| Step | Hyp | Ref
| Expression |
| 1 | | fveq2 6906 |
. . . . 5
⊢ (𝑥 = 0 → (𝑆‘𝑥) = (𝑆‘0)) |
| 2 | | id 22 |
. . . . 5
⊢ (𝑥 = 0 → 𝑥 = 0) |
| 3 | 1, 2 | breq12d 5156 |
. . . 4
⊢ (𝑥 = 0 → ((𝑆‘𝑥)𝐺𝑥 ↔ (𝑆‘0)𝐺0)) |
| 4 | 3 | imbi2d 340 |
. . 3
⊢ (𝑥 = 0 → ((𝜑 → (𝑆‘𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆‘0)𝐺0))) |
| 5 | | fveq2 6906 |
. . . . 5
⊢ (𝑥 = 𝑘 → (𝑆‘𝑥) = (𝑆‘𝑘)) |
| 6 | | id 22 |
. . . . 5
⊢ (𝑥 = 𝑘 → 𝑥 = 𝑘) |
| 7 | 5, 6 | breq12d 5156 |
. . . 4
⊢ (𝑥 = 𝑘 → ((𝑆‘𝑥)𝐺𝑥 ↔ (𝑆‘𝑘)𝐺𝑘)) |
| 8 | 7 | imbi2d 340 |
. . 3
⊢ (𝑥 = 𝑘 → ((𝜑 → (𝑆‘𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆‘𝑘)𝐺𝑘))) |
| 9 | | fveq2 6906 |
. . . . 5
⊢ (𝑥 = (𝑘 + 1) → (𝑆‘𝑥) = (𝑆‘(𝑘 + 1))) |
| 10 | | id 22 |
. . . . 5
⊢ (𝑥 = (𝑘 + 1) → 𝑥 = (𝑘 + 1)) |
| 11 | 9, 10 | breq12d 5156 |
. . . 4
⊢ (𝑥 = (𝑘 + 1) → ((𝑆‘𝑥)𝐺𝑥 ↔ (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))) |
| 12 | 11 | imbi2d 340 |
. . 3
⊢ (𝑥 = (𝑘 + 1) → ((𝜑 → (𝑆‘𝑥)𝐺𝑥) ↔ (𝜑 → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1)))) |
| 13 | | fveq2 6906 |
. . . . 5
⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴)) |
| 14 | | id 22 |
. . . . 5
⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) |
| 15 | 13, 14 | breq12d 5156 |
. . . 4
⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥)𝐺𝑥 ↔ (𝑆‘𝐴)𝐺𝐴)) |
| 16 | 15 | imbi2d 340 |
. . 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 6907 |
. . . . . 6
⊢ (𝑆‘0) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘0) |
| 19 | | 0z 12624 |
. . . . . . 7
⊢ 0 ∈
ℤ |
| 20 | | seq1 14055 |
. . . . . . 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 2765 |
. . . . 5
⊢ (𝑆‘0) = ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0) |
| 23 | | 0nn0 12541 |
. . . . . 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 | relopabiv 5830 |
. . . . . . . 8
⊢ Rel 𝐺 |
| 27 | 26 | brrelex1i 5741 |
. . . . . . 7
⊢ (𝐶𝐺0 → 𝐶 ∈ V) |
| 28 | 24, 27 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐶 ∈ V) |
| 29 | | iftrue 4531 |
. . . . . . 7
⊢ (𝑚 = 0 → if(𝑚 = 0, 𝐶, (𝑚 − 1)) = 𝐶) |
| 30 | | eqid 2737 |
. . . . . . 7
⊢ (𝑚 ∈ ℕ0
↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))) |
| 31 | 29, 30 | fvmptg 7014 |
. . . . . 6
⊢ ((0
∈ ℕ0 ∧ 𝐶 ∈ V) → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0) = 𝐶) |
| 32 | 23, 28, 31 | sylancr 587 |
. . . . 5
⊢ (𝜑 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘0) = 𝐶) |
| 33 | 22, 32 | eqtrid 2789 |
. . . 4
⊢ (𝜑 → (𝑆‘0) = 𝐶) |
| 34 | 33, 24 | eqbrtrd 5165 |
. . 3
⊢ (𝜑 → (𝑆‘0)𝐺0) |
| 35 | | df-br 5144 |
. . . . . 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 6906 |
. . . . . . . . . . 11
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → (𝑇‘𝑥) = (𝑇‘〈(𝑆‘𝑘), 𝑘〉)) |
| 38 | | df-ov 7434 |
. . . . . . . . . . 11
⊢ ((𝑆‘𝑘)𝑇𝑘) = (𝑇‘〈(𝑆‘𝑘), 𝑘〉) |
| 39 | 37, 38 | eqtr4di 2795 |
. . . . . . . . . 10
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → (𝑇‘𝑥) = ((𝑆‘𝑘)𝑇𝑘)) |
| 40 | | fvex 6919 |
. . . . . . . . . . . 12
⊢ (𝑆‘𝑘) ∈ V |
| 41 | | vex 3484 |
. . . . . . . . . . . 12
⊢ 𝑘 ∈ V |
| 42 | 40, 41 | op2ndd 8025 |
. . . . . . . . . . 11
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → (2nd ‘𝑥) = 𝑘) |
| 43 | 42 | oveq1d 7446 |
. . . . . . . . . 10
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → ((2nd ‘𝑥) + 1) = (𝑘 + 1)) |
| 44 | 39, 43 | breq12d 5156 |
. . . . . . . . 9
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ↔ ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1))) |
| 45 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → (𝐵‘𝑥) = (𝐵‘〈(𝑆‘𝑘), 𝑘〉)) |
| 46 | | df-ov 7434 |
. . . . . . . . . . . 12
⊢ ((𝑆‘𝑘)𝐵𝑘) = (𝐵‘〈(𝑆‘𝑘), 𝑘〉) |
| 47 | 45, 46 | eqtr4di 2795 |
. . . . . . . . . . 11
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → (𝐵‘𝑥) = ((𝑆‘𝑘)𝐵𝑘)) |
| 48 | 39, 43 | oveq12d 7449 |
. . . . . . . . . . 11
⊢ (𝑥 = 〈(𝑆‘𝑘), 𝑘〉 → ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1)) = (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) |
| 49 | 47, 48 | ineq12d 4221 |
. . . . . . . . . 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 2826 |
. . . . . . . . 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 632 |
. . . . . . . 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 3619 |
. . . . . . 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 | biimtrid 242 |
. . . . 5
⊢ (𝜑 → ((𝑆‘𝑘)𝐺𝑘 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾))) |
| 55 | | seqp1 14057 |
. . . . . . . . . . 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 12920 |
. . . . . . . . . . 11
⊢
ℕ0 = (ℤ≥‘0) |
| 57 | 55, 56 | eleq2s 2859 |
. . . . . . . . . 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 6907 |
. . . . . . . . . 10
⊢ (𝑆‘(𝑘 + 1)) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1)) |
| 59 | 17 | fveq1i 6907 |
. . . . . . . . . . 11
⊢ (𝑆‘𝑘) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘) |
| 60 | 59 | oveq1i 7441 |
. . . . . . . . . 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 2802 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ0
→ (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)))) |
| 62 | | eqeq1 2741 |
. . . . . . . . . . . . 13
⊢ (𝑚 = (𝑘 + 1) → (𝑚 = 0 ↔ (𝑘 + 1) = 0)) |
| 63 | | oveq1 7438 |
. . . . . . . . . . . . 13
⊢ (𝑚 = (𝑘 + 1) → (𝑚 − 1) = ((𝑘 + 1) − 1)) |
| 64 | 62, 63 | ifbieq2d 4552 |
. . . . . . . . . . . 12
⊢ (𝑚 = (𝑘 + 1) → if(𝑚 = 0, 𝐶, (𝑚 − 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1))) |
| 65 | | peano2nn0 12566 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ0
→ (𝑘 + 1) ∈
ℕ0) |
| 66 | | nn0p1nn 12565 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ0
→ (𝑘 + 1) ∈
ℕ) |
| 67 | | nnne0 12300 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 + 1) ∈ ℕ →
(𝑘 + 1) ≠
0) |
| 68 | 67 | neneqd 2945 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 + 1) ∈ ℕ →
¬ (𝑘 + 1) =
0) |
| 69 | | iffalse 4534 |
. . . . . . . . . . . . . 14
⊢ (¬
(𝑘 + 1) = 0 →
if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) = ((𝑘 + 1) − 1)) |
| 70 | 66, 68, 69 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ0
→ if((𝑘 + 1) = 0,
𝐶, ((𝑘 + 1) − 1)) = ((𝑘 + 1) − 1)) |
| 71 | | ovex 7464 |
. . . . . . . . . . . . 13
⊢ ((𝑘 + 1) − 1) ∈
V |
| 72 | 70, 71 | eqeltrdi 2849 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ0
→ if((𝑘 + 1) = 0,
𝐶, ((𝑘 + 1) − 1)) ∈ V) |
| 73 | 30, 64, 65, 72 | fvmptd3 7039 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ0
→ ((𝑚 ∈
ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1))) |
| 74 | | nn0cn 12536 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℂ) |
| 75 | | ax-1cn 11213 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℂ |
| 76 | | pncan 11514 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑘 + 1)
− 1) = 𝑘) |
| 77 | 74, 75, 76 | sylancl 586 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ0
→ ((𝑘 + 1) − 1)
= 𝑘) |
| 78 | 73, 70, 77 | 3eqtrd 2781 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ0
→ ((𝑚 ∈
ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = 𝑘) |
| 79 | 78 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ0
→ ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) = ((𝑆‘𝑘)𝑇𝑘)) |
| 80 | 61, 79 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ0
→ (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇𝑘)) |
| 81 | 80 | breq1d 5153 |
. . . . . . 7
⊢ (𝑘 ∈ ℕ0
→ ((𝑆‘(𝑘 + 1))𝐺(𝑘 + 1) ↔ ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1))) |
| 82 | 81 | biimprd 248 |
. . . . . 6
⊢ (𝑘 ∈ ℕ0
→ (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))) |
| 83 | 82 | adantrd 491 |
. . . . 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))) |
| 84 | 54, 83 | syl9r 78 |
. . . 4
⊢ (𝑘 ∈ ℕ0
→ (𝜑 → ((𝑆‘𝑘)𝐺𝑘 → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1)))) |
| 85 | 84 | a2d 29 |
. . 3
⊢ (𝑘 ∈ ℕ0
→ ((𝜑 → (𝑆‘𝑘)𝐺𝑘) → (𝜑 → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1)))) |
| 86 | 4, 8, 12, 16, 34, 85 | nn0ind 12713 |
. 2
⊢ (𝐴 ∈ ℕ0
→ (𝜑 → (𝑆‘𝐴)𝐺𝐴)) |
| 87 | 86 | impcom 407 |
1
⊢ ((𝜑 ∧ 𝐴 ∈ ℕ0) → (𝑆‘𝐴)𝐺𝐴) |