ) ) = ( 𝑧 ∈ 𝑊 ↦ ( ( 𝐺 ↾ 𝑊 ) ‘ 𝑧 ) ) ) |
86 |
4 22 23
|
3syl |
⊢ ( 𝜑 → 𝐹 : 𝐵 ⟶ 𝑋 ) |
87 |
86
|
adantr |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐹 : 𝐵 ⟶ 𝑋 ) |
88 |
87
|
feqmptd |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐹 = ( 𝑦 ∈ 𝐵 ↦ ( 𝐹 ‘ 𝑦 ) ) ) |
89 |
|
fveq2 |
⊢ ( 𝑦 = ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ) ) |
90 |
70 74 88 89
|
fmptco |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝐹 ∘ ( 𝑄 ‘ 𝑀 ) ) = ( 𝑧 ∈ 𝑊 ↦ ( 𝐹 ‘ ( ◡ ( 𝐹 ↾ ( ℩ 𝑏 ∈ ( 2nd ‘ ( 𝑇 ‘ 𝑀 ) ) ( ( 𝑄 ‘ ( 𝑀 − 1 ) ) ‘ ( ( 𝑀 − 1 ) / 𝑁 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺 ‘ 𝑧 ) ) ) ) ) |
91 |
|
iiuni |
⊢ ( 0 [,] 1 ) = ∪ II |
92 |
91 3
|
cnf |
⊢ ( 𝐺 ∈ ( II Cn 𝐽 ) → 𝐺 : ( 0 [,] 1 ) ⟶ 𝑋 ) |
93 |
5 92
|
syl |
⊢ ( 𝜑 → 𝐺 : ( 0 [,] 1 ) ⟶ 𝑋 ) |
94 |
93
|
adantr |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐺 : ( 0 [,] 1 ) ⟶ 𝑋 ) |
95 |
1 2 3 4 5 6 7 8 9 10 11 14 13
|
cvmliftlem2 |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑊 ⊆ ( 0 [,] 1 ) ) |
96 |
94 95
|
fssresd |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝐺 ↾ 𝑊 ) : 𝑊 ⟶ 𝑋 ) |
97 |
96
|
feqmptd |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝐺 ↾ 𝑊 ) = ( 𝑧 ∈ 𝑊 ↦ ( ( 𝐺 ↾ 𝑊 ) ‘ 𝑧 ) ) ) |
98 |
85 90 97
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝐹 ∘ ( 𝑄 ‘ 𝑀 ) ) = ( 𝐺 ↾ 𝑊 ) ) |
99 |
76 98
|
jca |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝑄 ‘ 𝑀 ) : 𝑊 ⟶ 𝐵 ∧ ( 𝐹 ∘ ( 𝑄 ‘ 𝑀 ) ) = ( 𝐺 ↾ 𝑊 ) ) ) |