Theorem lincresunitlem2

Description: Lemma for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019)

	Ref	Expression
Hypotheses	lincresunit.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
	lincresunit.r	⊢ 𝑅 = ( Scalar ‘ 𝑀 )
	lincresunit.e	⊢ 𝐸 = ( Base ‘ 𝑅 )
	lincresunit.u	⊢ 𝑈 = ( Unit ‘ 𝑅 )
	lincresunit.0	⊢ 0 = ( 0g ‘ 𝑅 )
	lincresunit.z	⊢ 𝑍 = ( 0g ‘ 𝑀 )
	lincresunit.n	⊢ 𝑁 = ( invg ‘ 𝑅 )
	lincresunit.i	⊢ 𝐼 = ( invr ‘ 𝑅 )
	lincresunit.t	⊢ · = ( .r ‘ 𝑅 )
	lincresunit.g	⊢ 𝐺 = ( 𝑠 ∈ ( 𝑆 ∖ { 𝑋 } ) ↦ ( ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) · ( 𝐹 ‘ 𝑠 ) ) )
Assertion	lincresunitlem2	⊢ ( ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) ∧ 𝑌 ∈ 𝑆 ) → ( ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) · ( 𝐹 ‘ 𝑌 ) ) ∈ 𝐸 )

Proof

Step	Hyp	Ref	Expression
1		lincresunit.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
2		lincresunit.r	⊢ 𝑅 = ( Scalar ‘ 𝑀 )
3		lincresunit.e	⊢ 𝐸 = ( Base ‘ 𝑅 )
4		lincresunit.u	⊢ 𝑈 = ( Unit ‘ 𝑅 )
5		lincresunit.0	⊢ 0 = ( 0g ‘ 𝑅 )
6		lincresunit.z	⊢ 𝑍 = ( 0g ‘ 𝑀 )
7		lincresunit.n	⊢ 𝑁 = ( invg ‘ 𝑅 )
8		lincresunit.i	⊢ 𝐼 = ( invr ‘ 𝑅 )
9		lincresunit.t	⊢ · = ( .r ‘ 𝑅 )
10		lincresunit.g	⊢ 𝐺 = ( 𝑠 ∈ ( 𝑆 ∖ { 𝑋 } ) ↦ ( ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) · ( 𝐹 ‘ 𝑠 ) ) )
11	2	lmodring	⊢ ( 𝑀 ∈ LMod → 𝑅 ∈ Ring )
12	11	3ad2ant2	⊢ ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) → 𝑅 ∈ Ring )
13	12	adantr	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) → 𝑅 ∈ Ring )
14	13	adantr	⊢ ( ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) ∧ 𝑌 ∈ 𝑆 ) → 𝑅 ∈ Ring )
15	1 2 3 4 5 6 7 8 9 10	lincresunitlem1	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) → ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) ∈ 𝐸 )
16	15	adantr	⊢ ( ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) ∧ 𝑌 ∈ 𝑆 ) → ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) ∈ 𝐸 )
17		elmapi	⊢ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) → 𝐹 : 𝑆 ⟶ 𝐸 )
18		ffvelcdm	⊢ ( ( 𝐹 : 𝑆 ⟶ 𝐸 ∧ 𝑌 ∈ 𝑆 ) → ( 𝐹 ‘ 𝑌 ) ∈ 𝐸 )
19	18	ex	⊢ ( 𝐹 : 𝑆 ⟶ 𝐸 → ( 𝑌 ∈ 𝑆 → ( 𝐹 ‘ 𝑌 ) ∈ 𝐸 ) )
20	17 19	syl	⊢ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) → ( 𝑌 ∈ 𝑆 → ( 𝐹 ‘ 𝑌 ) ∈ 𝐸 ) )
21	20	ad2antrl	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) → ( 𝑌 ∈ 𝑆 → ( 𝐹 ‘ 𝑌 ) ∈ 𝐸 ) )
22	21	imp	⊢ ( ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) ∧ 𝑌 ∈ 𝑆 ) → ( 𝐹 ‘ 𝑌 ) ∈ 𝐸 )
23	3 9	ringcl	⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) ∈ 𝐸 ∧ ( 𝐹 ‘ 𝑌 ) ∈ 𝐸 ) → ( ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) · ( 𝐹 ‘ 𝑌 ) ) ∈ 𝐸 )
24	14 16 22 23	syl3anc	⊢ ( ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) ∧ 𝑌 ∈ 𝑆 ) → ( ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) · ( 𝐹 ‘ 𝑌 ) ) ∈ 𝐸 )