Step |
Hyp |
Ref |
Expression |
1 |
|
smat.s |
|- S = ( K ( subMat1 ` A ) L ) |
2 |
|
smat.m |
|- ( ph -> M e. NN ) |
3 |
|
smat.n |
|- ( ph -> N e. NN ) |
4 |
|
smat.k |
|- ( ph -> K e. ( 1 ... M ) ) |
5 |
|
smat.l |
|- ( ph -> L e. ( 1 ... N ) ) |
6 |
|
smat.a |
|- ( ph -> A e. ( B ^m ( ( 1 ... M ) X. ( 1 ... N ) ) ) ) |
7 |
|
smatbr.i |
|- ( ph -> I e. ( K ... M ) ) |
8 |
|
smatbr.j |
|- ( ph -> J e. ( L ... N ) ) |
9 |
|
fz1ssnn |
|- ( 1 ... M ) C_ NN |
10 |
9 4
|
sselid |
|- ( ph -> K e. NN ) |
11 |
|
fzssnn |
|- ( K e. NN -> ( K ... M ) C_ NN ) |
12 |
10 11
|
syl |
|- ( ph -> ( K ... M ) C_ NN ) |
13 |
12 7
|
sseldd |
|- ( ph -> I e. NN ) |
14 |
|
fz1ssnn |
|- ( 1 ... N ) C_ NN |
15 |
14 5
|
sselid |
|- ( ph -> L e. NN ) |
16 |
|
fzssnn |
|- ( L e. NN -> ( L ... N ) C_ NN ) |
17 |
15 16
|
syl |
|- ( ph -> ( L ... N ) C_ NN ) |
18 |
17 8
|
sseldd |
|- ( ph -> J e. NN ) |
19 |
|
elfzle1 |
|- ( I e. ( K ... M ) -> K <_ I ) |
20 |
7 19
|
syl |
|- ( ph -> K <_ I ) |
21 |
10
|
nnred |
|- ( ph -> K e. RR ) |
22 |
13
|
nnred |
|- ( ph -> I e. RR ) |
23 |
21 22
|
lenltd |
|- ( ph -> ( K <_ I <-> -. I < K ) ) |
24 |
20 23
|
mpbid |
|- ( ph -> -. I < K ) |
25 |
24
|
iffalsed |
|- ( ph -> if ( I < K , I , ( I + 1 ) ) = ( I + 1 ) ) |
26 |
|
elfzle1 |
|- ( J e. ( L ... N ) -> L <_ J ) |
27 |
8 26
|
syl |
|- ( ph -> L <_ J ) |
28 |
15
|
nnred |
|- ( ph -> L e. RR ) |
29 |
18
|
nnred |
|- ( ph -> J e. RR ) |
30 |
28 29
|
lenltd |
|- ( ph -> ( L <_ J <-> -. J < L ) ) |
31 |
27 30
|
mpbid |
|- ( ph -> -. J < L ) |
32 |
31
|
iffalsed |
|- ( ph -> if ( J < L , J , ( J + 1 ) ) = ( J + 1 ) ) |
33 |
1 2 3 4 5 6 13 18 25 32
|
smatlem |
|- ( ph -> ( I S J ) = ( ( I + 1 ) A ( J + 1 ) ) ) |