Theorem lincreslvec3

Description: Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019) (Proof shortened by AV, 30-Jul-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	lincreslvec3	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ ( 𝐹 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ( 𝐺 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑋 } ) ) = 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		lveclmod	⊢ ( 𝑀 ∈ LVec → 𝑀 ∈ LMod )
12	11	3anim2i	⊢ ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆 ) → ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) )
13	12	3ad2ant1	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ ( 𝐹 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) )
14		simp21	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ ( 𝐹 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) )
15		elmapi	⊢ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) → 𝐹 : 𝑆 ⟶ 𝐸 )
16	15	3ad2ant1	⊢ ( ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ 0 ∧ 𝐹 finSupp 0 ) → 𝐹 : 𝑆 ⟶ 𝐸 )
17		simp3	⊢ ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆 ) → 𝑋 ∈ 𝑆 )
18		ffvelcdm	⊢ ( ( 𝐹 : 𝑆 ⟶ 𝐸 ∧ 𝑋 ∈ 𝑆 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐸 )
19	16 17 18	syl2anr	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ 0 ∧ 𝐹 finSupp 0 ) ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐸 )
20		simpr2	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ 0 ∧ 𝐹 finSupp 0 ) ) → ( 𝐹 ‘ 𝑋 ) ≠ 0 )
21	2	lvecdrng	⊢ ( 𝑀 ∈ LVec → 𝑅 ∈ DivRing )
22	21	3ad2ant2	⊢ ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆 ) → 𝑅 ∈ DivRing )
23	22	adantr	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ 0 ∧ 𝐹 finSupp 0 ) ) → 𝑅 ∈ DivRing )
24	3 4 5	drngunit	⊢ ( 𝑅 ∈ DivRing → ( ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ↔ ( ( 𝐹 ‘ 𝑋 ) ∈ 𝐸 ∧ ( 𝐹 ‘ 𝑋 ) ≠ 0 ) ) )
25	23 24	syl	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ 0 ∧ 𝐹 finSupp 0 ) ) → ( ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ↔ ( ( 𝐹 ‘ 𝑋 ) ∈ 𝐸 ∧ ( 𝐹 ‘ 𝑋 ) ≠ 0 ) ) )
26	19 20 25	mpbir2and	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ 0 ∧ 𝐹 finSupp 0 ) ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 )
27	26	3adant3	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ ( 𝐹 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 )
28		simp3	⊢ ( ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ 0 ∧ 𝐹 finSupp 0 ) → 𝐹 finSupp 0 )
29	28	3ad2ant2	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ ( 𝐹 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → 𝐹 finSupp 0 )
30		simp3	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ ( 𝐹 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ( 𝐹 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 )
31	1 2 3 4 5 6 7 8 9 10	lincresunit3	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ ( 𝐹 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ( 𝐺 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑋 } ) ) = 𝑋 )
32	13 14 27 29 30 31	syl131anc	⊢ ( ( ( 𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆 ) ∧ ( 𝐹 ∈ ( 𝐸 ↑m 𝑆 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ ( 𝐹 ( linC ‘ 𝑀 ) 𝑆 ) = 𝑍 ) → ( 𝐺 ( linC ‘ 𝑀 ) ( 𝑆 ∖ { 𝑋 } ) ) = 𝑋 )