Theorem smadiadetlem3lem0

Description: Lemma 0 for smadiadetlem3 . (Contributed by AV, 12-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 )
	smadiadetlem.w	⊢ 𝑊 = ( Base ‘ ( SymGrp ‘ ( 𝑁 ∖ { 𝐾 } ) ) )
	smadiadetlem.z	⊢ 𝑍 = ( pmSgn ‘ ( 𝑁 ∖ { 𝐾 } ) )
Assertion	smadiadetlem3lem0	⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ) ∧ 𝑄 ∈ 𝑊 ) → ( ( ( 𝑌 ∘ 𝑍 ) ‘ 𝑄 ) ( .r ‘ 𝑅 ) ( 𝐺 Σg ( 𝑛 ∈ ( 𝑁 ∖ { 𝐾 } ) ↦ ( 𝑛 ( 𝑖 ∈ ( 𝑁 ∖ { 𝐾 } ) , 𝑗 ∈ ( 𝑁 ∖ { 𝐾 } ) ↦ ( 𝑖 𝑀 𝑗 ) ) ( 𝑄 ‘ 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		smadiadetlem.w	⊢ 𝑊 = ( Base ‘ ( SymGrp ‘ ( 𝑁 ∖ { 𝐾 } ) ) )
12		smadiadetlem.z	⊢ 𝑍 = ( pmSgn ‘ ( 𝑁 ∖ { 𝐾 } ) )
13		difssd	⊢ ( 𝐾 ∈ 𝑁 → ( 𝑁 ∖ { 𝐾 } ) ⊆ 𝑁 )
14	13	anim2i	⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ) → ( 𝑀 ∈ 𝐵 ∧ ( 𝑁 ∖ { 𝐾 } ) ⊆ 𝑁 ) )
15	14	adantr	⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ) ∧ 𝑄 ∈ 𝑊 ) → ( 𝑀 ∈ 𝐵 ∧ ( 𝑁 ∖ { 𝐾 } ) ⊆ 𝑁 ) )
16	1 2	submabas	⊢ ( ( 𝑀 ∈ 𝐵 ∧ ( 𝑁 ∖ { 𝐾 } ) ⊆ 𝑁 ) → ( 𝑖 ∈ ( 𝑁 ∖ { 𝐾 } ) , 𝑗 ∈ ( 𝑁 ∖ { 𝐾 } ) ↦ ( 𝑖 𝑀 𝑗 ) ) ∈ ( Base ‘ ( ( 𝑁 ∖ { 𝐾 } ) Mat 𝑅 ) ) )
17	15 16	syl	⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ) ∧ 𝑄 ∈ 𝑊 ) → ( 𝑖 ∈ ( 𝑁 ∖ { 𝐾 } ) , 𝑗 ∈ ( 𝑁 ∖ { 𝐾 } ) ↦ ( 𝑖 𝑀 𝑗 ) ) ∈ ( Base ‘ ( ( 𝑁 ∖ { 𝐾 } ) Mat 𝑅 ) ) )
18		simpr	⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ) ∧ 𝑄 ∈ 𝑊 ) → 𝑄 ∈ 𝑊 )
19		eqid	⊢ ( ( 𝑁 ∖ { 𝐾 } ) Mat 𝑅 ) = ( ( 𝑁 ∖ { 𝐾 } ) Mat 𝑅 )
20		eqid	⊢ ( Base ‘ ( ( 𝑁 ∖ { 𝐾 } ) Mat 𝑅 ) ) = ( Base ‘ ( ( 𝑁 ∖ { 𝐾 } ) Mat 𝑅 ) )
21	11 12 8 19 20 7	madetsmelbas2	⊢ ( ( 𝑅 ∈ CRing ∧ ( 𝑖 ∈ ( 𝑁 ∖ { 𝐾 } ) , 𝑗 ∈ ( 𝑁 ∖ { 𝐾 } ) ↦ ( 𝑖 𝑀 𝑗 ) ) ∈ ( Base ‘ ( ( 𝑁 ∖ { 𝐾 } ) Mat 𝑅 ) ) ∧ 𝑄 ∈ 𝑊 ) → ( ( ( 𝑌 ∘ 𝑍 ) ‘ 𝑄 ) ( .r ‘ 𝑅 ) ( 𝐺 Σg ( 𝑛 ∈ ( 𝑁 ∖ { 𝐾 } ) ↦ ( 𝑛 ( 𝑖 ∈ ( 𝑁 ∖ { 𝐾 } ) , 𝑗 ∈ ( 𝑁 ∖ { 𝐾 } ) ↦ ( 𝑖 𝑀 𝑗 ) ) ( 𝑄 ‘ 𝑛 ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
22	3 17 18 21	mp3an2i	⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ) ∧ 𝑄 ∈ 𝑊 ) → ( ( ( 𝑌 ∘ 𝑍 ) ‘ 𝑄 ) ( .r ‘ 𝑅 ) ( 𝐺 Σg ( 𝑛 ∈ ( 𝑁 ∖ { 𝐾 } ) ↦ ( 𝑛 ( 𝑖 ∈ ( 𝑁 ∖ { 𝐾 } ) , 𝑗 ∈ ( 𝑁 ∖ { 𝐾 } ) ↦ ( 𝑖 𝑀 𝑗 ) ) ( 𝑄 ‘ 𝑛 ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) )