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	⊢ ( 𝜑 → 𝑀 ∈ ℕ )
	smat.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
	smat.k	⊢ ( 𝜑 → 𝐾 ∈ ( 1 ... 𝑀 ) )
	smat.l	⊢ ( 𝜑 → 𝐿 ∈ ( 1 ... 𝑁 ) )
	smat.a	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 ↑m ( ( 1 ... 𝑀 ) × ( 1 ... 𝑁 ) ) ) )
	smatbl.i	⊢ ( 𝜑 → 𝐼 ∈ ( 1 ..^ 𝐾 ) )
	smatbl.j	⊢ ( 𝜑 → 𝐽 ∈ ( 𝐿 ... 𝑁 ) )
Assertion	smatbl	⊢ ( 𝜑 → ( 𝐼 𝑆 𝐽 ) = ( 𝐼 𝐴 ( 𝐽 + 1 ) ) )

Proof

Step	Hyp	Ref	Expression
1		smat.s	⊢ 𝑆 = ( 𝐾 ( subMat1 ‘ 𝐴 ) 𝐿 )
2		smat.m	⊢ ( 𝜑 → 𝑀 ∈ ℕ )
3		smat.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
4		smat.k	⊢ ( 𝜑 → 𝐾 ∈ ( 1 ... 𝑀 ) )
5		smat.l	⊢ ( 𝜑 → 𝐿 ∈ ( 1 ... 𝑁 ) )
6		smat.a	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 ↑m ( ( 1 ... 𝑀 ) × ( 1 ... 𝑁 ) ) ) )
7		smatbl.i	⊢ ( 𝜑 → 𝐼 ∈ ( 1 ..^ 𝐾 ) )
8		smatbl.j	⊢ ( 𝜑 → 𝐽 ∈ ( 𝐿 ... 𝑁 ) )
9		fzossnn	⊢ ( 1 ..^ 𝐾 ) ⊆ ℕ
10	9 7	sselid	⊢ ( 𝜑 → 𝐼 ∈ ℕ )
11		fz1ssnn	⊢ ( 1 ... 𝑁 ) ⊆ ℕ
12	11 5	sselid	⊢ ( 𝜑 → 𝐿 ∈ ℕ )
13		fzssnn	⊢ ( 𝐿 ∈ ℕ → ( 𝐿 ... 𝑁 ) ⊆ ℕ )
14	12 13	syl	⊢ ( 𝜑 → ( 𝐿 ... 𝑁 ) ⊆ ℕ )
15	14 8	sseldd	⊢ ( 𝜑 → 𝐽 ∈ ℕ )
16		elfzolt2	⊢ ( 𝐼 ∈ ( 1 ..^ 𝐾 ) → 𝐼 < 𝐾 )
17	7 16	syl	⊢ ( 𝜑 → 𝐼 < 𝐾 )
18	17	iftrued	⊢ ( 𝜑 → if ( 𝐼 < 𝐾 , 𝐼 , ( 𝐼 + 1 ) ) = 𝐼 )
19		elfzle1	⊢ ( 𝐽 ∈ ( 𝐿 ... 𝑁 ) → 𝐿 ≤ 𝐽 )
20	8 19	syl	⊢ ( 𝜑 → 𝐿 ≤ 𝐽 )
21	12	nnred	⊢ ( 𝜑 → 𝐿 ∈ ℝ )
22	15	nnred	⊢ ( 𝜑 → 𝐽 ∈ ℝ )
23	21 22	lenltd	⊢ ( 𝜑 → ( 𝐿 ≤ 𝐽 ↔ ¬ 𝐽 < 𝐿 ) )
24	20 23	mpbid	⊢ ( 𝜑 → ¬ 𝐽 < 𝐿 )
25	24	iffalsed	⊢ ( 𝜑 → if ( 𝐽 < 𝐿 , 𝐽 , ( 𝐽 + 1 ) ) = ( 𝐽 + 1 ) )
26	1 2 3 4 5 6 10 15 18 25	smatlem	⊢ ( 𝜑 → ( 𝐼 𝑆 𝐽 ) = ( 𝐼 𝐴 ( 𝐽 + 1 ) ) )