Theorem smadiadetlem1

Description: Lemma 1 for smadiadet : A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019)

	Ref	Expression
Hypotheses	marep01ma.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
	marep01ma.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
	marep01ma.r	⊢ 𝑅 ∈ CRing
	marep01ma.0	⊢ 0 = ( 0g ‘ 𝑅 )
	marep01ma.1	⊢ 1 = ( 1r ‘ 𝑅 )
	smadiadetlem.p	⊢ 𝑃 = ( Base ‘ ( SymGrp ‘ 𝑁 ) )
	smadiadetlem.g	⊢ 𝐺 = ( mulGrp ‘ 𝑅 )
	madetminlem.y	⊢ 𝑌 = ( ℤRHom ‘ 𝑅 )
	madetminlem.s	⊢ 𝑆 = ( pmSgn ‘ 𝑁 )
	madetminlem.t	⊢ · = ( .r ‘ 𝑅 )
Assertion	smadiadetlem1	⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ) ∧ 𝑝 ∈ 𝑃 ) → ( ( ( 𝑌 ∘ 𝑆 ) ‘ 𝑝 ) ( .r ‘ 𝑅 ) ( 𝐺 Σg ( 𝑛 ∈ 𝑁 ↦ ( 𝑛 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐾 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑝 ‘ 𝑛 ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		marep01ma.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
2		marep01ma.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
3		marep01ma.r	⊢ 𝑅 ∈ CRing
4		marep01ma.0	⊢ 0 = ( 0g ‘ 𝑅 )
5		marep01ma.1	⊢ 1 = ( 1r ‘ 𝑅 )
6		smadiadetlem.p	⊢ 𝑃 = ( Base ‘ ( SymGrp ‘ 𝑁 ) )
7		smadiadetlem.g	⊢ 𝐺 = ( mulGrp ‘ 𝑅 )
8		madetminlem.y	⊢ 𝑌 = ( ℤRHom ‘ 𝑅 )
9		madetminlem.s	⊢ 𝑆 = ( pmSgn ‘ 𝑁 )
10		madetminlem.t	⊢ · = ( .r ‘ 𝑅 )
11	1 2 3 4 5	marep01ma	⊢ ( 𝑀 ∈ 𝐵 → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐾 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ∈ 𝐵 )
12	11	ad2antrr	⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ) ∧ 𝑝 ∈ 𝑃 ) → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐾 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ∈ 𝐵 )
13		simpr	⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ) ∧ 𝑝 ∈ 𝑃 ) → 𝑝 ∈ 𝑃 )
14	6 9 8 1 2 7	madetsmelbas2	⊢ ( ( 𝑅 ∈ CRing ∧ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐾 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ∈ 𝐵 ∧ 𝑝 ∈ 𝑃 ) → ( ( ( 𝑌 ∘ 𝑆 ) ‘ 𝑝 ) ( .r ‘ 𝑅 ) ( 𝐺 Σg ( 𝑛 ∈ 𝑁 ↦ ( 𝑛 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐾 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑝 ‘ 𝑛 ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
15	3 12 13 14	mp3an2i	⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ) ∧ 𝑝 ∈ 𝑃 ) → ( ( ( 𝑌 ∘ 𝑆 ) ‘ 𝑝 ) ( .r ‘ 𝑅 ) ( 𝐺 Σg ( 𝑛 ∈ 𝑁 ↦ ( 𝑛 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐾 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑝 ‘ 𝑛 ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) )