Metamath Proof Explorer


Theorem smatbr

Description: Entries of a submatrix, bottom right. (Contributed by Thierry Arnoux, 19-Aug-2020)

Ref Expression
Hypotheses smat.s
|- S = ( K ( subMat1 ` A ) L )
smat.m
|- ( ph -> M e. NN )
smat.n
|- ( ph -> N e. NN )
smat.k
|- ( ph -> K e. ( 1 ... M ) )
smat.l
|- ( ph -> L e. ( 1 ... N ) )
smat.a
|- ( ph -> A e. ( B ^m ( ( 1 ... M ) X. ( 1 ... N ) ) ) )
smatbr.i
|- ( ph -> I e. ( K ... M ) )
smatbr.j
|- ( ph -> J e. ( L ... N ) )
Assertion smatbr
|- ( ph -> ( I S J ) = ( ( I + 1 ) A ( J + 1 ) ) )

Proof

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 ) ) )