Theorem lincresunit3 49261

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.)

Hypotheses

Ref	Expression
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

Ref	Expression
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)

Distinct variable groups:   𝐵,𝑠   𝐸,𝑠   𝐹,𝑠   𝑀,𝑠   𝑆,𝑠   𝑋,𝑠   𝑈,𝑠   𝐼,𝑠   𝑁,𝑠   · ,𝑠   0 ,𝑠   𝐺,𝑠   𝑅,𝑠   𝑍,𝑠

Proof of Theorem lincresunit3

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1155	. . . 4 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑀 ∈ LMod)
2	1	3ad2ant1 1151	. . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → 𝑀 ∈ LMod)
3		simp1 1154	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆))
4		3simpa 1166	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) → (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈))
5	4	3ad2ant2 1152	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈))
6	3, 5	jca 520	. . . . . . 7 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)))
7		eldifi 4085	. . . . . . 7 ⊢ (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠 ∈ 𝑆)
8		lincresunit.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑀)
9		lincresunit.r	. . . . . . . 8 ⊢ 𝑅 = (Scalar‘𝑀)
10		lincresunit.e	. . . . . . . 8 ⊢ 𝐸 = (Base‘𝑅)
11		lincresunit.u	. . . . . . . 8 ⊢ 𝑈 = (Unit‘𝑅)
12		lincresunit.0	. . . . . . . 8 ⊢ 0 = (0g‘𝑅)
13		lincresunit.z	. . . . . . . 8 ⊢ 𝑍 = (0g‘𝑀)
14		lincresunit.n	. . . . . . . 8 ⊢ 𝑁 = (invg‘𝑅)
15		lincresunit.i	. . . . . . . 8 ⊢ 𝐼 = (invr‘𝑅)
16		lincresunit.t	. . . . . . . 8 ⊢ · = (.r‘𝑅)
17		lincresunit.g	. . . . . . . 8 ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)))
18	8, 9, 10, 11, 12, 13, 14, 15, 16, 17	lincresunitlem2 49256	. . . . . . 7 ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) ∧ 𝑠 ∈ 𝑆) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)) ∈ 𝐸)
19	6, 7, 18	syl2an 607	. . . . . 6 ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)) ∈ 𝐸)
20	9	fveq2i 6884	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(Scalar‘𝑀))
21	10, 20	eqtri 2786	. . . . . 6 ⊢ 𝐸 = (Base‘(Scalar‘𝑀))
22	19, 21	eleqtrdi 2873	. . . . 5 ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)) ∈ (Base‘(Scalar‘𝑀)))
23	22, 17	fmptd 7109	. . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → 𝐺:(𝑆 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀)))
24		fvex 6894	. . . . 5 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
25		difexg 5300	. . . . . . 7 ⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∖ {𝑋}) ∈ V)
26	25	3ad2ant1 1151	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ V)
27	26	3ad2ant1 1151	. . . . 5 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑆 ∖ {𝑋}) ∈ V)
28		elmapg 8832	. . . . 5 ⊢ (((Base‘(Scalar‘𝑀)) ∈ V ∧ (𝑆 ∖ {𝑋}) ∈ V) → (𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑋})) ↔ 𝐺:(𝑆 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀))))
29	24, 27, 28	sylancr 598	. . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑋})) ↔ 𝐺:(𝑆 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀))))
30	23, 29	mpbird 260	. . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → 𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑋})))
31		difexg 5300	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑋}) ∈ V)
32	31	adantl 486	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ∈ V)
33		ssdifss 4094	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
34	33	a1i 11	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑆 → (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀)))
35		elpwi 4569	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
36	34, 35	impel 514	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
37	32, 36	elpwd 4568	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
38	37	expcom 418	. . . . . . 7 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑋 ∈ 𝑆 → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
39	8	pweqi 4578	. . . . . . 7 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
40	38, 39	eleq2s 2881	. . . . . 6 ⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑋 ∈ 𝑆 → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
41	40	imp 411	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
42	41	3adant2 1149	. . . 4 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
43	42	3ad2ant1 1151	. . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
44		lincval 49189	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑋})) ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))))
45	2, 30, 43, 44	syl3anc 1398	. 2 ⊢ (((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})) = (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))))
46		simp1 1154	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑆 ∈ 𝒫 𝐵)
47		simp3 1156	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
48	1, 46, 47	3jca 1146	. . . . . . 7 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆))
49	48	adantr 485	. . . . . 6 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆))
50		3simpb 1167	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) → (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ 𝐹 finSupp 0 ))
51	50	adantl 486	. . . . . 6 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ 𝐹 finSupp 0 ))
52		eqidd 2764	. . . . . 6 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝐹 ↾ (𝑆 ∖ {𝑋})) = (𝐹 ↾ (𝑆 ∖ {𝑋})))
53		eqid 2763	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
54		eqid 2763	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
55	8, 9, 10, 53, 54, 12	lincdifsn 49204	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ 𝐹 finSupp 0 ) ∧ (𝐹 ↾ (𝑆 ∖ {𝑋})) = (𝐹 ↾ (𝑆 ∖ {𝑋}))) → (𝐹( linC ‘𝑀)𝑆) = (((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋)))
56	49, 51, 52, 55	syl3anc 1398	. . . . 5 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝐹( linC ‘𝑀)𝑆) = (((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋)))
57	56	eqeq1d 2765	. . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((𝐹( linC ‘𝑀)𝑆) = 𝑍 ↔ (((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋)) = 𝑍))
58		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑧 → (𝐺‘𝑠) = (𝐺‘𝑧))
59		id 23	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑧 → 𝑠 = 𝑧)
60	58, 59	oveq12d 7428	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑧 → ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠) = ((𝐺‘𝑧)( ·𝑠 ‘𝑀)𝑧))
61	60	cbvmptv 5215	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)) = (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑧)( ·𝑠 ‘𝑀)𝑧))
62	61	a1i 11	. . . . . . . . . 10 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)) = (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑧)( ·𝑠 ‘𝑀)𝑧)))
63	62	oveq2d 7426	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))) = (𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑧)( ·𝑠 ‘𝑀)𝑧))))
64	63	oveq2d 7426	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)))) = ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑧)( ·𝑠 ‘𝑀)𝑧)))))
65	8, 9, 10, 11, 12, 13, 14, 15, 16, 17	lincresunit3lem2 49260	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑧)( ·𝑠 ‘𝑀)𝑧)))) = ((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋})))
66	64, 65	eqtr2d 2799	. . . . . . 7 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋})) = ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)))))
67	66	oveq1d 7425	. . . . . 6 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋)) = (((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋)))
68	67	eqeq1d 2765	. . . . 5 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋)) = 𝑍 ↔ (((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋)) = 𝑍))
69		lmodgrp 20988	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
70	69	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑀 ∈ Grp)
71	70	adantr 485	. . . . . . 7 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝑀 ∈ Grp)
72	1	adantr 485	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝑀 ∈ LMod)
73		elmapi 8842	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐸 ↑m 𝑆) → 𝐹:𝑆⟶𝐸)
74	73	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) → 𝐹:𝑆⟶𝐸)
75		ffvelcdm 7076	. . . . . . . . 9 ⊢ ((𝐹:𝑆⟶𝐸 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ 𝐸)
76	74, 47, 75	syl2anr 608	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝐹‘𝑋) ∈ 𝐸)
77		elpwi 4569	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝒫 𝐵 → 𝑆 ⊆ 𝐵)
78	77	sselda 3937	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵)
79	78	3adant2 1149	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵)
80	79	adantr 485	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝑋 ∈ 𝐵)
81	8, 9, 53, 10	lmodvscl 20999	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ (𝐹‘𝑋) ∈ 𝐸 ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵)
82	72, 76, 80, 81	syl3anc 1398	. . . . . . 7 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵)
83	9	lmodfgrp 20990	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Grp)
84	83	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑅 ∈ Grp)
85	10, 14	grpinvcl 19049	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ (𝐹‘𝑋) ∈ 𝐸) → (𝑁‘(𝐹‘𝑋)) ∈ 𝐸)
86	84, 76, 85	syl2an2r 697	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑁‘(𝐹‘𝑋)) ∈ 𝐸)
87		lmodcmn 21031	. . . . . . . . . . 11 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
88	87	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑀 ∈ CMnd)
89	88	adantr 485	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝑀 ∈ CMnd)
90	26	adantr 485	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑆 ∖ {𝑋}) ∈ V)
91		simpll2 1232	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → 𝑀 ∈ LMod)
92	8, 9, 10, 11, 12, 13, 14, 15, 16, 17	lincresunit1 49257	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋})))
93	92	3adantr3 1190	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋})))
94		elmapi 8842	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋})) → 𝐺:(𝑆 ∖ {𝑋})⟶𝐸)
95	93, 94	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝐺:(𝑆 ∖ {𝑋})⟶𝐸)
96	95	ffvelcdmda 7079	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → (𝐺‘𝑠) ∈ 𝐸)
97		ssel2 3932	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝐵)
98	97	expcom 418	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ 𝑆 → (𝑆 ⊆ 𝐵 → 𝑠 ∈ 𝐵))
99	7, 77, 98	syl2imc 42	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠 ∈ 𝐵))
100	99	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠 ∈ 𝐵))
101	100	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠 ∈ 𝐵))
102	101	imp 411	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → 𝑠 ∈ 𝐵)
103	8, 9, 53, 10	lmodvscl 20999	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ LMod ∧ (𝐺‘𝑠) ∈ 𝐸 ∧ 𝑠 ∈ 𝐵) → ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠) ∈ 𝐵)
104	91, 96, 102, 103	syl3anc 1398	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠) ∈ 𝐵)
105	104	fmpttd 7110	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)):(𝑆 ∖ {𝑋})⟶𝐵)
106	25	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ V)
107		ssdifss 4094	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ⊆ 𝐵 → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
108	77, 107	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
109	108	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
110	109, 8	sseqtrdi 3977	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
111	106, 110	elpwd 4568	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
112	111	3adant2 1149	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
113	1, 112	jca 520	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
114	113	adantr 485	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
115	8, 9, 10, 11, 12, 13, 14, 15, 16, 17	lincresunit2 49258	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝐺 finSupp 0 )
116	115, 12	breqtrdi 5152	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝐺 finSupp (0g‘𝑅))
117	9, 10	scmfsupp 49155	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋})) ∧ 𝐺 finSupp (0g‘𝑅)) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)) finSupp (0g‘𝑀))
118	114, 93, 116, 117	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)) finSupp (0g‘𝑀))
119	118, 13	breqtrrdi 5153	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)) finSupp 𝑍)
120	8, 13, 89, 90, 105, 119	gsumcl 19980	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))) ∈ 𝐵)
121	8, 9, 53, 10	lmodvscl 20999	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ (𝑁‘(𝐹‘𝑋)) ∈ 𝐸 ∧ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))) ∈ 𝐵) → ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)))) ∈ 𝐵)
122	72, 86, 120, 121	syl3anc 1398	. . . . . . 7 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)))) ∈ 𝐵)
123		eqid 2763	. . . . . . . 8 ⊢ (invg‘𝑀) = (invg‘𝑀)
124	8, 54, 13, 123	grpinvid2 19054	. . . . . . 7 ⊢ ((𝑀 ∈ Grp ∧ ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵 ∧ ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)))) ∈ 𝐵) → (((invg‘𝑀)‘((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋)) = ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)))) ↔ (((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋)) = 𝑍))
125	71, 82, 122, 124	syl3anc 1398	. . . . . 6 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (((invg‘𝑀)‘((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋)) = ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)))) ↔ (((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋)) = 𝑍))
126	8, 9, 53, 123, 10, 14, 72, 80, 76	lmodvsneg 21027	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((invg‘𝑀)‘((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋)) = ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)𝑋))
127	126	eqeq1d 2765	. . . . . . 7 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (((invg‘𝑀)‘((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋)) = ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)))) ↔ ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)𝑋) = ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))))))
128		simpr2 1214	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝐹‘𝑋) ∈ 𝑈)
129	8, 9, 10, 11, 14, 53	lincresunit3lem3 49254	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))) ∈ 𝐵) ∧ (𝐹‘𝑋) ∈ 𝑈) → (((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)𝑋) = ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)))) ↔ 𝑋 = (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)))))
130		eqcom 2770	. . . . . . . . . 10 ⊢ (𝑋 = (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))) ↔ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))) = 𝑋)
131	129, 130	bitrdi 290	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))) ∈ 𝐵) ∧ (𝐹‘𝑋) ∈ 𝑈) → (((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)𝑋) = ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)))) ↔ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))) = 𝑋))
132	72, 80, 120, 128, 131	syl31anc 1400	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)𝑋) = ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)))) ↔ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))) = 𝑋))
133	132	biimpd 232	. . . . . . 7 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)𝑋) = ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)))) → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))) = 𝑋))
134	127, 133	sylbid 243	. . . . . 6 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (((invg‘𝑀)‘((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋)) = ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠)))) → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))) = 𝑋))
135	125, 134	sylbird 263	. . . . 5 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋)) = 𝑍 → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))) = 𝑋))
136	68, 135	sylbid 243	. . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋)) = 𝑍 → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))) = 𝑋))
137	57, 136	sylbid 243	. . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((𝐹( linC ‘𝑀)𝑆) = 𝑍 → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))) = 𝑋))
138	137	3impia 1135	. 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠))) = 𝑋)
139	45, 138	eqtrd 2798	1 ⊢ (((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)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  𝒫 cpw 4562  {csn 4589   class class class wbr 5109   ↦ cmpt 5192   ↾ cres 5663  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8820   finSupp cfsupp 9317  Basecbs 17264  +_gcplusg 17305  ._rcmulr 17306  Scalarcsca 17308   ·_𝑠 cvsca 17309  0_gc0g 17487   Σ_g cgsu 17488  Grpcgrp 18995  inv_gcminusg 18996  CMndccmn 19845  Unitcui 20433  inv_rcinvr 20465  LModclmod 20981   linC clinc 49184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-oi 9468  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-n0 12500  df-z 12587  df-uz 12858  df-fz 13531  df-fzo 13679  df-seq 14034  df-hash 14363  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-mulr 17319  df-0g 17489  df-gsum 17490  df-mre 17633  df-mrc 17634  df-acs 17636  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-mhm 18836  df-submnd 18837  df-grp 18998  df-minusg 18999  df-mulg 19129  df-ghm 19279  df-cntz 19382  df-cmn 19847  df-abl 19848  df-mgp 20212  df-rng 20226  df-ur 20259  df-ring 20312  df-oppr 20415  df-dvdsr 20435  df-unit 20436  df-invr 20466  df-lmod 20983  df-linc 49186

This theorem is referenced by:  lincreslvec3  49262