Theorem smatbl

Description: Entries of a submatrix, bottom left. (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 ) ) ) )
	smatbl.i	|- ( ph -> I e. ( 1 ..^ K ) )
	smatbl.j	|- ( ph -> J e. ( L ... N ) )
Assertion	smatbl	|- ( ph -> ( I S J ) = ( I 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		smatbl.i	|- ( ph -> I e. ( 1 ..^ K ) )
8		smatbl.j	|- ( ph -> J e. ( L ... N ) )
9		fzossnn	|- ( 1 ..^ K ) C_ NN
10	9 7	sseldi	|- ( ph -> I e. NN )
11		fz1ssnn	|- ( 1 ... N ) C_ NN
12	11 5	sseldi	|- ( ph -> L e. NN )
13		fzssnn	|- ( L e. NN -> ( L ... N ) C_ NN )
14	12 13	syl	|- ( ph -> ( L ... N ) C_ NN )
15	14 8	sseldd	|- ( ph -> J e. NN )
16		elfzolt2	|- ( I e. ( 1 ..^ K ) -> I < K )
17	7 16	syl	|- ( ph -> I < K )
18	17	iftrued	|- ( ph -> if ( I < K , I , ( I + 1 ) ) = I )
19		elfzle1	|- ( J e. ( L ... N ) -> L <_ J )
20	8 19	syl	|- ( ph -> L <_ J )
21	12	nnred	|- ( ph -> L e. RR )
22	15	nnred	|- ( ph -> J e. RR )
23	21 22	lenltd	|- ( ph -> ( L <_ J <-> -. J < L ) )
24	20 23	mpbid	|- ( ph -> -. J < L )
25	24	iffalsed	|- ( ph -> if ( J < L , J , ( J + 1 ) ) = ( J + 1 ) )
26	1 2 3 4 5 6 10 15 18 25	smatlem	|- ( ph -> ( I S J ) = ( I A ( J + 1 ) ) )