Step | Hyp | Ref
| Expression |
1 | | fveq2 6103 |
. . . . 5
⊢ (𝑥 = 0 → (𝑆‘𝑥) = (𝑆‘0)) |
2 | | id 22 |
. . . . 5
⊢ (𝑥 = 0 → 𝑥 = 0) |
3 | 1, 2 | breq12d 4596 |
. . . 4
⊢ (𝑥 = 0 → ((𝑆‘𝑥)𝐺𝑥 ↔ (𝑆‘0)𝐺0)) |
4 | 3 | imbi2d 329 |
. . 3
⊢ (𝑥 = 0 → ((𝜑 → (𝑆‘𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆‘0)𝐺0))) |
5 | | fveq2 6103 |
. . . . 5
⊢ (𝑥 = 𝑘 → (𝑆‘𝑥) = (𝑆‘𝑘)) |
6 | | id 22 |
. . . . 5
⊢ (𝑥 = 𝑘 → 𝑥 = 𝑘) |
7 | 5, 6 | breq12d 4596 |
. . . 4
⊢ (𝑥 = 𝑘 → ((𝑆‘𝑥)𝐺𝑥 ↔ (𝑆‘𝑘)𝐺𝑘)) |
8 | 7 | imbi2d 329 |
. . 3
⊢ (𝑥 = 𝑘 → ((𝜑 → (𝑆‘𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆‘𝑘)𝐺𝑘))) |
9 | | fveq2 6103 |
. . . . 5
⊢ (𝑥 = (𝑘 + 1) → (𝑆‘𝑥) = (𝑆‘(𝑘 + 1))) |
10 | | id 22 |
. . . . 5
⊢ (𝑥 = (𝑘 + 1) → 𝑥 = (𝑘 + 1)) |
11 | 9, 10 | breq12d 4596 |
. . . 4
⊢ (𝑥 = (𝑘 + 1) → ((𝑆‘𝑥)𝐺𝑥 ↔ (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))) |
12 | 11 | imbi2d 329 |
. . 3
⊢ (𝑥 = (𝑘 + 1) → ((𝜑 → (𝑆‘𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1)))) |
13 | | fveq2 6103 |
. . . . 5
⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴)) |
14 | | id 22 |
. . . . 5
⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) |
15 | 13, 14 | breq12d 4596 |
. . . 4
⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥)𝐺𝑥 ↔ (𝑆‘𝐴)𝐺𝐴)) |
16 | 15 | imbi2d 329 |
. . 3
⊢ (𝑥 = 𝐴 → ((𝜑 → (𝑆‘𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆‘𝐴)𝐺𝐴))) |
17 | | heibor.11 |
. . . . . . 7
⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))) |
18 | 17 | fveq1i 6104 |
. . . . . 6
⊢ (𝑆‘0) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘0) |
19 | | 0z 11265 |
. . . . . . 7
⊢ 0 ∈
ℤ |
20 | | seq1 12676 |
. . . . . . 7
⊢ (0 ∈
ℤ → (seq0(𝑇,
(𝑚 ∈
ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘0) = ((𝑚 ∈ ℕ0
↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0)) |
21 | 19, 20 | ax-mp 5 |
. . . . . 6
⊢
(seq0(𝑇, (𝑚 ∈ ℕ0
↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘0) = ((𝑚 ∈ ℕ0
↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0) |
22 | 18, 21 | eqtri 2632 |
. . . . 5
⊢ (𝑆‘0) = ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0) |
23 | | 0nn0 11184 |
. . . . . 6
⊢ 0 ∈
ℕ0 |
24 | | heibor.10 |
. . . . . . 7
⊢ (𝜑 → 𝐶𝐺0) |
25 | | heibor.4 |
. . . . . . . . 9
⊢ 𝐺 = {〈𝑦, 𝑛〉 ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} |
26 | 25 | relopabi 5167 |
. . . . . . . 8
⊢ Rel 𝐺 |
27 | 26 | brrelexi 5082 |
. . . . . . 7
⊢ (𝐶𝐺0 → 𝐶 ∈ V) |
28 | 24, 27 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐶 ∈ V) |
29 | | iftrue 4042 |
. . . . . . 7
⊢ (𝑚 = 0 → if(𝑚 = 0, 𝐶, (𝑚 − 1)) = 𝐶) |
30 | | eqid 2610 |
. . . . . . 7
⊢ (𝑚 ∈ ℕ0
↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))) |
31 | 29, 30 | fvmptg 6189 |
. . . . . 6
⊢ ((0
∈ ℕ0 ∧ 𝐶 ∈ V) → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0) = 𝐶) |
32 | 23, 28, 31 | sylancr 694 |
. . . . 5
⊢ (𝜑 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0) = 𝐶) |
33 | 22, 32 | syl5eq 2656 |
. . . 4
⊢ (𝜑 → (𝑆‘0) = 𝐶) |
34 | 33, 24 | eqbrtrd 4605 |
. . 3
⊢ (𝜑 → (𝑆‘0)𝐺0) |
35 | | df-br 4584 |
. . . . . 6
⊢ ((𝑆‘𝑘)𝐺𝑘 ↔ 〈(𝑆‘𝑘), 𝑘〉 ∈ 𝐺) |
36 | | heibor.9 |
. . . . . . 7
⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) |
37 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → (𝑇‘𝑥) = (𝑇‘〈(𝑆‘𝑘), 𝑘〉)) |
38 | | df-ov 6552 |
. . . . . . . . . . 11
⊢ ((𝑆‘𝑘)𝑇𝑘) = (𝑇‘〈(𝑆‘𝑘), 𝑘〉) |
39 | 37, 38 | syl6eqr 2662 |
. . . . . . . . . 10
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → (𝑇‘𝑥) = ((𝑆‘𝑘)𝑇𝑘)) |
40 | | fvex 6113 |
. . . . . . . . . . . 12
⊢ (𝑆‘𝑘) ∈ V |
41 | | vex 3176 |
. . . . . . . . . . . 12
⊢ 𝑘 ∈ V |
42 | 40, 41 | op2ndd 7070 |
. . . . . . . . . . 11
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → (2nd ‘𝑥) = 𝑘) |
43 | 42 | oveq1d 6564 |
. . . . . . . . . 10
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → ((2nd ‘𝑥) + 1) = (𝑘 + 1)) |
44 | 39, 43 | breq12d 4596 |
. . . . . . . . 9
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ↔ ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1))) |
45 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → (𝐵‘𝑥) = (𝐵‘〈(𝑆‘𝑘), 𝑘〉)) |
46 | | df-ov 6552 |
. . . . . . . . . . . 12
⊢ ((𝑆‘𝑘)𝐵𝑘) = (𝐵‘〈(𝑆‘𝑘), 𝑘〉) |
47 | 45, 46 | syl6eqr 2662 |
. . . . . . . . . . 11
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → (𝐵‘𝑥) = ((𝑆‘𝑘)𝐵𝑘)) |
48 | 39, 43 | oveq12d 6567 |
. . . . . . . . . . 11
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1)) = (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) |
49 | 47, 48 | ineq12d 3777 |
. . . . . . . . . 10
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) = (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)))) |
50 | 49 | eleq1d 2672 |
. . . . . . . . 9
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → (((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾 ↔ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)) |
51 | 44, 50 | anbi12d 743 |
. . . . . . . 8
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → (((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾) ↔ (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾))) |
52 | 51 | rspccv 3279 |
. . . . . . 7
⊢
(∀𝑥 ∈
𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾) → (〈(𝑆‘𝑘), 𝑘〉 ∈ 𝐺 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾))) |
53 | 36, 52 | syl 17 |
. . . . . 6
⊢ (𝜑 → (〈(𝑆‘𝑘), 𝑘〉 ∈ 𝐺 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾))) |
54 | 35, 53 | syl5bi 231 |
. . . . 5
⊢ (𝜑 → ((𝑆‘𝑘)𝐺𝑘 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾))) |
55 | | seqp1 12678 |
. . . . . . . . . . 11
⊢ (𝑘 ∈
(ℤ≥‘0) → (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1)) = ((seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)))) |
56 | | nn0uz 11598 |
. . . . . . . . . . 11
⊢
ℕ0 = (ℤ≥‘0) |
57 | 55, 56 | eleq2s 2706 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ0
→ (seq0(𝑇, (𝑚 ∈ ℕ0
↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1)) = ((seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)))) |
58 | 17 | fveq1i 6104 |
. . . . . . . . . 10
⊢ (𝑆‘(𝑘 + 1)) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1)) |
59 | 17 | fveq1i 6104 |
. . . . . . . . . . 11
⊢ (𝑆‘𝑘) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘) |
60 | 59 | oveq1i 6559 |
. . . . . . . . . 10
⊢ ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) = ((seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) |
61 | 57, 58, 60 | 3eqtr4g 2669 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ0
→ (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)))) |
62 | | peano2nn0 11210 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ0
→ (𝑘 + 1) ∈
ℕ0) |
63 | | nn0p1nn 11209 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ0
→ (𝑘 + 1) ∈
ℕ) |
64 | | nnne0 10930 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 + 1) ∈ ℕ →
(𝑘 + 1) ≠
0) |
65 | 64 | neneqd 2787 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 + 1) ∈ ℕ →
¬ (𝑘 + 1) =
0) |
66 | | iffalse 4045 |
. . . . . . . . . . . . . 14
⊢ (¬
(𝑘 + 1) = 0 →
if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) = ((𝑘 + 1) − 1)) |
67 | 63, 65, 66 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ0
→ if((𝑘 + 1) = 0,
𝐶, ((𝑘 + 1) − 1)) = ((𝑘 + 1) − 1)) |
68 | | ovex 6577 |
. . . . . . . . . . . . 13
⊢ ((𝑘 + 1) − 1) ∈
V |
69 | 67, 68 | syl6eqel 2696 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ0
→ if((𝑘 + 1) = 0,
𝐶, ((𝑘 + 1) − 1)) ∈ V) |
70 | | eqeq1 2614 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = (𝑘 + 1) → (𝑚 = 0 ↔ (𝑘 + 1) = 0)) |
71 | | oveq1 6556 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = (𝑘 + 1) → (𝑚 − 1) = ((𝑘 + 1) − 1)) |
72 | 70, 71 | ifbieq2d 4061 |
. . . . . . . . . . . . 13
⊢ (𝑚 = (𝑘 + 1) → if(𝑚 = 0, 𝐶, (𝑚 − 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1))) |
73 | 72, 30 | fvmptg 6189 |
. . . . . . . . . . . 12
⊢ (((𝑘 + 1) ∈ ℕ0
∧ if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) ∈ V) → ((𝑚 ∈ ℕ0
↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1))) |
74 | 62, 69, 73 | syl2anc 691 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ0
→ ((𝑚 ∈
ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1))) |
75 | | nn0cn 11179 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℂ) |
76 | | ax-1cn 9873 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℂ |
77 | | pncan 10166 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑘 + 1)
− 1) = 𝑘) |
78 | 75, 76, 77 | sylancl 693 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ0
→ ((𝑘 + 1) − 1)
= 𝑘) |
79 | 74, 67, 78 | 3eqtrd 2648 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ0
→ ((𝑚 ∈
ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = 𝑘) |
80 | 79 | oveq2d 6565 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ0
→ ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) = ((𝑆‘𝑘)𝑇𝑘)) |
81 | 61, 80 | eqtrd 2644 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ0
→ (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇𝑘)) |
82 | 81 | breq1d 4593 |
. . . . . . 7
⊢ (𝑘 ∈ ℕ0
→ ((𝑆‘(𝑘 + 1))𝐺(𝑘 + 1) ↔ ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1))) |
83 | 82 | biimprd 237 |
. . . . . 6
⊢ (𝑘 ∈ ℕ0
→ (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))) |
84 | 83 | adantrd 483 |
. . . . 5
⊢ (𝑘 ∈ ℕ0
→ ((((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾) → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))) |
85 | 54, 84 | syl9r 76 |
. . . 4
⊢ (𝑘 ∈ ℕ0
→ (𝜑 → ((𝑆‘𝑘)𝐺𝑘 → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1)))) |
86 | 85 | a2d 29 |
. . 3
⊢ (𝑘 ∈ ℕ0
→ ((𝜑 → (𝑆‘𝑘)𝐺𝑘) → (𝜑 → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1)))) |
87 | 4, 8, 12, 16, 34, 86 | nn0ind 11348 |
. 2
⊢ (𝐴 ∈ ℕ0
→ (𝜑 → (𝑆‘𝐴)𝐺𝐴)) |
88 | 87 | impcom 445 |
1
⊢ ((𝜑 ∧ 𝐴 ∈ ℕ0) → (𝑆‘𝐴)𝐺𝐴) |