Step |
Hyp |
Ref |
Expression |
1 |
|
heibor.1 |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
2 |
|
heibor.3 |
⊢ 𝐾 = { 𝑢 ∣ ¬ ∃ 𝑣 ∈ ( 𝒫 𝑈 ∩ Fin ) 𝑢 ⊆ ∪ 𝑣 } |
3 |
|
heibor.4 |
⊢ 𝐺 = { 〈 𝑦 , 𝑛 〉 ∣ ( 𝑛 ∈ ℕ0 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑛 ) ∧ ( 𝑦 𝐵 𝑛 ) ∈ 𝐾 ) } |
4 |
|
heibor.5 |
⊢ 𝐵 = ( 𝑧 ∈ 𝑋 , 𝑚 ∈ ℕ0 ↦ ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / ( 2 ↑ 𝑚 ) ) ) ) |
5 |
|
heibor.6 |
⊢ ( 𝜑 → 𝐷 ∈ ( CMet ‘ 𝑋 ) ) |
6 |
|
heibor.7 |
⊢ ( 𝜑 → 𝐹 : ℕ0 ⟶ ( 𝒫 𝑋 ∩ Fin ) ) |
7 |
|
heibor.8 |
⊢ ( 𝜑 → ∀ 𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑛 ) ( 𝑦 𝐵 𝑛 ) ) |
8 |
|
heibor.9 |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐺 ( ( 𝑇 ‘ 𝑥 ) 𝐺 ( ( 2nd ‘ 𝑥 ) + 1 ) ∧ ( ( 𝐵 ‘ 𝑥 ) ∩ ( ( 𝑇 ‘ 𝑥 ) 𝐵 ( ( 2nd ‘ 𝑥 ) + 1 ) ) ) ∈ 𝐾 ) ) |
9 |
|
heibor.10 |
⊢ ( 𝜑 → 𝐶 𝐺 0 ) |
10 |
|
heibor.11 |
⊢ 𝑆 = seq 0 ( 𝑇 , ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ) |
11 |
|
fveq2 |
⊢ ( 𝑥 = 0 → ( 𝑆 ‘ 𝑥 ) = ( 𝑆 ‘ 0 ) ) |
12 |
|
id |
⊢ ( 𝑥 = 0 → 𝑥 = 0 ) |
13 |
11 12
|
breq12d |
⊢ ( 𝑥 = 0 → ( ( 𝑆 ‘ 𝑥 ) 𝐺 𝑥 ↔ ( 𝑆 ‘ 0 ) 𝐺 0 ) ) |
14 |
13
|
imbi2d |
⊢ ( 𝑥 = 0 → ( ( 𝜑 → ( 𝑆 ‘ 𝑥 ) 𝐺 𝑥 ) ↔ ( 𝜑 → ( 𝑆 ‘ 0 ) 𝐺 0 ) ) ) |
15 |
|
fveq2 |
⊢ ( 𝑥 = 𝑘 → ( 𝑆 ‘ 𝑥 ) = ( 𝑆 ‘ 𝑘 ) ) |
16 |
|
id |
⊢ ( 𝑥 = 𝑘 → 𝑥 = 𝑘 ) |
17 |
15 16
|
breq12d |
⊢ ( 𝑥 = 𝑘 → ( ( 𝑆 ‘ 𝑥 ) 𝐺 𝑥 ↔ ( 𝑆 ‘ 𝑘 ) 𝐺 𝑘 ) ) |
18 |
17
|
imbi2d |
⊢ ( 𝑥 = 𝑘 → ( ( 𝜑 → ( 𝑆 ‘ 𝑥 ) 𝐺 𝑥 ) ↔ ( 𝜑 → ( 𝑆 ‘ 𝑘 ) 𝐺 𝑘 ) ) ) |
19 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → ( 𝑆 ‘ 𝑥 ) = ( 𝑆 ‘ ( 𝑘 + 1 ) ) ) |
20 |
|
id |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → 𝑥 = ( 𝑘 + 1 ) ) |
21 |
19 20
|
breq12d |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → ( ( 𝑆 ‘ 𝑥 ) 𝐺 𝑥 ↔ ( 𝑆 ‘ ( 𝑘 + 1 ) ) 𝐺 ( 𝑘 + 1 ) ) ) |
22 |
21
|
imbi2d |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → ( ( 𝜑 → ( 𝑆 ‘ 𝑥 ) 𝐺 𝑥 ) ↔ ( 𝜑 → ( 𝑆 ‘ ( 𝑘 + 1 ) ) 𝐺 ( 𝑘 + 1 ) ) ) ) |
23 |
|
fveq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝑆 ‘ 𝑥 ) = ( 𝑆 ‘ 𝐴 ) ) |
24 |
|
id |
⊢ ( 𝑥 = 𝐴 → 𝑥 = 𝐴 ) |
25 |
23 24
|
breq12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑆 ‘ 𝑥 ) 𝐺 𝑥 ↔ ( 𝑆 ‘ 𝐴 ) 𝐺 𝐴 ) ) |
26 |
25
|
imbi2d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝜑 → ( 𝑆 ‘ 𝑥 ) 𝐺 𝑥 ) ↔ ( 𝜑 → ( 𝑆 ‘ 𝐴 ) 𝐺 𝐴 ) ) ) |
27 |
10
|
fveq1i |
⊢ ( 𝑆 ‘ 0 ) = ( seq 0 ( 𝑇 , ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ) ‘ 0 ) |
28 |
|
0z |
⊢ 0 ∈ ℤ |
29 |
|
seq1 |
⊢ ( 0 ∈ ℤ → ( seq 0 ( 𝑇 , ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ) ‘ 0 ) = ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ‘ 0 ) ) |
30 |
28 29
|
ax-mp |
⊢ ( seq 0 ( 𝑇 , ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ) ‘ 0 ) = ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ‘ 0 ) |
31 |
27 30
|
eqtri |
⊢ ( 𝑆 ‘ 0 ) = ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ‘ 0 ) |
32 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
33 |
3
|
relopabi |
⊢ Rel 𝐺 |
34 |
33
|
brrelex1i |
⊢ ( 𝐶 𝐺 0 → 𝐶 ∈ V ) |
35 |
9 34
|
syl |
⊢ ( 𝜑 → 𝐶 ∈ V ) |
36 |
|
iftrue |
⊢ ( 𝑚 = 0 → if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) = 𝐶 ) |
37 |
|
eqid |
⊢ ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) = ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) |
38 |
36 37
|
fvmptg |
⊢ ( ( 0 ∈ ℕ0 ∧ 𝐶 ∈ V ) → ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ‘ 0 ) = 𝐶 ) |
39 |
32 35 38
|
sylancr |
⊢ ( 𝜑 → ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ‘ 0 ) = 𝐶 ) |
40 |
31 39
|
syl5eq |
⊢ ( 𝜑 → ( 𝑆 ‘ 0 ) = 𝐶 ) |
41 |
40 9
|
eqbrtrd |
⊢ ( 𝜑 → ( 𝑆 ‘ 0 ) 𝐺 0 ) |
42 |
|
df-br |
⊢ ( ( 𝑆 ‘ 𝑘 ) 𝐺 𝑘 ↔ 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 ∈ 𝐺 ) |
43 |
|
fveq2 |
⊢ ( 𝑥 = 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 → ( 𝑇 ‘ 𝑥 ) = ( 𝑇 ‘ 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 ) ) |
44 |
|
df-ov |
⊢ ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) = ( 𝑇 ‘ 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 ) |
45 |
43 44
|
eqtr4di |
⊢ ( 𝑥 = 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 → ( 𝑇 ‘ 𝑥 ) = ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) ) |
46 |
|
fvex |
⊢ ( 𝑆 ‘ 𝑘 ) ∈ V |
47 |
|
vex |
⊢ 𝑘 ∈ V |
48 |
46 47
|
op2ndd |
⊢ ( 𝑥 = 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 → ( 2nd ‘ 𝑥 ) = 𝑘 ) |
49 |
48
|
oveq1d |
⊢ ( 𝑥 = 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 → ( ( 2nd ‘ 𝑥 ) + 1 ) = ( 𝑘 + 1 ) ) |
50 |
45 49
|
breq12d |
⊢ ( 𝑥 = 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 → ( ( 𝑇 ‘ 𝑥 ) 𝐺 ( ( 2nd ‘ 𝑥 ) + 1 ) ↔ ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) 𝐺 ( 𝑘 + 1 ) ) ) |
51 |
|
fveq2 |
⊢ ( 𝑥 = 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 → ( 𝐵 ‘ 𝑥 ) = ( 𝐵 ‘ 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 ) ) |
52 |
|
df-ov |
⊢ ( ( 𝑆 ‘ 𝑘 ) 𝐵 𝑘 ) = ( 𝐵 ‘ 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 ) |
53 |
51 52
|
eqtr4di |
⊢ ( 𝑥 = 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 → ( 𝐵 ‘ 𝑥 ) = ( ( 𝑆 ‘ 𝑘 ) 𝐵 𝑘 ) ) |
54 |
45 49
|
oveq12d |
⊢ ( 𝑥 = 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 → ( ( 𝑇 ‘ 𝑥 ) 𝐵 ( ( 2nd ‘ 𝑥 ) + 1 ) ) = ( ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) 𝐵 ( 𝑘 + 1 ) ) ) |
55 |
53 54
|
ineq12d |
⊢ ( 𝑥 = 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 → ( ( 𝐵 ‘ 𝑥 ) ∩ ( ( 𝑇 ‘ 𝑥 ) 𝐵 ( ( 2nd ‘ 𝑥 ) + 1 ) ) ) = ( ( ( 𝑆 ‘ 𝑘 ) 𝐵 𝑘 ) ∩ ( ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) 𝐵 ( 𝑘 + 1 ) ) ) ) |
56 |
55
|
eleq1d |
⊢ ( 𝑥 = 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 → ( ( ( 𝐵 ‘ 𝑥 ) ∩ ( ( 𝑇 ‘ 𝑥 ) 𝐵 ( ( 2nd ‘ 𝑥 ) + 1 ) ) ) ∈ 𝐾 ↔ ( ( ( 𝑆 ‘ 𝑘 ) 𝐵 𝑘 ) ∩ ( ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) 𝐵 ( 𝑘 + 1 ) ) ) ∈ 𝐾 ) ) |
57 |
50 56
|
anbi12d |
⊢ ( 𝑥 = 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 → ( ( ( 𝑇 ‘ 𝑥 ) 𝐺 ( ( 2nd ‘ 𝑥 ) + 1 ) ∧ ( ( 𝐵 ‘ 𝑥 ) ∩ ( ( 𝑇 ‘ 𝑥 ) 𝐵 ( ( 2nd ‘ 𝑥 ) + 1 ) ) ) ∈ 𝐾 ) ↔ ( ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) 𝐺 ( 𝑘 + 1 ) ∧ ( ( ( 𝑆 ‘ 𝑘 ) 𝐵 𝑘 ) ∩ ( ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) 𝐵 ( 𝑘 + 1 ) ) ) ∈ 𝐾 ) ) ) |
58 |
57
|
rspccv |
⊢ ( ∀ 𝑥 ∈ 𝐺 ( ( 𝑇 ‘ 𝑥 ) 𝐺 ( ( 2nd ‘ 𝑥 ) + 1 ) ∧ ( ( 𝐵 ‘ 𝑥 ) ∩ ( ( 𝑇 ‘ 𝑥 ) 𝐵 ( ( 2nd ‘ 𝑥 ) + 1 ) ) ) ∈ 𝐾 ) → ( 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 ∈ 𝐺 → ( ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) 𝐺 ( 𝑘 + 1 ) ∧ ( ( ( 𝑆 ‘ 𝑘 ) 𝐵 𝑘 ) ∩ ( ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) 𝐵 ( 𝑘 + 1 ) ) ) ∈ 𝐾 ) ) ) |
59 |
8 58
|
syl |
⊢ ( 𝜑 → ( 〈 ( 𝑆 ‘ 𝑘 ) , 𝑘 〉 ∈ 𝐺 → ( ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) 𝐺 ( 𝑘 + 1 ) ∧ ( ( ( 𝑆 ‘ 𝑘 ) 𝐵 𝑘 ) ∩ ( ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) 𝐵 ( 𝑘 + 1 ) ) ) ∈ 𝐾 ) ) ) |
60 |
42 59
|
syl5bi |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑘 ) 𝐺 𝑘 → ( ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) 𝐺 ( 𝑘 + 1 ) ∧ ( ( ( 𝑆 ‘ 𝑘 ) 𝐵 𝑘 ) ∩ ( ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) 𝐵 ( 𝑘 + 1 ) ) ) ∈ 𝐾 ) ) ) |
61 |
|
seqp1 |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 0 ) → ( seq 0 ( 𝑇 , ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( seq 0 ( 𝑇 , ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ) ‘ 𝑘 ) 𝑇 ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ‘ ( 𝑘 + 1 ) ) ) ) |
62 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
63 |
61 62
|
eleq2s |
⊢ ( 𝑘 ∈ ℕ0 → ( seq 0 ( 𝑇 , ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( seq 0 ( 𝑇 , ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ) ‘ 𝑘 ) 𝑇 ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ‘ ( 𝑘 + 1 ) ) ) ) |
64 |
10
|
fveq1i |
⊢ ( 𝑆 ‘ ( 𝑘 + 1 ) ) = ( seq 0 ( 𝑇 , ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ) ‘ ( 𝑘 + 1 ) ) |
65 |
10
|
fveq1i |
⊢ ( 𝑆 ‘ 𝑘 ) = ( seq 0 ( 𝑇 , ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ) ‘ 𝑘 ) |
66 |
65
|
oveq1i |
⊢ ( ( 𝑆 ‘ 𝑘 ) 𝑇 ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ‘ ( 𝑘 + 1 ) ) ) = ( ( seq 0 ( 𝑇 , ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ) ‘ 𝑘 ) 𝑇 ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ‘ ( 𝑘 + 1 ) ) ) |
67 |
63 64 66
|
3eqtr4g |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝑆 ‘ ( 𝑘 + 1 ) ) = ( ( 𝑆 ‘ 𝑘 ) 𝑇 ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ‘ ( 𝑘 + 1 ) ) ) ) |
68 |
|
eqeq1 |
⊢ ( 𝑚 = ( 𝑘 + 1 ) → ( 𝑚 = 0 ↔ ( 𝑘 + 1 ) = 0 ) ) |
69 |
|
oveq1 |
⊢ ( 𝑚 = ( 𝑘 + 1 ) → ( 𝑚 − 1 ) = ( ( 𝑘 + 1 ) − 1 ) ) |
70 |
68 69
|
ifbieq2d |
⊢ ( 𝑚 = ( 𝑘 + 1 ) → if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) = if ( ( 𝑘 + 1 ) = 0 , 𝐶 , ( ( 𝑘 + 1 ) − 1 ) ) ) |
71 |
|
peano2nn0 |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝑘 + 1 ) ∈ ℕ0 ) |
72 |
|
nn0p1nn |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝑘 + 1 ) ∈ ℕ ) |
73 |
|
nnne0 |
⊢ ( ( 𝑘 + 1 ) ∈ ℕ → ( 𝑘 + 1 ) ≠ 0 ) |
74 |
73
|
neneqd |
⊢ ( ( 𝑘 + 1 ) ∈ ℕ → ¬ ( 𝑘 + 1 ) = 0 ) |
75 |
|
iffalse |
⊢ ( ¬ ( 𝑘 + 1 ) = 0 → if ( ( 𝑘 + 1 ) = 0 , 𝐶 , ( ( 𝑘 + 1 ) − 1 ) ) = ( ( 𝑘 + 1 ) − 1 ) ) |
76 |
72 74 75
|
3syl |
⊢ ( 𝑘 ∈ ℕ0 → if ( ( 𝑘 + 1 ) = 0 , 𝐶 , ( ( 𝑘 + 1 ) − 1 ) ) = ( ( 𝑘 + 1 ) − 1 ) ) |
77 |
|
ovex |
⊢ ( ( 𝑘 + 1 ) − 1 ) ∈ V |
78 |
76 77
|
eqeltrdi |
⊢ ( 𝑘 ∈ ℕ0 → if ( ( 𝑘 + 1 ) = 0 , 𝐶 , ( ( 𝑘 + 1 ) − 1 ) ) ∈ V ) |
79 |
37 70 71 78
|
fvmptd3 |
⊢ ( 𝑘 ∈ ℕ0 → ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ‘ ( 𝑘 + 1 ) ) = if ( ( 𝑘 + 1 ) = 0 , 𝐶 , ( ( 𝑘 + 1 ) − 1 ) ) ) |
80 |
|
nn0cn |
⊢ ( 𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ ) |
81 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
82 |
|
pncan |
⊢ ( ( 𝑘 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑘 + 1 ) − 1 ) = 𝑘 ) |
83 |
80 81 82
|
sylancl |
⊢ ( 𝑘 ∈ ℕ0 → ( ( 𝑘 + 1 ) − 1 ) = 𝑘 ) |
84 |
79 76 83
|
3eqtrd |
⊢ ( 𝑘 ∈ ℕ0 → ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ‘ ( 𝑘 + 1 ) ) = 𝑘 ) |
85 |
84
|
oveq2d |
⊢ ( 𝑘 ∈ ℕ0 → ( ( 𝑆 ‘ 𝑘 ) 𝑇 ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ‘ ( 𝑘 + 1 ) ) ) = ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) ) |
86 |
67 85
|
eqtrd |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝑆 ‘ ( 𝑘 + 1 ) ) = ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) ) |
87 |
86
|
breq1d |
⊢ ( 𝑘 ∈ ℕ0 → ( ( 𝑆 ‘ ( 𝑘 + 1 ) ) 𝐺 ( 𝑘 + 1 ) ↔ ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) 𝐺 ( 𝑘 + 1 ) ) ) |
88 |
87
|
biimprd |
⊢ ( 𝑘 ∈ ℕ0 → ( ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) 𝐺 ( 𝑘 + 1 ) → ( 𝑆 ‘ ( 𝑘 + 1 ) ) 𝐺 ( 𝑘 + 1 ) ) ) |
89 |
88
|
adantrd |
⊢ ( 𝑘 ∈ ℕ0 → ( ( ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) 𝐺 ( 𝑘 + 1 ) ∧ ( ( ( 𝑆 ‘ 𝑘 ) 𝐵 𝑘 ) ∩ ( ( ( 𝑆 ‘ 𝑘 ) 𝑇 𝑘 ) 𝐵 ( 𝑘 + 1 ) ) ) ∈ 𝐾 ) → ( 𝑆 ‘ ( 𝑘 + 1 ) ) 𝐺 ( 𝑘 + 1 ) ) ) |
90 |
60 89
|
syl9r |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝜑 → ( ( 𝑆 ‘ 𝑘 ) 𝐺 𝑘 → ( 𝑆 ‘ ( 𝑘 + 1 ) ) 𝐺 ( 𝑘 + 1 ) ) ) ) |
91 |
90
|
a2d |
⊢ ( 𝑘 ∈ ℕ0 → ( ( 𝜑 → ( 𝑆 ‘ 𝑘 ) 𝐺 𝑘 ) → ( 𝜑 → ( 𝑆 ‘ ( 𝑘 + 1 ) ) 𝐺 ( 𝑘 + 1 ) ) ) ) |
92 |
14 18 22 26 41 91
|
nn0ind |
⊢ ( 𝐴 ∈ ℕ0 → ( 𝜑 → ( 𝑆 ‘ 𝐴 ) 𝐺 𝐴 ) ) |
93 |
92
|
impcom |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℕ0 ) → ( 𝑆 ‘ 𝐴 ) 𝐺 𝐴 ) |