| Step | Hyp | Ref
| Expression |
| 1 | | efgredlem.5 |
. . . . 5
⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0)) |
| 2 | | efgredlem.3 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ dom 𝑆) |
| 3 | | efgval.w |
. . . . . . . . . 10
⊢ 𝑊 = ( I ‘Word (𝐼 ×
2o)) |
| 4 | | efgval.r |
. . . . . . . . . 10
⊢ ∼ = (
~FG ‘𝐼) |
| 5 | | efgval2.m |
. . . . . . . . . 10
⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) |
| 6 | | efgval2.t |
. . . . . . . . . 10
⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) |
| 7 | | efgred.d |
. . . . . . . . . 10
⊢ 𝐷 = (𝑊 ∖ ∪
𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) |
| 8 | | efgred.s |
. . . . . . . . . 10
⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈
(1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) |
| 9 | 3, 4, 5, 6, 7, 8 | efgsval 19749 |
. . . . . . . . 9
⊢ (𝐵 ∈ dom 𝑆 → (𝑆‘𝐵) = (𝐵‘((♯‘𝐵) − 1))) |
| 10 | 2, 9 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑆‘𝐵) = (𝐵‘((♯‘𝐵) − 1))) |
| 11 | | efgredlem.4 |
. . . . . . . . 9
⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵)) |
| 12 | | efgredlem.2 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈ dom 𝑆) |
| 13 | 3, 4, 5, 6, 7, 8 | efgsval 19749 |
. . . . . . . . . 10
⊢ (𝐴 ∈ dom 𝑆 → (𝑆‘𝐴) = (𝐴‘((♯‘𝐴) − 1))) |
| 14 | 12, 13 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝑆‘𝐴) = (𝐴‘((♯‘𝐴) − 1))) |
| 15 | 11, 14 | eqtr3d 2779 |
. . . . . . . 8
⊢ (𝜑 → (𝑆‘𝐵) = (𝐴‘((♯‘𝐴) − 1))) |
| 16 | 10, 15 | eqtr3d 2779 |
. . . . . . 7
⊢ (𝜑 → (𝐵‘((♯‘𝐵) − 1)) = (𝐴‘((♯‘𝐴) − 1))) |
| 17 | | oveq1 7438 |
. . . . . . . . 9
⊢
((♯‘𝐴) =
1 → ((♯‘𝐴)
− 1) = (1 − 1)) |
| 18 | | 1m1e0 12338 |
. . . . . . . . 9
⊢ (1
− 1) = 0 |
| 19 | 17, 18 | eqtrdi 2793 |
. . . . . . . 8
⊢
((♯‘𝐴) =
1 → ((♯‘𝐴)
− 1) = 0) |
| 20 | 19 | fveq2d 6910 |
. . . . . . 7
⊢
((♯‘𝐴) =
1 → (𝐴‘((♯‘𝐴) − 1)) = (𝐴‘0)) |
| 21 | 16, 20 | sylan9eq 2797 |
. . . . . 6
⊢ ((𝜑 ∧ (♯‘𝐴) = 1) → (𝐵‘((♯‘𝐵) − 1)) = (𝐴‘0)) |
| 22 | 11 | eleq1d 2826 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑆‘𝐴) ∈ 𝐷 ↔ (𝑆‘𝐵) ∈ 𝐷)) |
| 23 | 3, 4, 5, 6, 7, 8 | efgs1b 19754 |
. . . . . . . . . 10
⊢ (𝐴 ∈ dom 𝑆 → ((𝑆‘𝐴) ∈ 𝐷 ↔ (♯‘𝐴) = 1)) |
| 24 | 12, 23 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑆‘𝐴) ∈ 𝐷 ↔ (♯‘𝐴) = 1)) |
| 25 | 3, 4, 5, 6, 7, 8 | efgs1b 19754 |
. . . . . . . . . 10
⊢ (𝐵 ∈ dom 𝑆 → ((𝑆‘𝐵) ∈ 𝐷 ↔ (♯‘𝐵) = 1)) |
| 26 | 2, 25 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑆‘𝐵) ∈ 𝐷 ↔ (♯‘𝐵) = 1)) |
| 27 | 22, 24, 26 | 3bitr3d 309 |
. . . . . . . 8
⊢ (𝜑 → ((♯‘𝐴) = 1 ↔
(♯‘𝐵) =
1)) |
| 28 | 27 | biimpa 476 |
. . . . . . 7
⊢ ((𝜑 ∧ (♯‘𝐴) = 1) →
(♯‘𝐵) =
1) |
| 29 | | oveq1 7438 |
. . . . . . . . 9
⊢
((♯‘𝐵) =
1 → ((♯‘𝐵)
− 1) = (1 − 1)) |
| 30 | 29, 18 | eqtrdi 2793 |
. . . . . . . 8
⊢
((♯‘𝐵) =
1 → ((♯‘𝐵)
− 1) = 0) |
| 31 | 30 | fveq2d 6910 |
. . . . . . 7
⊢
((♯‘𝐵) =
1 → (𝐵‘((♯‘𝐵) − 1)) = (𝐵‘0)) |
| 32 | 28, 31 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ (♯‘𝐴) = 1) → (𝐵‘((♯‘𝐵) − 1)) = (𝐵‘0)) |
| 33 | 21, 32 | eqtr3d 2779 |
. . . . 5
⊢ ((𝜑 ∧ (♯‘𝐴) = 1) → (𝐴‘0) = (𝐵‘0)) |
| 34 | 1, 33 | mtand 816 |
. . . 4
⊢ (𝜑 → ¬ (♯‘𝐴) = 1) |
| 35 | 3, 4, 5, 6, 7, 8 | efgsdm 19748 |
. . . . . . . 8
⊢ (𝐴 ∈ dom 𝑆 ↔ (𝐴 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐴‘0) ∈ 𝐷 ∧ ∀𝑢 ∈ (1..^(♯‘𝐴))(𝐴‘𝑢) ∈ ran (𝑇‘(𝐴‘(𝑢 − 1))))) |
| 36 | 35 | simp1bi 1146 |
. . . . . . 7
⊢ (𝐴 ∈ dom 𝑆 → 𝐴 ∈ (Word 𝑊 ∖ {∅})) |
| 37 | | eldifsn 4786 |
. . . . . . . 8
⊢ (𝐴 ∈ (Word 𝑊 ∖ {∅}) ↔ (𝐴 ∈ Word 𝑊 ∧ 𝐴 ≠ ∅)) |
| 38 | | lennncl 14572 |
. . . . . . . 8
⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐴 ≠ ∅) → (♯‘𝐴) ∈
ℕ) |
| 39 | 37, 38 | sylbi 217 |
. . . . . . 7
⊢ (𝐴 ∈ (Word 𝑊 ∖ {∅}) →
(♯‘𝐴) ∈
ℕ) |
| 40 | 12, 36, 39 | 3syl 18 |
. . . . . 6
⊢ (𝜑 → (♯‘𝐴) ∈
ℕ) |
| 41 | | elnn1uz2 12967 |
. . . . . 6
⊢
((♯‘𝐴)
∈ ℕ ↔ ((♯‘𝐴) = 1 ∨ (♯‘𝐴) ∈
(ℤ≥‘2))) |
| 42 | 40, 41 | sylib 218 |
. . . . 5
⊢ (𝜑 → ((♯‘𝐴) = 1 ∨ (♯‘𝐴) ∈
(ℤ≥‘2))) |
| 43 | 42 | ord 865 |
. . . 4
⊢ (𝜑 → (¬
(♯‘𝐴) = 1
→ (♯‘𝐴)
∈ (ℤ≥‘2))) |
| 44 | 34, 43 | mpd 15 |
. . 3
⊢ (𝜑 → (♯‘𝐴) ∈
(ℤ≥‘2)) |
| 45 | | uz2m1nn 12965 |
. . 3
⊢
((♯‘𝐴)
∈ (ℤ≥‘2) → ((♯‘𝐴) − 1) ∈
ℕ) |
| 46 | 44, 45 | syl 17 |
. 2
⊢ (𝜑 → ((♯‘𝐴) − 1) ∈
ℕ) |
| 47 | 34, 27 | mtbid 324 |
. . . 4
⊢ (𝜑 → ¬ (♯‘𝐵) = 1) |
| 48 | 3, 4, 5, 6, 7, 8 | efgsdm 19748 |
. . . . . . . 8
⊢ (𝐵 ∈ dom 𝑆 ↔ (𝐵 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐵‘0) ∈ 𝐷 ∧ ∀𝑢 ∈ (1..^(♯‘𝐵))(𝐵‘𝑢) ∈ ran (𝑇‘(𝐵‘(𝑢 − 1))))) |
| 49 | 48 | simp1bi 1146 |
. . . . . . 7
⊢ (𝐵 ∈ dom 𝑆 → 𝐵 ∈ (Word 𝑊 ∖ {∅})) |
| 50 | | eldifsn 4786 |
. . . . . . . 8
⊢ (𝐵 ∈ (Word 𝑊 ∖ {∅}) ↔ (𝐵 ∈ Word 𝑊 ∧ 𝐵 ≠ ∅)) |
| 51 | | lennncl 14572 |
. . . . . . . 8
⊢ ((𝐵 ∈ Word 𝑊 ∧ 𝐵 ≠ ∅) → (♯‘𝐵) ∈
ℕ) |
| 52 | 50, 51 | sylbi 217 |
. . . . . . 7
⊢ (𝐵 ∈ (Word 𝑊 ∖ {∅}) →
(♯‘𝐵) ∈
ℕ) |
| 53 | 2, 49, 52 | 3syl 18 |
. . . . . 6
⊢ (𝜑 → (♯‘𝐵) ∈
ℕ) |
| 54 | | elnn1uz2 12967 |
. . . . . 6
⊢
((♯‘𝐵)
∈ ℕ ↔ ((♯‘𝐵) = 1 ∨ (♯‘𝐵) ∈
(ℤ≥‘2))) |
| 55 | 53, 54 | sylib 218 |
. . . . 5
⊢ (𝜑 → ((♯‘𝐵) = 1 ∨ (♯‘𝐵) ∈
(ℤ≥‘2))) |
| 56 | 55 | ord 865 |
. . . 4
⊢ (𝜑 → (¬
(♯‘𝐵) = 1
→ (♯‘𝐵)
∈ (ℤ≥‘2))) |
| 57 | 47, 56 | mpd 15 |
. . 3
⊢ (𝜑 → (♯‘𝐵) ∈
(ℤ≥‘2)) |
| 58 | | uz2m1nn 12965 |
. . 3
⊢
((♯‘𝐵)
∈ (ℤ≥‘2) → ((♯‘𝐵) − 1) ∈
ℕ) |
| 59 | 57, 58 | syl 17 |
. 2
⊢ (𝜑 → ((♯‘𝐵) − 1) ∈
ℕ) |
| 60 | 46, 59 | jca 511 |
1
⊢ (𝜑 → (((♯‘𝐴) − 1) ∈ ℕ
∧ ((♯‘𝐵)
− 1) ∈ ℕ)) |