Theorem lincresunit3 48969

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 1138	. . . 4 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑀 ∈ LMod)
2	1	3ad2ant1 1134	. . 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 1137	. . . . . . . 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 1149	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) → (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈))
5	4	3ad2ant2 1135	. . . . . . . 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 511	. . . . . . 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 4072	. . . . . . 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 48964	. . . . . . 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 597	. . . . . 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 6837	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(Scalar‘𝑀))
21	10, 20	eqtri 2760	. . . . . 6 ⊢ 𝐸 = (Base‘(Scalar‘𝑀))
22	19, 21	eleqtrdi 2847	. . . . 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 7060	. . . 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 6847	. . . . 5 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
25		difexg 5266	. . . . . . 7 ⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∖ {𝑋}) ∈ V)
26	25	3ad2ant1 1134	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ V)
27	26	3ad2ant1 1134	. . . . 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 8779	. . . . 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 588	. . . 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 257	. . 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 5266	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑋}) ∈ V)
32	31	adantl 481	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ∈ V)
33		ssdifss 4081	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
34	33	a1i 11	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑆 → (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀)))
35		elpwi 4549	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
36	34, 35	impel 505	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
37	32, 36	elpwd 4548	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
38	37	expcom 413	. . . . . . 7 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑋 ∈ 𝑆 → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
39	8	pweqi 4558	. . . . . . 7 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
40	38, 39	eleq2s 2855	. . . . . 6 ⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑋 ∈ 𝑆 → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
41	40	imp 406	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
42	41	3adant2 1132	. . . 4 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
43	42	3ad2ant1 1134	. . 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 48897	. . 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 1374	. 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 1137	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑆 ∈ 𝒫 𝐵)
47		simp3 1139	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
48	1, 46, 47	3jca 1129	. . . . . . 7 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆))
49	48	adantr 480	. . . . . 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 1150	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) → (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ 𝐹 finSupp 0 ))
51	50	adantl 481	. . . . . 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 2738	. . . . . 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 2737	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
54		eqid 2737	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
55	8, 9, 10, 53, 54, 12	lincdifsn 48912	. . . . . 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 1374	. . . . 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 2739	. . . 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 6834	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑧 → (𝐺‘𝑠) = (𝐺‘𝑧))
59		id 22	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑧 → 𝑠 = 𝑧)
60	58, 59	oveq12d 7378	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑧 → ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠) = ((𝐺‘𝑧)( ·𝑠 ‘𝑀)𝑧))
61	60	cbvmptv 5190	. . . . . . . . . . 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 7376	. . . . . . . . 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 7376	. . . . . . . 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 48968	. . . . . . . 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 2773	. . . . . . 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 7375	. . . . . 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 2739	. . . . 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 20853	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
70	69	3ad2ant2 1135	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑀 ∈ Grp)
71	70	adantr 480	. . . . . . 7 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝑀 ∈ Grp)
72	1	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝑀 ∈ LMod)
73		elmapi 8789	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐸 ↑m 𝑆) → 𝐹:𝑆⟶𝐸)
74	73	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) → 𝐹:𝑆⟶𝐸)
75		ffvelcdm 7027	. . . . . . . . 9 ⊢ ((𝐹:𝑆⟶𝐸 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ 𝐸)
76	74, 47, 75	syl2anr 598	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝐹‘𝑋) ∈ 𝐸)
77		elpwi 4549	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝒫 𝐵 → 𝑆 ⊆ 𝐵)
78	77	sselda 3922	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵)
79	78	3adant2 1132	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵)
80	79	adantr 480	. . . . . . . 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 20864	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ (𝐹‘𝑋) ∈ 𝐸 ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵)
82	72, 76, 80, 81	syl3anc 1374	. . . . . . 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 20855	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Grp)
84	83	3ad2ant2 1135	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑅 ∈ Grp)
85	10, 14	grpinvcl 18954	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ (𝐹‘𝑋) ∈ 𝐸) → (𝑁‘(𝐹‘𝑋)) ∈ 𝐸)
86	84, 76, 85	syl2an2r 686	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑁‘(𝐹‘𝑋)) ∈ 𝐸)
87		lmodcmn 20896	. . . . . . . . . . 11 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
88	87	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑀 ∈ CMnd)
89	88	adantr 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝑀 ∈ CMnd)
90	26	adantr 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑆 ∖ {𝑋}) ∈ V)
91		simpll2 1215	. . . . . . . . . . 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 48965	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋})))
93	92	3adantr3 1173	. . . . . . . . . . . . 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 8789	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋})) → 𝐺:(𝑆 ∖ {𝑋})⟶𝐸)
95	93, 94	syl 17	. . . . . . . . . . . 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 7030	. . . . . . . . . . 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 3917	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝐵)
98	97	expcom 413	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ 𝑆 → (𝑆 ⊆ 𝐵 → 𝑠 ∈ 𝐵))
99	7, 77, 98	syl2imc 41	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠 ∈ 𝐵))
100	99	3ad2ant1 1134	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠 ∈ 𝐵))
101	100	adantr 480	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 20864	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ LMod ∧ (𝐺‘𝑠) ∈ 𝐸 ∧ 𝑠 ∈ 𝐵) → ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠) ∈ 𝐵)
104	91, 96, 102, 103	syl3anc 1374	. . . . . . . . . 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 7061	. . . . . . . . 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 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ V)
107		ssdifss 4081	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ⊆ 𝐵 → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
108	77, 107	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
109	108	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
110	109, 8	sseqtrdi 3963	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
111	106, 110	elpwd 4548	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
112	111	3adant2 1132	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
113	1, 112	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
114	113	adantr 480	. . . . . . . . . . 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 48966	. . . . . . . . . . . 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 5127	. . . . . . . . . . 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 48863	. . . . . . . . . . 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 1374	. . . . . . . . . 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 5128	. . . . . . . . 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 19881	. . . . . . . 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 20864	. . . . . . . 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 1374	. . . . . . 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 2737	. . . . . . . 8 ⊢ (invg‘𝑀) = (invg‘𝑀)
124	8, 54, 13, 123	grpinvid2 18959	. . . . . . 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 1374	. . . . . 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 20892	. . . . . . . 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 2739	. . . . . . 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 1197	. . . . . . . . 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 48962	. . . . . . . . . 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 2744	. . . . . . . . . 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 287	. . . . . . . . 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 1376	. . . . . . . 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 229	. . . . . . 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 240	. . . . . 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 260	. . . . 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 240	. . . 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 240	. . 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 1118	. 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 2772	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  𝒫 cpw 4542  {csn 4568   class class class wbr 5086   ↦ cmpt 5167   ↾ cres 5626  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360   ↑_m cmap 8766   finSupp cfsupp 9267  Basecbs 17170  +_gcplusg 17211  ._rcmulr 17212  Scalarcsca 17214   ·_𝑠 cvsca 17215  0_gc0g 17393   Σ_g cgsu 17394  Grpcgrp 18900  inv_gcminusg 18901  CMndccmn 19746  Unitcui 20326  inv_rcinvr 20358  LModclmod 20846   linC clinc 48892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8104  df-tpos 8169  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-er 8636  df-map 8768  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9268  df-oi 9418  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-fzo 13600  df-seq 13955  df-hash 14284  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-0g 17395  df-gsum 17396  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-submnd 18743  df-grp 18903  df-minusg 18904  df-mulg 19035  df-ghm 19179  df-cntz 19283  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-oppr 20308  df-dvdsr 20328  df-unit 20329  df-invr 20359  df-lmod 20848  df-linc 48894

This theorem is referenced by:  lincreslvec3  48970