| Step | Hyp | Ref
| Expression |
| 1 | | chnwrd.1 |
. . . 4
⊢ (𝜑 → 𝐶 ∈ ( < Chain𝐴)) |
| 2 | 1 | chnwrd 32997 |
. . 3
⊢ (𝜑 → 𝐶 ∈ Word 𝐴) |
| 3 | | pfxcl 14715 |
. . 3
⊢ (𝐶 ∈ Word 𝐴 → (𝐶 prefix 𝐿) ∈ Word 𝐴) |
| 4 | 2, 3 | syl 17 |
. 2
⊢ (𝜑 → (𝐶 prefix 𝐿) ∈ Word 𝐴) |
| 5 | 1 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → 𝐶 ∈ ( < Chain𝐴)) |
| 6 | | pfxchn.2 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐿 ∈ (0...(♯‘𝐶))) |
| 7 | 6 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → 𝐿 ∈ (0...(♯‘𝐶))) |
| 8 | | elfzuz3 13561 |
. . . . . . . . 9
⊢ (𝐿 ∈
(0...(♯‘𝐶))
→ (♯‘𝐶)
∈ (ℤ≥‘𝐿)) |
| 9 | | fzoss2 13727 |
. . . . . . . . 9
⊢
((♯‘𝐶)
∈ (ℤ≥‘𝐿) → (0..^𝐿) ⊆ (0..^(♯‘𝐶))) |
| 10 | 7, 8, 9 | 3syl 18 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → (0..^𝐿) ⊆
(0..^(♯‘𝐶))) |
| 11 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) |
| 12 | 11 | eldifad 3963 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → 𝑛 ∈ dom (𝐶 prefix 𝐿)) |
| 13 | 2 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → 𝐶 ∈ Word 𝐴) |
| 14 | | pfxlen 14721 |
. . . . . . . . . . . . 13
⊢ ((𝐶 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝐶))) → (♯‘(𝐶 prefix 𝐿)) = 𝐿) |
| 15 | 13, 7, 14 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → (♯‘(𝐶 prefix 𝐿)) = 𝐿) |
| 16 | 15 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → 𝐿 = (♯‘(𝐶 prefix 𝐿))) |
| 17 | 4 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → (𝐶 prefix 𝐿) ∈ Word 𝐴) |
| 18 | 16, 17 | wrdfd 32918 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → (𝐶 prefix 𝐿):(0..^𝐿)⟶𝐴) |
| 19 | 18 | fdmd 6746 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → dom (𝐶 prefix 𝐿) = (0..^𝐿)) |
| 20 | 12, 19 | eleqtrd 2843 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → 𝑛 ∈ (0..^𝐿)) |
| 21 | 10, 20 | sseldd 3984 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → 𝑛 ∈ (0..^(♯‘𝐶))) |
| 22 | | eqidd 2738 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → (♯‘𝐶) = (♯‘𝐶)) |
| 23 | 22, 13 | wrdfd 32918 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → 𝐶:(0..^(♯‘𝐶))⟶𝐴) |
| 24 | 23 | fdmd 6746 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → dom 𝐶 = (0..^(♯‘𝐶))) |
| 25 | 21, 24 | eleqtrrd 2844 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → 𝑛 ∈ dom 𝐶) |
| 26 | | eldifsni 4790 |
. . . . . . 7
⊢ (𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0}) → 𝑛 ≠ 0) |
| 27 | 11, 26 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → 𝑛 ≠ 0) |
| 28 | 25, 27 | eldifsnd 4787 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → 𝑛 ∈ (dom 𝐶 ∖ {0})) |
| 29 | 5, 28 | chnltm1 32998 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → (𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)) |
| 30 | 7 | elfzelzd 13565 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → 𝐿 ∈ ℤ) |
| 31 | | fzossrbm1 13728 |
. . . . . . 7
⊢ (𝐿 ∈ ℤ →
(0..^(𝐿 − 1)) ⊆
(0..^𝐿)) |
| 32 | 30, 31 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → (0..^(𝐿 − 1)) ⊆ (0..^𝐿)) |
| 33 | | fzom1ne1 32803 |
. . . . . . 7
⊢ ((𝑛 ∈ (0..^𝐿) ∧ 𝑛 ≠ 0) → (𝑛 − 1) ∈ (0..^(𝐿 − 1))) |
| 34 | 20, 27, 33 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → (𝑛 − 1) ∈ (0..^(𝐿 − 1))) |
| 35 | 32, 34 | sseldd 3984 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → (𝑛 − 1) ∈ (0..^𝐿)) |
| 36 | | pfxfv 14720 |
. . . . 5
⊢ ((𝐶 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝐶)) ∧ (𝑛 − 1) ∈ (0..^𝐿)) → ((𝐶 prefix 𝐿)‘(𝑛 − 1)) = (𝐶‘(𝑛 − 1))) |
| 37 | 13, 7, 35, 36 | syl3anc 1373 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → ((𝐶 prefix 𝐿)‘(𝑛 − 1)) = (𝐶‘(𝑛 − 1))) |
| 38 | | pfxfv 14720 |
. . . . 5
⊢ ((𝐶 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝐶)) ∧ 𝑛 ∈ (0..^𝐿)) → ((𝐶 prefix 𝐿)‘𝑛) = (𝐶‘𝑛)) |
| 39 | 13, 7, 20, 38 | syl3anc 1373 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → ((𝐶 prefix 𝐿)‘𝑛) = (𝐶‘𝑛)) |
| 40 | 29, 37, 39 | 3brtr4d 5175 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})) → ((𝐶 prefix 𝐿)‘(𝑛 − 1)) < ((𝐶 prefix 𝐿)‘𝑛)) |
| 41 | 40 | ralrimiva 3146 |
. 2
⊢ (𝜑 → ∀𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})((𝐶 prefix 𝐿)‘(𝑛 − 1)) < ((𝐶 prefix 𝐿)‘𝑛)) |
| 42 | | ischn 32996 |
. 2
⊢ ((𝐶 prefix 𝐿) ∈ ( < Chain𝐴) ↔ ((𝐶 prefix 𝐿) ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom (𝐶 prefix 𝐿) ∖ {0})((𝐶 prefix 𝐿)‘(𝑛 − 1)) < ((𝐶 prefix 𝐿)‘𝑛))) |
| 43 | 4, 41, 42 | sylanbrc 583 |
1
⊢ (𝜑 → (𝐶 prefix 𝐿) ∈ ( < Chain𝐴)) |