Step |
Hyp |
Ref |
Expression |
1 |
|
lmatfval.m |
⊢ 𝑀 = ( litMat ‘ 𝑊 ) |
2 |
|
lmatfval.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
3 |
|
lmatfval.w |
⊢ ( 𝜑 → 𝑊 ∈ Word Word 𝑉 ) |
4 |
|
lmatfval.1 |
⊢ ( 𝜑 → ( ♯ ‘ 𝑊 ) = 𝑁 ) |
5 |
|
lmatfval.2 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑁 ) ) → ( ♯ ‘ ( 𝑊 ‘ 𝑖 ) ) = 𝑁 ) |
6 |
|
lmatfval.i |
⊢ ( 𝜑 → 𝐼 ∈ ( 1 ... 𝑁 ) ) |
7 |
|
lmatfval.j |
⊢ ( 𝜑 → 𝐽 ∈ ( 1 ... 𝑁 ) ) |
8 |
|
lmatval |
⊢ ( 𝑊 ∈ Word Word 𝑉 → ( litMat ‘ 𝑊 ) = ( 𝑖 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑗 ∈ ( 1 ... ( ♯ ‘ ( 𝑊 ‘ 0 ) ) ) ↦ ( ( 𝑊 ‘ ( 𝑖 − 1 ) ) ‘ ( 𝑗 − 1 ) ) ) ) |
9 |
3 8
|
syl |
⊢ ( 𝜑 → ( litMat ‘ 𝑊 ) = ( 𝑖 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑗 ∈ ( 1 ... ( ♯ ‘ ( 𝑊 ‘ 0 ) ) ) ↦ ( ( 𝑊 ‘ ( 𝑖 − 1 ) ) ‘ ( 𝑗 − 1 ) ) ) ) |
10 |
1 9
|
syl5eq |
⊢ ( 𝜑 → 𝑀 = ( 𝑖 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) , 𝑗 ∈ ( 1 ... ( ♯ ‘ ( 𝑊 ‘ 0 ) ) ) ↦ ( ( 𝑊 ‘ ( 𝑖 − 1 ) ) ‘ ( 𝑗 − 1 ) ) ) ) |
11 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐽 ) ) → 𝑖 = 𝐼 ) |
12 |
11
|
fvoveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐽 ) ) → ( 𝑊 ‘ ( 𝑖 − 1 ) ) = ( 𝑊 ‘ ( 𝐼 − 1 ) ) ) |
13 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐽 ) ) → 𝑗 = 𝐽 ) |
14 |
13
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐽 ) ) → ( 𝑗 − 1 ) = ( 𝐽 − 1 ) ) |
15 |
12 14
|
fveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐽 ) ) → ( ( 𝑊 ‘ ( 𝑖 − 1 ) ) ‘ ( 𝑗 − 1 ) ) = ( ( 𝑊 ‘ ( 𝐼 − 1 ) ) ‘ ( 𝐽 − 1 ) ) ) |
16 |
4
|
oveq2d |
⊢ ( 𝜑 → ( 1 ... ( ♯ ‘ 𝑊 ) ) = ( 1 ... 𝑁 ) ) |
17 |
6 16
|
eleqtrrd |
⊢ ( 𝜑 → 𝐼 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) |
18 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
19 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
20 |
2 19
|
eleqtrdi |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 1 ) ) |
21 |
|
eluzfz1 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 1 ) → 1 ∈ ( 1 ... 𝑁 ) ) |
22 |
20 21
|
syl |
⊢ ( 𝜑 → 1 ∈ ( 1 ... 𝑁 ) ) |
23 |
|
fz1fzo0m1 |
⊢ ( 1 ∈ ( 1 ... 𝑁 ) → ( 1 − 1 ) ∈ ( 0 ..^ 𝑁 ) ) |
24 |
22 23
|
syl |
⊢ ( 𝜑 → ( 1 − 1 ) ∈ ( 0 ..^ 𝑁 ) ) |
25 |
18 24
|
eqeltrrid |
⊢ ( 𝜑 → 0 ∈ ( 0 ..^ 𝑁 ) ) |
26 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑖 = 0 ) → 𝑖 = 0 ) |
27 |
26
|
eleq1d |
⊢ ( ( 𝜑 ∧ 𝑖 = 0 ) → ( 𝑖 ∈ ( 0 ..^ 𝑁 ) ↔ 0 ∈ ( 0 ..^ 𝑁 ) ) ) |
28 |
26
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 = 0 ) → ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) |
29 |
28
|
fveqeq2d |
⊢ ( ( 𝜑 ∧ 𝑖 = 0 ) → ( ( ♯ ‘ ( 𝑊 ‘ 𝑖 ) ) = 𝑁 ↔ ( ♯ ‘ ( 𝑊 ‘ 0 ) ) = 𝑁 ) ) |
30 |
27 29
|
imbi12d |
⊢ ( ( 𝜑 ∧ 𝑖 = 0 ) → ( ( 𝑖 ∈ ( 0 ..^ 𝑁 ) → ( ♯ ‘ ( 𝑊 ‘ 𝑖 ) ) = 𝑁 ) ↔ ( 0 ∈ ( 0 ..^ 𝑁 ) → ( ♯ ‘ ( 𝑊 ‘ 0 ) ) = 𝑁 ) ) ) |
31 |
5
|
ex |
⊢ ( 𝜑 → ( 𝑖 ∈ ( 0 ..^ 𝑁 ) → ( ♯ ‘ ( 𝑊 ‘ 𝑖 ) ) = 𝑁 ) ) |
32 |
25 30 31
|
vtocld |
⊢ ( 𝜑 → ( 0 ∈ ( 0 ..^ 𝑁 ) → ( ♯ ‘ ( 𝑊 ‘ 0 ) ) = 𝑁 ) ) |
33 |
25 32
|
mpd |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝑊 ‘ 0 ) ) = 𝑁 ) |
34 |
33
|
oveq2d |
⊢ ( 𝜑 → ( 1 ... ( ♯ ‘ ( 𝑊 ‘ 0 ) ) ) = ( 1 ... 𝑁 ) ) |
35 |
7 34
|
eleqtrrd |
⊢ ( 𝜑 → 𝐽 ∈ ( 1 ... ( ♯ ‘ ( 𝑊 ‘ 0 ) ) ) ) |
36 |
|
fz1fzo0m1 |
⊢ ( 𝐼 ∈ ( 1 ... 𝑁 ) → ( 𝐼 − 1 ) ∈ ( 0 ..^ 𝑁 ) ) |
37 |
6 36
|
syl |
⊢ ( 𝜑 → ( 𝐼 − 1 ) ∈ ( 0 ..^ 𝑁 ) ) |
38 |
4
|
oveq2d |
⊢ ( 𝜑 → ( 0 ..^ ( ♯ ‘ 𝑊 ) ) = ( 0 ..^ 𝑁 ) ) |
39 |
37 38
|
eleqtrrd |
⊢ ( 𝜑 → ( 𝐼 − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
40 |
|
wrdsymbcl |
⊢ ( ( 𝑊 ∈ Word Word 𝑉 ∧ ( 𝐼 − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 ‘ ( 𝐼 − 1 ) ) ∈ Word 𝑉 ) |
41 |
3 39 40
|
syl2anc |
⊢ ( 𝜑 → ( 𝑊 ‘ ( 𝐼 − 1 ) ) ∈ Word 𝑉 ) |
42 |
|
fz1fzo0m1 |
⊢ ( 𝐽 ∈ ( 1 ... 𝑁 ) → ( 𝐽 − 1 ) ∈ ( 0 ..^ 𝑁 ) ) |
43 |
7 42
|
syl |
⊢ ( 𝜑 → ( 𝐽 − 1 ) ∈ ( 0 ..^ 𝑁 ) ) |
44 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑖 = ( 𝐼 − 1 ) ) → 𝑖 = ( 𝐼 − 1 ) ) |
45 |
44
|
eleq1d |
⊢ ( ( 𝜑 ∧ 𝑖 = ( 𝐼 − 1 ) ) → ( 𝑖 ∈ ( 0 ..^ 𝑁 ) ↔ ( 𝐼 − 1 ) ∈ ( 0 ..^ 𝑁 ) ) ) |
46 |
44
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 = ( 𝐼 − 1 ) ) → ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ ( 𝐼 − 1 ) ) ) |
47 |
46
|
fveqeq2d |
⊢ ( ( 𝜑 ∧ 𝑖 = ( 𝐼 − 1 ) ) → ( ( ♯ ‘ ( 𝑊 ‘ 𝑖 ) ) = 𝑁 ↔ ( ♯ ‘ ( 𝑊 ‘ ( 𝐼 − 1 ) ) ) = 𝑁 ) ) |
48 |
45 47
|
imbi12d |
⊢ ( ( 𝜑 ∧ 𝑖 = ( 𝐼 − 1 ) ) → ( ( 𝑖 ∈ ( 0 ..^ 𝑁 ) → ( ♯ ‘ ( 𝑊 ‘ 𝑖 ) ) = 𝑁 ) ↔ ( ( 𝐼 − 1 ) ∈ ( 0 ..^ 𝑁 ) → ( ♯ ‘ ( 𝑊 ‘ ( 𝐼 − 1 ) ) ) = 𝑁 ) ) ) |
49 |
37 48 31
|
vtocld |
⊢ ( 𝜑 → ( ( 𝐼 − 1 ) ∈ ( 0 ..^ 𝑁 ) → ( ♯ ‘ ( 𝑊 ‘ ( 𝐼 − 1 ) ) ) = 𝑁 ) ) |
50 |
37 49
|
mpd |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝑊 ‘ ( 𝐼 − 1 ) ) ) = 𝑁 ) |
51 |
50
|
oveq2d |
⊢ ( 𝜑 → ( 0 ..^ ( ♯ ‘ ( 𝑊 ‘ ( 𝐼 − 1 ) ) ) ) = ( 0 ..^ 𝑁 ) ) |
52 |
43 51
|
eleqtrrd |
⊢ ( 𝜑 → ( 𝐽 − 1 ) ∈ ( 0 ..^ ( ♯ ‘ ( 𝑊 ‘ ( 𝐼 − 1 ) ) ) ) ) |
53 |
|
wrdsymbcl |
⊢ ( ( ( 𝑊 ‘ ( 𝐼 − 1 ) ) ∈ Word 𝑉 ∧ ( 𝐽 − 1 ) ∈ ( 0 ..^ ( ♯ ‘ ( 𝑊 ‘ ( 𝐼 − 1 ) ) ) ) ) → ( ( 𝑊 ‘ ( 𝐼 − 1 ) ) ‘ ( 𝐽 − 1 ) ) ∈ 𝑉 ) |
54 |
41 52 53
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑊 ‘ ( 𝐼 − 1 ) ) ‘ ( 𝐽 − 1 ) ) ∈ 𝑉 ) |
55 |
10 15 17 35 54
|
ovmpod |
⊢ ( 𝜑 → ( 𝐼 𝑀 𝐽 ) = ( ( 𝑊 ‘ ( 𝐼 − 1 ) ) ‘ ( 𝐽 − 1 ) ) ) |