Proof of Theorem repsdf2
Step | Hyp | Ref
| Expression |
1 | | repsconst 14337 |
. . 3
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = ((0..^𝑁) × {𝑆})) |
2 | 1 | eqeq2d 2748 |
. 2
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑊 = (𝑆 repeatS 𝑁) ↔ 𝑊 = ((0..^𝑁) × {𝑆}))) |
3 | | fconst2g 7018 |
. . 3
⊢ (𝑆 ∈ 𝑉 → (𝑊:(0..^𝑁)⟶{𝑆} ↔ 𝑊 = ((0..^𝑁) × {𝑆}))) |
4 | 3 | adantr 484 |
. 2
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑊:(0..^𝑁)⟶{𝑆} ↔ 𝑊 = ((0..^𝑁) × {𝑆}))) |
5 | | fconstfv 7028 |
. . . . . . . . 9
⊢ (𝑊:(0..^𝑁)⟶{𝑆} ↔ (𝑊 Fn (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆)) |
6 | | snssi 4721 |
. . . . . . . . . . . . . 14
⊢ (𝑆 ∈ 𝑉 → {𝑆} ⊆ 𝑉) |
7 | 6 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → {𝑆} ⊆ 𝑉) |
8 | 7 | anim1ci 619 |
. . . . . . . . . . . 12
⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ 𝑊:(0..^𝑁)⟶{𝑆}) → (𝑊:(0..^𝑁)⟶{𝑆} ∧ {𝑆} ⊆ 𝑉)) |
9 | | fss 6562 |
. . . . . . . . . . . 12
⊢ ((𝑊:(0..^𝑁)⟶{𝑆} ∧ {𝑆} ⊆ 𝑉) → 𝑊:(0..^𝑁)⟶𝑉) |
10 | | iswrdi 14073 |
. . . . . . . . . . . 12
⊢ (𝑊:(0..^𝑁)⟶𝑉 → 𝑊 ∈ Word 𝑉) |
11 | 8, 9, 10 | 3syl 18 |
. . . . . . . . . . 11
⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ 𝑊:(0..^𝑁)⟶{𝑆}) → 𝑊 ∈ Word 𝑉) |
12 | | ffzo0hash 14013 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ0
∧ 𝑊 Fn (0..^𝑁)) → (♯‘𝑊) = 𝑁) |
13 | 12 | expcom 417 |
. . . . . . . . . . . . . 14
⊢ (𝑊 Fn (0..^𝑁) → (𝑁 ∈ ℕ0 →
(♯‘𝑊) = 𝑁)) |
14 | | ffn 6545 |
. . . . . . . . . . . . . 14
⊢ (𝑊:(0..^𝑁)⟶{𝑆} → 𝑊 Fn (0..^𝑁)) |
15 | 13, 14 | syl11 33 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ0
→ (𝑊:(0..^𝑁)⟶{𝑆} → (♯‘𝑊) = 𝑁)) |
16 | 15 | adantl 485 |
. . . . . . . . . . . 12
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑊:(0..^𝑁)⟶{𝑆} → (♯‘𝑊) = 𝑁)) |
17 | 16 | imp 410 |
. . . . . . . . . . 11
⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ 𝑊:(0..^𝑁)⟶{𝑆}) → (♯‘𝑊) = 𝑁) |
18 | 11, 17 | jca 515 |
. . . . . . . . . 10
⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ 𝑊:(0..^𝑁)⟶{𝑆}) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁)) |
19 | 18 | ex 416 |
. . . . . . . . 9
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑊:(0..^𝑁)⟶{𝑆} → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁))) |
20 | 5, 19 | syl5bir 246 |
. . . . . . . 8
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ((𝑊 Fn (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁))) |
21 | 20 | expcomd 420 |
. . . . . . 7
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) →
(∀𝑖 ∈
(0..^𝑁)(𝑊‘𝑖) = 𝑆 → (𝑊 Fn (0..^𝑁) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁)))) |
22 | 21 | imp 410 |
. . . . . 6
⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧
∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆) → (𝑊 Fn (0..^𝑁) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁))) |
23 | | wrdf 14074 |
. . . . . . . . . 10
⊢ (𝑊 ∈ Word 𝑉 → 𝑊:(0..^(♯‘𝑊))⟶𝑉) |
24 | | ffn 6545 |
. . . . . . . . . 10
⊢ (𝑊:(0..^(♯‘𝑊))⟶𝑉 → 𝑊 Fn (0..^(♯‘𝑊))) |
25 | | oveq2 7221 |
. . . . . . . . . . . . . 14
⊢
((♯‘𝑊) =
𝑁 →
(0..^(♯‘𝑊)) =
(0..^𝑁)) |
26 | 25 | fneq2d 6473 |
. . . . . . . . . . . . 13
⊢
((♯‘𝑊) =
𝑁 → (𝑊 Fn (0..^(♯‘𝑊)) ↔ 𝑊 Fn (0..^𝑁))) |
27 | 26 | biimpd 232 |
. . . . . . . . . . . 12
⊢
((♯‘𝑊) =
𝑁 → (𝑊 Fn (0..^(♯‘𝑊)) → 𝑊 Fn (0..^𝑁))) |
28 | 27 | a1d 25 |
. . . . . . . . . . 11
⊢
((♯‘𝑊) =
𝑁 → ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑊 Fn (0..^(♯‘𝑊)) → 𝑊 Fn (0..^𝑁)))) |
29 | 28 | com13 88 |
. . . . . . . . . 10
⊢ (𝑊 Fn (0..^(♯‘𝑊)) → ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) →
((♯‘𝑊) = 𝑁 → 𝑊 Fn (0..^𝑁)))) |
30 | 23, 24, 29 | 3syl 18 |
. . . . . . . . 9
⊢ (𝑊 ∈ Word 𝑉 → ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) →
((♯‘𝑊) = 𝑁 → 𝑊 Fn (0..^𝑁)))) |
31 | 30 | com12 32 |
. . . . . . . 8
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑊 ∈ Word 𝑉 → ((♯‘𝑊) = 𝑁 → 𝑊 Fn (0..^𝑁)))) |
32 | 31 | impd 414 |
. . . . . . 7
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) → 𝑊 Fn (0..^𝑁))) |
33 | 32 | adantr 484 |
. . . . . 6
⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧
∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) → 𝑊 Fn (0..^𝑁))) |
34 | 22, 33 | impbid 215 |
. . . . 5
⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧
∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆) → (𝑊 Fn (0..^𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁))) |
35 | 34 | ex 416 |
. . . 4
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) →
(∀𝑖 ∈
(0..^𝑁)(𝑊‘𝑖) = 𝑆 → (𝑊 Fn (0..^𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁)))) |
36 | 35 | pm5.32rd 581 |
. . 3
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ((𝑊 Fn (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆) ↔ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆))) |
37 | | df-3an 1091 |
. . 3
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆) ↔ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆)) |
38 | 36, 5, 37 | 3bitr4g 317 |
. 2
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑊:(0..^𝑁)⟶{𝑆} ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆))) |
39 | 2, 4, 38 | 3bitr2d 310 |
1
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑊 = (𝑆 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆))) |