| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ccatcl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( 𝑆 ++ 𝑇 ) ∈ Word 𝐴 ) |
| 2 |
|
revcl |
⊢ ( ( 𝑆 ++ 𝑇 ) ∈ Word 𝐴 → ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) ∈ Word 𝐴 ) |
| 3 |
|
wrdfn |
⊢ ( ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) ∈ Word 𝐴 → ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) Fn ( 0 ..^ ( ♯ ‘ ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) ) ) ) |
| 4 |
1 2 3
|
3syl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) Fn ( 0 ..^ ( ♯ ‘ ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) ) ) ) |
| 5 |
|
revlen |
⊢ ( ( 𝑆 ++ 𝑇 ) ∈ Word 𝐴 → ( ♯ ‘ ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) ) = ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) ) |
| 6 |
1 5
|
syl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ♯ ‘ ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) ) = ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) ) |
| 7 |
|
ccatlen |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) = ( ( ♯ ‘ 𝑆 ) + ( ♯ ‘ 𝑇 ) ) ) |
| 8 |
|
lencl |
⊢ ( 𝑆 ∈ Word 𝐴 → ( ♯ ‘ 𝑆 ) ∈ ℕ0 ) |
| 9 |
8
|
nn0cnd |
⊢ ( 𝑆 ∈ Word 𝐴 → ( ♯ ‘ 𝑆 ) ∈ ℂ ) |
| 10 |
|
lencl |
⊢ ( 𝑇 ∈ Word 𝐴 → ( ♯ ‘ 𝑇 ) ∈ ℕ0 ) |
| 11 |
10
|
nn0cnd |
⊢ ( 𝑇 ∈ Word 𝐴 → ( ♯ ‘ 𝑇 ) ∈ ℂ ) |
| 12 |
|
addcom |
⊢ ( ( ( ♯ ‘ 𝑆 ) ∈ ℂ ∧ ( ♯ ‘ 𝑇 ) ∈ ℂ ) → ( ( ♯ ‘ 𝑆 ) + ( ♯ ‘ 𝑇 ) ) = ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) |
| 13 |
9 11 12
|
syl2an |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ♯ ‘ 𝑆 ) + ( ♯ ‘ 𝑇 ) ) = ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) |
| 14 |
6 7 13
|
3eqtrd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ♯ ‘ ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) ) = ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) |
| 15 |
14
|
oveq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( 0 ..^ ( ♯ ‘ ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) ) ) = ( 0 ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) |
| 16 |
15
|
fneq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) Fn ( 0 ..^ ( ♯ ‘ ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) ) ) ↔ ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) Fn ( 0 ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) ) |
| 17 |
4 16
|
mpbid |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) Fn ( 0 ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) |
| 18 |
|
revcl |
⊢ ( 𝑇 ∈ Word 𝐴 → ( reverse ‘ 𝑇 ) ∈ Word 𝐴 ) |
| 19 |
|
revcl |
⊢ ( 𝑆 ∈ Word 𝐴 → ( reverse ‘ 𝑆 ) ∈ Word 𝐴 ) |
| 20 |
|
ccatcl |
⊢ ( ( ( reverse ‘ 𝑇 ) ∈ Word 𝐴 ∧ ( reverse ‘ 𝑆 ) ∈ Word 𝐴 ) → ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ∈ Word 𝐴 ) |
| 21 |
18 19 20
|
syl2anr |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ∈ Word 𝐴 ) |
| 22 |
|
wrdfn |
⊢ ( ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ∈ Word 𝐴 → ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) Fn ( 0 ..^ ( ♯ ‘ ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ) ) ) |
| 23 |
21 22
|
syl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) Fn ( 0 ..^ ( ♯ ‘ ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ) ) ) |
| 24 |
|
ccatlen |
⊢ ( ( ( reverse ‘ 𝑇 ) ∈ Word 𝐴 ∧ ( reverse ‘ 𝑆 ) ∈ Word 𝐴 ) → ( ♯ ‘ ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ) = ( ( ♯ ‘ ( reverse ‘ 𝑇 ) ) + ( ♯ ‘ ( reverse ‘ 𝑆 ) ) ) ) |
| 25 |
18 19 24
|
syl2anr |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ♯ ‘ ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ) = ( ( ♯ ‘ ( reverse ‘ 𝑇 ) ) + ( ♯ ‘ ( reverse ‘ 𝑆 ) ) ) ) |
| 26 |
|
revlen |
⊢ ( 𝑇 ∈ Word 𝐴 → ( ♯ ‘ ( reverse ‘ 𝑇 ) ) = ( ♯ ‘ 𝑇 ) ) |
| 27 |
|
revlen |
⊢ ( 𝑆 ∈ Word 𝐴 → ( ♯ ‘ ( reverse ‘ 𝑆 ) ) = ( ♯ ‘ 𝑆 ) ) |
| 28 |
26 27
|
oveqan12rd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ♯ ‘ ( reverse ‘ 𝑇 ) ) + ( ♯ ‘ ( reverse ‘ 𝑆 ) ) ) = ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) |
| 29 |
25 28
|
eqtrd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ♯ ‘ ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ) = ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) |
| 30 |
29
|
oveq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( 0 ..^ ( ♯ ‘ ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ) ) = ( 0 ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) |
| 31 |
30
|
fneq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) Fn ( 0 ..^ ( ♯ ‘ ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ) ) ↔ ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) Fn ( 0 ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) ) |
| 32 |
23 31
|
mpbid |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) Fn ( 0 ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) |
| 33 |
|
id |
⊢ ( 𝑥 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) → 𝑥 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) |
| 34 |
10
|
nn0zd |
⊢ ( 𝑇 ∈ Word 𝐴 → ( ♯ ‘ 𝑇 ) ∈ ℤ ) |
| 35 |
34
|
adantl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ♯ ‘ 𝑇 ) ∈ ℤ ) |
| 36 |
|
fzospliti |
⊢ ( ( 𝑥 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ∧ ( ♯ ‘ 𝑇 ) ∈ ℤ ) → ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ∨ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) ) |
| 37 |
33 35 36
|
syl2anr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ∨ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) ) |
| 38 |
|
simpll |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → 𝑆 ∈ Word 𝐴 ) |
| 39 |
|
simplr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → 𝑇 ∈ Word 𝐴 ) |
| 40 |
|
fzoval |
⊢ ( ( ♯ ‘ 𝑇 ) ∈ ℤ → ( 0 ..^ ( ♯ ‘ 𝑇 ) ) = ( 0 ... ( ( ♯ ‘ 𝑇 ) − 1 ) ) ) |
| 41 |
34 40
|
syl |
⊢ ( 𝑇 ∈ Word 𝐴 → ( 0 ..^ ( ♯ ‘ 𝑇 ) ) = ( 0 ... ( ( ♯ ‘ 𝑇 ) − 1 ) ) ) |
| 42 |
41
|
adantl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( 0 ..^ ( ♯ ‘ 𝑇 ) ) = ( 0 ... ( ( ♯ ‘ 𝑇 ) − 1 ) ) ) |
| 43 |
42
|
eleq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ↔ 𝑥 ∈ ( 0 ... ( ( ♯ ‘ 𝑇 ) − 1 ) ) ) ) |
| 44 |
43
|
biimpa |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → 𝑥 ∈ ( 0 ... ( ( ♯ ‘ 𝑇 ) − 1 ) ) ) |
| 45 |
|
fznn0sub2 |
⊢ ( 𝑥 ∈ ( 0 ... ( ( ♯ ‘ 𝑇 ) − 1 ) ) → ( ( ( ♯ ‘ 𝑇 ) − 1 ) − 𝑥 ) ∈ ( 0 ... ( ( ♯ ‘ 𝑇 ) − 1 ) ) ) |
| 46 |
44 45
|
syl |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( ( ( ♯ ‘ 𝑇 ) − 1 ) − 𝑥 ) ∈ ( 0 ... ( ( ♯ ‘ 𝑇 ) − 1 ) ) ) |
| 47 |
41
|
ad2antlr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( 0 ..^ ( ♯ ‘ 𝑇 ) ) = ( 0 ... ( ( ♯ ‘ 𝑇 ) − 1 ) ) ) |
| 48 |
46 47
|
eleqtrrd |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( ( ( ♯ ‘ 𝑇 ) − 1 ) − 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) |
| 49 |
|
ccatval3 |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ ( ( ( ♯ ‘ 𝑇 ) − 1 ) − 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( ( 𝑆 ++ 𝑇 ) ‘ ( ( ( ( ♯ ‘ 𝑇 ) − 1 ) − 𝑥 ) + ( ♯ ‘ 𝑆 ) ) ) = ( 𝑇 ‘ ( ( ( ♯ ‘ 𝑇 ) − 1 ) − 𝑥 ) ) ) |
| 50 |
38 39 48 49
|
syl3anc |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( ( 𝑆 ++ 𝑇 ) ‘ ( ( ( ( ♯ ‘ 𝑇 ) − 1 ) − 𝑥 ) + ( ♯ ‘ 𝑆 ) ) ) = ( 𝑇 ‘ ( ( ( ♯ ‘ 𝑇 ) − 1 ) − 𝑥 ) ) ) |
| 51 |
7 13
|
eqtrd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) = ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) |
| 52 |
51
|
oveq1d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) = ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) ) |
| 53 |
11
|
adantl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ♯ ‘ 𝑇 ) ∈ ℂ ) |
| 54 |
9
|
adantr |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ♯ ‘ 𝑆 ) ∈ ℂ ) |
| 55 |
|
1cnd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → 1 ∈ ℂ ) |
| 56 |
53 54 55
|
addsubd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) = ( ( ( ♯ ‘ 𝑇 ) − 1 ) + ( ♯ ‘ 𝑆 ) ) ) |
| 57 |
52 56
|
eqtrd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) = ( ( ( ♯ ‘ 𝑇 ) − 1 ) + ( ♯ ‘ 𝑆 ) ) ) |
| 58 |
57
|
oveq1d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) = ( ( ( ( ♯ ‘ 𝑇 ) − 1 ) + ( ♯ ‘ 𝑆 ) ) − 𝑥 ) ) |
| 59 |
58
|
adantr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) = ( ( ( ( ♯ ‘ 𝑇 ) − 1 ) + ( ♯ ‘ 𝑆 ) ) − 𝑥 ) ) |
| 60 |
|
peano2zm |
⊢ ( ( ♯ ‘ 𝑇 ) ∈ ℤ → ( ( ♯ ‘ 𝑇 ) − 1 ) ∈ ℤ ) |
| 61 |
34 60
|
syl |
⊢ ( 𝑇 ∈ Word 𝐴 → ( ( ♯ ‘ 𝑇 ) − 1 ) ∈ ℤ ) |
| 62 |
61
|
zcnd |
⊢ ( 𝑇 ∈ Word 𝐴 → ( ( ♯ ‘ 𝑇 ) − 1 ) ∈ ℂ ) |
| 63 |
62
|
ad2antlr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( ( ♯ ‘ 𝑇 ) − 1 ) ∈ ℂ ) |
| 64 |
9
|
ad2antrr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( ♯ ‘ 𝑆 ) ∈ ℂ ) |
| 65 |
|
elfzoelz |
⊢ ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) → 𝑥 ∈ ℤ ) |
| 66 |
65
|
zcnd |
⊢ ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) → 𝑥 ∈ ℂ ) |
| 67 |
66
|
adantl |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → 𝑥 ∈ ℂ ) |
| 68 |
63 64 67
|
addsubd |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( ( ( ( ♯ ‘ 𝑇 ) − 1 ) + ( ♯ ‘ 𝑆 ) ) − 𝑥 ) = ( ( ( ( ♯ ‘ 𝑇 ) − 1 ) − 𝑥 ) + ( ♯ ‘ 𝑆 ) ) ) |
| 69 |
59 68
|
eqtrd |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) = ( ( ( ( ♯ ‘ 𝑇 ) − 1 ) − 𝑥 ) + ( ♯ ‘ 𝑆 ) ) ) |
| 70 |
69
|
fveq2d |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( ( 𝑆 ++ 𝑇 ) ‘ ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) ) = ( ( 𝑆 ++ 𝑇 ) ‘ ( ( ( ( ♯ ‘ 𝑇 ) − 1 ) − 𝑥 ) + ( ♯ ‘ 𝑆 ) ) ) ) |
| 71 |
|
revfv |
⊢ ( ( 𝑇 ∈ Word 𝐴 ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( ( reverse ‘ 𝑇 ) ‘ 𝑥 ) = ( 𝑇 ‘ ( ( ( ♯ ‘ 𝑇 ) − 1 ) − 𝑥 ) ) ) |
| 72 |
71
|
adantll |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( ( reverse ‘ 𝑇 ) ‘ 𝑥 ) = ( 𝑇 ‘ ( ( ( ♯ ‘ 𝑇 ) − 1 ) − 𝑥 ) ) ) |
| 73 |
50 70 72
|
3eqtr4d |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( ( 𝑆 ++ 𝑇 ) ‘ ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) ) = ( ( reverse ‘ 𝑇 ) ‘ 𝑥 ) ) |
| 74 |
34
|
uzidd |
⊢ ( 𝑇 ∈ Word 𝐴 → ( ♯ ‘ 𝑇 ) ∈ ( ℤ≥ ‘ ( ♯ ‘ 𝑇 ) ) ) |
| 75 |
|
uzaddcl |
⊢ ( ( ( ♯ ‘ 𝑇 ) ∈ ( ℤ≥ ‘ ( ♯ ‘ 𝑇 ) ) ∧ ( ♯ ‘ 𝑆 ) ∈ ℕ0 ) → ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ∈ ( ℤ≥ ‘ ( ♯ ‘ 𝑇 ) ) ) |
| 76 |
74 8 75
|
syl2anr |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ∈ ( ℤ≥ ‘ ( ♯ ‘ 𝑇 ) ) ) |
| 77 |
51 76
|
eqeltrd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) ∈ ( ℤ≥ ‘ ( ♯ ‘ 𝑇 ) ) ) |
| 78 |
|
fzoss2 |
⊢ ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) ∈ ( ℤ≥ ‘ ( ♯ ‘ 𝑇 ) ) → ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ⊆ ( 0 ..^ ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) ) ) |
| 79 |
77 78
|
syl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ⊆ ( 0 ..^ ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) ) ) |
| 80 |
79
|
sselda |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → 𝑥 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) ) ) |
| 81 |
|
revfv |
⊢ ( ( ( 𝑆 ++ 𝑇 ) ∈ Word 𝐴 ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) ) ) → ( ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) ‘ 𝑥 ) = ( ( 𝑆 ++ 𝑇 ) ‘ ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) ) ) |
| 82 |
1 80 81
|
syl2an2r |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) ‘ 𝑥 ) = ( ( 𝑆 ++ 𝑇 ) ‘ ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) ) ) |
| 83 |
18
|
ad2antlr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( reverse ‘ 𝑇 ) ∈ Word 𝐴 ) |
| 84 |
19
|
ad2antrr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( reverse ‘ 𝑆 ) ∈ Word 𝐴 ) |
| 85 |
26
|
adantl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ♯ ‘ ( reverse ‘ 𝑇 ) ) = ( ♯ ‘ 𝑇 ) ) |
| 86 |
85
|
oveq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) = ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) |
| 87 |
86
|
eleq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) ↔ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) ) |
| 88 |
87
|
biimpar |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → 𝑥 ∈ ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) ) |
| 89 |
|
ccatval1 |
⊢ ( ( ( reverse ‘ 𝑇 ) ∈ Word 𝐴 ∧ ( reverse ‘ 𝑆 ) ∈ Word 𝐴 ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) ) → ( ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ‘ 𝑥 ) = ( ( reverse ‘ 𝑇 ) ‘ 𝑥 ) ) |
| 90 |
83 84 88 89
|
syl3anc |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ‘ 𝑥 ) = ( ( reverse ‘ 𝑇 ) ‘ 𝑥 ) ) |
| 91 |
73 82 90
|
3eqtr4d |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) ‘ 𝑥 ) = ( ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ‘ 𝑥 ) ) |
| 92 |
8
|
nn0zd |
⊢ ( 𝑆 ∈ Word 𝐴 → ( ♯ ‘ 𝑆 ) ∈ ℤ ) |
| 93 |
|
peano2zm |
⊢ ( ( ♯ ‘ 𝑆 ) ∈ ℤ → ( ( ♯ ‘ 𝑆 ) − 1 ) ∈ ℤ ) |
| 94 |
92 93
|
syl |
⊢ ( 𝑆 ∈ Word 𝐴 → ( ( ♯ ‘ 𝑆 ) − 1 ) ∈ ℤ ) |
| 95 |
94
|
zcnd |
⊢ ( 𝑆 ∈ Word 𝐴 → ( ( ♯ ‘ 𝑆 ) − 1 ) ∈ ℂ ) |
| 96 |
95
|
ad2antrr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( ( ♯ ‘ 𝑆 ) − 1 ) ∈ ℂ ) |
| 97 |
|
elfzoelz |
⊢ ( 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) → 𝑥 ∈ ℤ ) |
| 98 |
97
|
zcnd |
⊢ ( 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) → 𝑥 ∈ ℂ ) |
| 99 |
98
|
adantl |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → 𝑥 ∈ ℂ ) |
| 100 |
11
|
ad2antlr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( ♯ ‘ 𝑇 ) ∈ ℂ ) |
| 101 |
96 99 100
|
subsub3d |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( ( ( ♯ ‘ 𝑆 ) − 1 ) − ( 𝑥 − ( ♯ ‘ 𝑇 ) ) ) = ( ( ( ( ♯ ‘ 𝑆 ) − 1 ) + ( ♯ ‘ 𝑇 ) ) − 𝑥 ) ) |
| 102 |
26
|
oveq2d |
⊢ ( 𝑇 ∈ Word 𝐴 → ( 𝑥 − ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) = ( 𝑥 − ( ♯ ‘ 𝑇 ) ) ) |
| 103 |
102
|
oveq2d |
⊢ ( 𝑇 ∈ Word 𝐴 → ( ( ( ♯ ‘ 𝑆 ) − 1 ) − ( 𝑥 − ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) ) = ( ( ( ♯ ‘ 𝑆 ) − 1 ) − ( 𝑥 − ( ♯ ‘ 𝑇 ) ) ) ) |
| 104 |
103
|
ad2antlr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( ( ( ♯ ‘ 𝑆 ) − 1 ) − ( 𝑥 − ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) ) = ( ( ( ♯ ‘ 𝑆 ) − 1 ) − ( 𝑥 − ( ♯ ‘ 𝑇 ) ) ) ) |
| 105 |
7
|
oveq1d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) = ( ( ( ♯ ‘ 𝑆 ) + ( ♯ ‘ 𝑇 ) ) − 1 ) ) |
| 106 |
54 53 55
|
addsubd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ( ♯ ‘ 𝑆 ) + ( ♯ ‘ 𝑇 ) ) − 1 ) = ( ( ( ♯ ‘ 𝑆 ) − 1 ) + ( ♯ ‘ 𝑇 ) ) ) |
| 107 |
105 106
|
eqtrd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) = ( ( ( ♯ ‘ 𝑆 ) − 1 ) + ( ♯ ‘ 𝑇 ) ) ) |
| 108 |
107
|
oveq1d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) = ( ( ( ( ♯ ‘ 𝑆 ) − 1 ) + ( ♯ ‘ 𝑇 ) ) − 𝑥 ) ) |
| 109 |
108
|
adantr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) = ( ( ( ( ♯ ‘ 𝑆 ) − 1 ) + ( ♯ ‘ 𝑇 ) ) − 𝑥 ) ) |
| 110 |
101 104 109
|
3eqtr4rd |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) = ( ( ( ♯ ‘ 𝑆 ) − 1 ) − ( 𝑥 − ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) ) ) |
| 111 |
110
|
fveq2d |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑆 ‘ ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) ) = ( 𝑆 ‘ ( ( ( ♯ ‘ 𝑆 ) − 1 ) − ( 𝑥 − ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) ) ) ) |
| 112 |
|
simpll |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → 𝑆 ∈ Word 𝐴 ) |
| 113 |
|
simplr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → 𝑇 ∈ Word 𝐴 ) |
| 114 |
|
zaddcl |
⊢ ( ( ( ♯ ‘ 𝑇 ) ∈ ℤ ∧ ( ♯ ‘ 𝑆 ) ∈ ℤ ) → ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ∈ ℤ ) |
| 115 |
34 92 114
|
syl2anr |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ∈ ℤ ) |
| 116 |
|
peano2zm |
⊢ ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ∈ ℤ → ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) ∈ ℤ ) |
| 117 |
115 116
|
syl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) ∈ ℤ ) |
| 118 |
|
fzoval |
⊢ ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ∈ ℤ → ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) = ( ( ♯ ‘ 𝑇 ) ... ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) ) ) |
| 119 |
115 118
|
syl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) = ( ( ♯ ‘ 𝑇 ) ... ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) ) ) |
| 120 |
119
|
eleq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ↔ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ... ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) ) ) ) |
| 121 |
120
|
biimpa |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ... ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) ) ) |
| 122 |
|
fzrev2i |
⊢ ( ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) ∈ ℤ ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ... ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) ) ) → ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − 𝑥 ) ∈ ( ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) ) ... ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − ( ♯ ‘ 𝑇 ) ) ) ) |
| 123 |
117 121 122
|
syl2an2r |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − 𝑥 ) ∈ ( ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) ) ... ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − ( ♯ ‘ 𝑇 ) ) ) ) |
| 124 |
52
|
oveq1d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) = ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − 𝑥 ) ) |
| 125 |
124
|
adantr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) = ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − 𝑥 ) ) |
| 126 |
92
|
adantr |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ♯ ‘ 𝑆 ) ∈ ℤ ) |
| 127 |
|
fzoval |
⊢ ( ( ♯ ‘ 𝑆 ) ∈ ℤ → ( 0 ..^ ( ♯ ‘ 𝑆 ) ) = ( 0 ... ( ( ♯ ‘ 𝑆 ) − 1 ) ) ) |
| 128 |
126 127
|
syl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( 0 ..^ ( ♯ ‘ 𝑆 ) ) = ( 0 ... ( ( ♯ ‘ 𝑆 ) − 1 ) ) ) |
| 129 |
117
|
zcnd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) ∈ ℂ ) |
| 130 |
129
|
subidd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) ) = 0 ) |
| 131 |
|
addcl |
⊢ ( ( ( ♯ ‘ 𝑇 ) ∈ ℂ ∧ ( ♯ ‘ 𝑆 ) ∈ ℂ ) → ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ∈ ℂ ) |
| 132 |
11 9 131
|
syl2anr |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ∈ ℂ ) |
| 133 |
132 55 53
|
sub32d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − ( ♯ ‘ 𝑇 ) ) = ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − ( ♯ ‘ 𝑇 ) ) − 1 ) ) |
| 134 |
|
pncan2 |
⊢ ( ( ( ♯ ‘ 𝑇 ) ∈ ℂ ∧ ( ♯ ‘ 𝑆 ) ∈ ℂ ) → ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − ( ♯ ‘ 𝑇 ) ) = ( ♯ ‘ 𝑆 ) ) |
| 135 |
11 9 134
|
syl2anr |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − ( ♯ ‘ 𝑇 ) ) = ( ♯ ‘ 𝑆 ) ) |
| 136 |
135
|
oveq1d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − ( ♯ ‘ 𝑇 ) ) − 1 ) = ( ( ♯ ‘ 𝑆 ) − 1 ) ) |
| 137 |
133 136
|
eqtrd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − ( ♯ ‘ 𝑇 ) ) = ( ( ♯ ‘ 𝑆 ) − 1 ) ) |
| 138 |
130 137
|
oveq12d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) ) ... ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − ( ♯ ‘ 𝑇 ) ) ) = ( 0 ... ( ( ♯ ‘ 𝑆 ) − 1 ) ) ) |
| 139 |
128 138
|
eqtr4d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( 0 ..^ ( ♯ ‘ 𝑆 ) ) = ( ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) ) ... ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − ( ♯ ‘ 𝑇 ) ) ) ) |
| 140 |
139
|
adantr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( 0 ..^ ( ♯ ‘ 𝑆 ) ) = ( ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) ) ... ( ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) − 1 ) − ( ♯ ‘ 𝑇 ) ) ) ) |
| 141 |
123 125 140
|
3eltr4d |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) |
| 142 |
|
ccatval1 |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) → ( ( 𝑆 ++ 𝑇 ) ‘ ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) ) = ( 𝑆 ‘ ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) ) ) |
| 143 |
112 113 141 142
|
syl3anc |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( ( 𝑆 ++ 𝑇 ) ‘ ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) ) = ( 𝑆 ‘ ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) ) ) |
| 144 |
|
simpl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → 𝑆 ∈ Word 𝐴 ) |
| 145 |
102
|
ad2antlr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑥 − ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) = ( 𝑥 − ( ♯ ‘ 𝑇 ) ) ) |
| 146 |
|
id |
⊢ ( 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) → 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) |
| 147 |
|
fzosubel3 |
⊢ ( ( 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ∧ ( ♯ ‘ 𝑆 ) ∈ ℤ ) → ( 𝑥 − ( ♯ ‘ 𝑇 ) ) ∈ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) |
| 148 |
146 126 147
|
syl2anr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑥 − ( ♯ ‘ 𝑇 ) ) ∈ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) |
| 149 |
145 148
|
eqeltrd |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑥 − ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) ∈ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) |
| 150 |
|
revfv |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑥 − ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) ∈ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) → ( ( reverse ‘ 𝑆 ) ‘ ( 𝑥 − ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) ) = ( 𝑆 ‘ ( ( ( ♯ ‘ 𝑆 ) − 1 ) − ( 𝑥 − ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) ) ) ) |
| 151 |
144 149 150
|
syl2an2r |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( ( reverse ‘ 𝑆 ) ‘ ( 𝑥 − ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) ) = ( 𝑆 ‘ ( ( ( ♯ ‘ 𝑆 ) − 1 ) − ( 𝑥 − ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) ) ) ) |
| 152 |
111 143 151
|
3eqtr4d |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( ( 𝑆 ++ 𝑇 ) ‘ ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) ) = ( ( reverse ‘ 𝑆 ) ‘ ( 𝑥 − ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) ) ) |
| 153 |
|
fzoss1 |
⊢ ( ( ♯ ‘ 𝑇 ) ∈ ( ℤ≥ ‘ 0 ) → ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ⊆ ( 0 ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) |
| 154 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
| 155 |
153 154
|
eleq2s |
⊢ ( ( ♯ ‘ 𝑇 ) ∈ ℕ0 → ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ⊆ ( 0 ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) |
| 156 |
10 155
|
syl |
⊢ ( 𝑇 ∈ Word 𝐴 → ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ⊆ ( 0 ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) |
| 157 |
156
|
adantl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ⊆ ( 0 ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) |
| 158 |
51
|
oveq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( 0 ..^ ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) ) = ( 0 ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) |
| 159 |
157 158
|
sseqtrrd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ⊆ ( 0 ..^ ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) ) ) |
| 160 |
159
|
sselda |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → 𝑥 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) ) ) |
| 161 |
1 160 81
|
syl2an2r |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) ‘ 𝑥 ) = ( ( 𝑆 ++ 𝑇 ) ‘ ( ( ( ♯ ‘ ( 𝑆 ++ 𝑇 ) ) − 1 ) − 𝑥 ) ) ) |
| 162 |
18
|
ad2antlr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( reverse ‘ 𝑇 ) ∈ Word 𝐴 ) |
| 163 |
19
|
ad2antrr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( reverse ‘ 𝑆 ) ∈ Word 𝐴 ) |
| 164 |
85 28
|
oveq12d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ..^ ( ( ♯ ‘ ( reverse ‘ 𝑇 ) ) + ( ♯ ‘ ( reverse ‘ 𝑆 ) ) ) ) = ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) |
| 165 |
164
|
eleq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( 𝑥 ∈ ( ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ..^ ( ( ♯ ‘ ( reverse ‘ 𝑇 ) ) + ( ♯ ‘ ( reverse ‘ 𝑆 ) ) ) ) ↔ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) ) |
| 166 |
165
|
biimpar |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → 𝑥 ∈ ( ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ..^ ( ( ♯ ‘ ( reverse ‘ 𝑇 ) ) + ( ♯ ‘ ( reverse ‘ 𝑆 ) ) ) ) ) |
| 167 |
|
ccatval2 |
⊢ ( ( ( reverse ‘ 𝑇 ) ∈ Word 𝐴 ∧ ( reverse ‘ 𝑆 ) ∈ Word 𝐴 ∧ 𝑥 ∈ ( ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ..^ ( ( ♯ ‘ ( reverse ‘ 𝑇 ) ) + ( ♯ ‘ ( reverse ‘ 𝑆 ) ) ) ) ) → ( ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ‘ 𝑥 ) = ( ( reverse ‘ 𝑆 ) ‘ ( 𝑥 − ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) ) ) |
| 168 |
162 163 166 167
|
syl3anc |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ‘ 𝑥 ) = ( ( reverse ‘ 𝑆 ) ‘ ( 𝑥 − ( ♯ ‘ ( reverse ‘ 𝑇 ) ) ) ) ) |
| 169 |
152 161 168
|
3eqtr4d |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) ‘ 𝑥 ) = ( ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ‘ 𝑥 ) ) |
| 170 |
91 169
|
jaodan |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ∨ 𝑥 ∈ ( ( ♯ ‘ 𝑇 ) ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) ) → ( ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) ‘ 𝑥 ) = ( ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ‘ 𝑥 ) ) |
| 171 |
37 170
|
syldan |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) ∧ 𝑥 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 𝑆 ) ) ) ) → ( ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) ‘ 𝑥 ) = ( ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ‘ 𝑥 ) ) |
| 172 |
17 32 171
|
eqfnfvd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ) → ( reverse ‘ ( 𝑆 ++ 𝑇 ) ) = ( ( reverse ‘ 𝑇 ) ++ ( reverse ‘ 𝑆 ) ) ) |