Step |
Hyp |
Ref |
Expression |
1 |
|
efgval.w |
⊢ 𝑊 = ( I ‘ Word ( 𝐼 × 2o ) ) |
2 |
|
efgval.r |
⊢ ∼ = ( ~FG ‘ 𝐼 ) |
3 |
|
efgval2.m |
⊢ 𝑀 = ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ 〈 𝑦 , ( 1o ∖ 𝑧 ) 〉 ) |
4 |
|
efgval2.t |
⊢ 𝑇 = ( 𝑣 ∈ 𝑊 ↦ ( 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) , 𝑤 ∈ ( 𝐼 × 2o ) ↦ ( 𝑣 splice 〈 𝑛 , 𝑛 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 ) ) ) |
5 |
|
efgred.d |
⊢ 𝐷 = ( 𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran ( 𝑇 ‘ 𝑥 ) ) |
6 |
|
efgred.s |
⊢ 𝑆 = ( 𝑚 ∈ { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } ↦ ( 𝑚 ‘ ( ( ♯ ‘ 𝑚 ) − 1 ) ) ) |
7 |
|
efgredlem.1 |
⊢ ( 𝜑 → ∀ 𝑎 ∈ dom 𝑆 ∀ 𝑏 ∈ dom 𝑆 ( ( ♯ ‘ ( 𝑆 ‘ 𝑎 ) ) < ( ♯ ‘ ( 𝑆 ‘ 𝐴 ) ) → ( ( 𝑆 ‘ 𝑎 ) = ( 𝑆 ‘ 𝑏 ) → ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) ) ) ) |
8 |
|
efgredlem.2 |
⊢ ( 𝜑 → 𝐴 ∈ dom 𝑆 ) |
9 |
|
efgredlem.3 |
⊢ ( 𝜑 → 𝐵 ∈ dom 𝑆 ) |
10 |
|
efgredlem.4 |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐴 ) = ( 𝑆 ‘ 𝐵 ) ) |
11 |
|
efgredlem.5 |
⊢ ( 𝜑 → ¬ ( 𝐴 ‘ 0 ) = ( 𝐵 ‘ 0 ) ) |
12 |
1 2 3 4 5 6
|
efgsval |
⊢ ( 𝐵 ∈ dom 𝑆 → ( 𝑆 ‘ 𝐵 ) = ( 𝐵 ‘ ( ( ♯ ‘ 𝐵 ) − 1 ) ) ) |
13 |
9 12
|
syl |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐵 ) = ( 𝐵 ‘ ( ( ♯ ‘ 𝐵 ) − 1 ) ) ) |
14 |
1 2 3 4 5 6
|
efgsval |
⊢ ( 𝐴 ∈ dom 𝑆 → ( 𝑆 ‘ 𝐴 ) = ( 𝐴 ‘ ( ( ♯ ‘ 𝐴 ) − 1 ) ) ) |
15 |
8 14
|
syl |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐴 ) = ( 𝐴 ‘ ( ( ♯ ‘ 𝐴 ) − 1 ) ) ) |
16 |
10 15
|
eqtr3d |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐵 ) = ( 𝐴 ‘ ( ( ♯ ‘ 𝐴 ) − 1 ) ) ) |
17 |
13 16
|
eqtr3d |
⊢ ( 𝜑 → ( 𝐵 ‘ ( ( ♯ ‘ 𝐵 ) − 1 ) ) = ( 𝐴 ‘ ( ( ♯ ‘ 𝐴 ) − 1 ) ) ) |
18 |
|
oveq1 |
⊢ ( ( ♯ ‘ 𝐴 ) = 1 → ( ( ♯ ‘ 𝐴 ) − 1 ) = ( 1 − 1 ) ) |
19 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
20 |
18 19
|
eqtrdi |
⊢ ( ( ♯ ‘ 𝐴 ) = 1 → ( ( ♯ ‘ 𝐴 ) − 1 ) = 0 ) |
21 |
20
|
fveq2d |
⊢ ( ( ♯ ‘ 𝐴 ) = 1 → ( 𝐴 ‘ ( ( ♯ ‘ 𝐴 ) − 1 ) ) = ( 𝐴 ‘ 0 ) ) |
22 |
17 21
|
sylan9eq |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝐴 ) = 1 ) → ( 𝐵 ‘ ( ( ♯ ‘ 𝐵 ) − 1 ) ) = ( 𝐴 ‘ 0 ) ) |
23 |
10
|
eleq1d |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐴 ) ∈ 𝐷 ↔ ( 𝑆 ‘ 𝐵 ) ∈ 𝐷 ) ) |
24 |
1 2 3 4 5 6
|
efgs1b |
⊢ ( 𝐴 ∈ dom 𝑆 → ( ( 𝑆 ‘ 𝐴 ) ∈ 𝐷 ↔ ( ♯ ‘ 𝐴 ) = 1 ) ) |
25 |
8 24
|
syl |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐴 ) ∈ 𝐷 ↔ ( ♯ ‘ 𝐴 ) = 1 ) ) |
26 |
1 2 3 4 5 6
|
efgs1b |
⊢ ( 𝐵 ∈ dom 𝑆 → ( ( 𝑆 ‘ 𝐵 ) ∈ 𝐷 ↔ ( ♯ ‘ 𝐵 ) = 1 ) ) |
27 |
9 26
|
syl |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐵 ) ∈ 𝐷 ↔ ( ♯ ‘ 𝐵 ) = 1 ) ) |
28 |
23 25 27
|
3bitr3d |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝐴 ) = 1 ↔ ( ♯ ‘ 𝐵 ) = 1 ) ) |
29 |
28
|
biimpa |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝐴 ) = 1 ) → ( ♯ ‘ 𝐵 ) = 1 ) |
30 |
|
oveq1 |
⊢ ( ( ♯ ‘ 𝐵 ) = 1 → ( ( ♯ ‘ 𝐵 ) − 1 ) = ( 1 − 1 ) ) |
31 |
30 19
|
eqtrdi |
⊢ ( ( ♯ ‘ 𝐵 ) = 1 → ( ( ♯ ‘ 𝐵 ) − 1 ) = 0 ) |
32 |
31
|
fveq2d |
⊢ ( ( ♯ ‘ 𝐵 ) = 1 → ( 𝐵 ‘ ( ( ♯ ‘ 𝐵 ) − 1 ) ) = ( 𝐵 ‘ 0 ) ) |
33 |
29 32
|
syl |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝐴 ) = 1 ) → ( 𝐵 ‘ ( ( ♯ ‘ 𝐵 ) − 1 ) ) = ( 𝐵 ‘ 0 ) ) |
34 |
22 33
|
eqtr3d |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝐴 ) = 1 ) → ( 𝐴 ‘ 0 ) = ( 𝐵 ‘ 0 ) ) |
35 |
11 34
|
mtand |
⊢ ( 𝜑 → ¬ ( ♯ ‘ 𝐴 ) = 1 ) |
36 |
1 2 3 4 5 6
|
efgsdm |
⊢ ( 𝐴 ∈ dom 𝑆 ↔ ( 𝐴 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( 𝐴 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑢 ∈ ( 1 ..^ ( ♯ ‘ 𝐴 ) ) ( 𝐴 ‘ 𝑢 ) ∈ ran ( 𝑇 ‘ ( 𝐴 ‘ ( 𝑢 − 1 ) ) ) ) ) |
37 |
36
|
simp1bi |
⊢ ( 𝐴 ∈ dom 𝑆 → 𝐴 ∈ ( Word 𝑊 ∖ { ∅ } ) ) |
38 |
|
eldifsn |
⊢ ( 𝐴 ∈ ( Word 𝑊 ∖ { ∅ } ) ↔ ( 𝐴 ∈ Word 𝑊 ∧ 𝐴 ≠ ∅ ) ) |
39 |
|
lennncl |
⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐴 ≠ ∅ ) → ( ♯ ‘ 𝐴 ) ∈ ℕ ) |
40 |
38 39
|
sylbi |
⊢ ( 𝐴 ∈ ( Word 𝑊 ∖ { ∅ } ) → ( ♯ ‘ 𝐴 ) ∈ ℕ ) |
41 |
8 37 40
|
3syl |
⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) ∈ ℕ ) |
42 |
|
elnn1uz2 |
⊢ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ↔ ( ( ♯ ‘ 𝐴 ) = 1 ∨ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) ) |
43 |
41 42
|
sylib |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝐴 ) = 1 ∨ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) ) |
44 |
43
|
ord |
⊢ ( 𝜑 → ( ¬ ( ♯ ‘ 𝐴 ) = 1 → ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) ) |
45 |
35 44
|
mpd |
⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) |
46 |
|
uz2m1nn |
⊢ ( ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) → ( ( ♯ ‘ 𝐴 ) − 1 ) ∈ ℕ ) |
47 |
45 46
|
syl |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝐴 ) − 1 ) ∈ ℕ ) |
48 |
35 28
|
mtbid |
⊢ ( 𝜑 → ¬ ( ♯ ‘ 𝐵 ) = 1 ) |
49 |
1 2 3 4 5 6
|
efgsdm |
⊢ ( 𝐵 ∈ dom 𝑆 ↔ ( 𝐵 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( 𝐵 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑢 ∈ ( 1 ..^ ( ♯ ‘ 𝐵 ) ) ( 𝐵 ‘ 𝑢 ) ∈ ran ( 𝑇 ‘ ( 𝐵 ‘ ( 𝑢 − 1 ) ) ) ) ) |
50 |
49
|
simp1bi |
⊢ ( 𝐵 ∈ dom 𝑆 → 𝐵 ∈ ( Word 𝑊 ∖ { ∅ } ) ) |
51 |
|
eldifsn |
⊢ ( 𝐵 ∈ ( Word 𝑊 ∖ { ∅ } ) ↔ ( 𝐵 ∈ Word 𝑊 ∧ 𝐵 ≠ ∅ ) ) |
52 |
|
lennncl |
⊢ ( ( 𝐵 ∈ Word 𝑊 ∧ 𝐵 ≠ ∅ ) → ( ♯ ‘ 𝐵 ) ∈ ℕ ) |
53 |
51 52
|
sylbi |
⊢ ( 𝐵 ∈ ( Word 𝑊 ∖ { ∅ } ) → ( ♯ ‘ 𝐵 ) ∈ ℕ ) |
54 |
9 50 53
|
3syl |
⊢ ( 𝜑 → ( ♯ ‘ 𝐵 ) ∈ ℕ ) |
55 |
|
elnn1uz2 |
⊢ ( ( ♯ ‘ 𝐵 ) ∈ ℕ ↔ ( ( ♯ ‘ 𝐵 ) = 1 ∨ ( ♯ ‘ 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) ) ) |
56 |
54 55
|
sylib |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝐵 ) = 1 ∨ ( ♯ ‘ 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) ) ) |
57 |
56
|
ord |
⊢ ( 𝜑 → ( ¬ ( ♯ ‘ 𝐵 ) = 1 → ( ♯ ‘ 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) ) ) |
58 |
48 57
|
mpd |
⊢ ( 𝜑 → ( ♯ ‘ 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) ) |
59 |
|
uz2m1nn |
⊢ ( ( ♯ ‘ 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) → ( ( ♯ ‘ 𝐵 ) − 1 ) ∈ ℕ ) |
60 |
58 59
|
syl |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝐵 ) − 1 ) ∈ ℕ ) |
61 |
47 60
|
jca |
⊢ ( 𝜑 → ( ( ( ♯ ‘ 𝐴 ) − 1 ) ∈ ℕ ∧ ( ( ♯ ‘ 𝐵 ) − 1 ) ∈ ℕ ) ) |