Theorem lincresunit1

Description: Property 1 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	lincresunit1	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) → 𝐺 ∈ ( 𝐸 ↑m ( 𝑆 ∖ { 𝑋 } ) ) )

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		eldifi	⊢ ( 𝑠 ∈ ( 𝑆 ∖ { 𝑋 } ) → 𝑠 ∈ 𝑆 )
12	1 2 3 4 5 6 7 8 9 10	lincresunitlem2	⊢ ( ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) ∧ 𝑠 ∈ 𝑆 ) → ( ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) · ( 𝐹 ‘ 𝑠 ) ) ∈ 𝐸 )
13	11 12	sylan2	⊢ ( ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) ∧ 𝑠 ∈ ( 𝑆 ∖ { 𝑋 } ) ) → ( ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) · ( 𝐹 ‘ 𝑠 ) ) ∈ 𝐸 )
14	13	fmpttd	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) → ( 𝑠 ∈ ( 𝑆 ∖ { 𝑋 } ) ↦ ( ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) · ( 𝐹 ‘ 𝑠 ) ) ) : ( 𝑆 ∖ { 𝑋 } ) ⟶ 𝐸 )
15	3	fvexi	⊢ 𝐸 ∈ V
16		difexg	⊢ ( 𝑆 ∈ 𝒫 𝐵 → ( 𝑆 ∖ { 𝑋 } ) ∈ V )
17	16	3ad2ant1	⊢ ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) → ( 𝑆 ∖ { 𝑋 } ) ∈ V )
18	17	adantr	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) → ( 𝑆 ∖ { 𝑋 } ) ∈ V )
19		elmapg	⊢ ( ( 𝐸 ∈ V ∧ ( 𝑆 ∖ { 𝑋 } ) ∈ V ) → ( ( 𝑠 ∈ ( 𝑆 ∖ { 𝑋 } ) ↦ ( ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) · ( 𝐹 ‘ 𝑠 ) ) ) ∈ ( 𝐸 ↑m ( 𝑆 ∖ { 𝑋 } ) ) ↔ ( 𝑠 ∈ ( 𝑆 ∖ { 𝑋 } ) ↦ ( ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) · ( 𝐹 ‘ 𝑠 ) ) ) : ( 𝑆 ∖ { 𝑋 } ) ⟶ 𝐸 ) )
20	15 18 19	sylancr	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) → ( ( 𝑠 ∈ ( 𝑆 ∖ { 𝑋 } ) ↦ ( ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) · ( 𝐹 ‘ 𝑠 ) ) ) ∈ ( 𝐸 ↑m ( 𝑆 ∖ { 𝑋 } ) ) ↔ ( 𝑠 ∈ ( 𝑆 ∖ { 𝑋 } ) ↦ ( ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) · ( 𝐹 ‘ 𝑠 ) ) ) : ( 𝑆 ∖ { 𝑋 } ) ⟶ 𝐸 ) )
21	14 20	mpbird	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) → ( 𝑠 ∈ ( 𝑆 ∖ { 𝑋 } ) ↦ ( ( 𝐼 ‘ ( 𝑁 ‘ ( 𝐹 ‘ 𝑋 ) ) ) · ( 𝐹 ‘ 𝑠 ) ) ) ∈ ( 𝐸 ↑m ( 𝑆 ∖ { 𝑋 } ) ) )
22	10 21	eqeltrid	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ) ) → 𝐺 ∈ ( 𝐸 ↑m ( 𝑆 ∖ { 𝑋 } ) ) )