Theorem lincresunit3 48669

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 1137	. . . 4 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑀 ∈ LMod)
2	1	3ad2ant1 1133	. . 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 1136	. . . . . . . 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 1148	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) → (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈))
5	4	3ad2ant2 1134	. . . . . . . 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 4081	. . . . . . 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 48664	. . . . . . 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 596	. . . . . 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 6835	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(Scalar‘𝑀))
21	10, 20	eqtri 2757	. . . . . 6 ⊢ 𝐸 = (Base‘(Scalar‘𝑀))
22	19, 21	eleqtrdi 2844	. . . . 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 7057	. . . 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 6845	. . . . 5 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
25		difexg 5272	. . . . . . 7 ⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∖ {𝑋}) ∈ V)
26	25	3ad2ant1 1133	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ V)
27	26	3ad2ant1 1133	. . . . 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 8774	. . . . 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 587	. . . 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 5272	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑋}) ∈ V)
32	31	adantl 481	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ∈ V)
33		ssdifss 4090	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
34	33	a1i 11	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑆 → (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀)))
35		elpwi 4559	. . . . . . . . . 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 4558	. . . . . . . 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 4568	. . . . . . 7 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
40	38, 39	eleq2s 2852	. . . . . 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 1131	. . . 4 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
43	42	3ad2ant1 1133	. . 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 48597	. . 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 1373	. 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 1136	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑆 ∈ 𝒫 𝐵)
47		simp3 1138	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
48	1, 46, 47	3jca 1128	. . . . . . 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 1149	. . . . . . 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 2735	. . . . . 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 2734	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
54		eqid 2734	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
55	8, 9, 10, 53, 54, 12	lincdifsn 48612	. . . . . 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 1373	. . . . 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 2736	. . . 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 6832	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑧 → (𝐺‘𝑠) = (𝐺‘𝑧))
59		id 22	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑧 → 𝑠 = 𝑧)
60	58, 59	oveq12d 7374	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑧 → ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠) = ((𝐺‘𝑧)( ·𝑠 ‘𝑀)𝑧))
61	60	cbvmptv 5200	. . . . . . . . . . 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 7372	. . . . . . . . 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 7372	. . . . . . . 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 48668	. . . . . . . 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 2770	. . . . . . 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 7371	. . . . . 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 2736	. . . . 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 20816	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
70	69	3ad2ant2 1134	. . . . . . . 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 8784	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐸 ↑m 𝑆) → 𝐹:𝑆⟶𝐸)
74	73	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) → 𝐹:𝑆⟶𝐸)
75		ffvelcdm 7024	. . . . . . . . 9 ⊢ ((𝐹:𝑆⟶𝐸 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ 𝐸)
76	74, 47, 75	syl2anr 597	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝐹‘𝑋) ∈ 𝐸)
77		elpwi 4559	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝒫 𝐵 → 𝑆 ⊆ 𝐵)
78	77	sselda 3931	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵)
79	78	3adant2 1131	. . . . . . . . 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 20827	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ (𝐹‘𝑋) ∈ 𝐸 ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵)
82	72, 76, 80, 81	syl3anc 1373	. . . . . . 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 20818	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Grp)
84	83	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑅 ∈ Grp)
85	10, 14	grpinvcl 18915	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ (𝐹‘𝑋) ∈ 𝐸) → (𝑁‘(𝐹‘𝑋)) ∈ 𝐸)
86	84, 76, 85	syl2an2r 685	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑁‘(𝐹‘𝑋)) ∈ 𝐸)
87		lmodcmn 20859	. . . . . . . . . . 11 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
88	87	3ad2ant2 1134	. . . . . . . . . 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 1214	. . . . . . . . . . 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 48665	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋})))
93	92	3adantr3 1172	. . . . . . . . . . . . 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 8784	. . . . . . . . . . . . 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 7027	. . . . . . . . . . 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 3926	. . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . 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 20827	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ LMod ∧ (𝐺‘𝑠) ∈ 𝐸 ∧ 𝑠 ∈ 𝐵) → ((𝐺‘𝑠)( ·𝑠 ‘𝑀)𝑠) ∈ 𝐵)
104	91, 96, 102, 103	syl3anc 1373	. . . . . . . . . 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 7058	. . . . . . . . 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 4090	. . . . . . . . . . . . . . . . . 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 3972	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
111	106, 110	elpwd 4558	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
112	111	3adant2 1131	. . . . . . . . . . . . 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 48666	. . . . . . . . . . . 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 5137	. . . . . . . . . . 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 48563	. . . . . . . . . . 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 1373	. . . . . . . . . 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 5138	. . . . . . . . 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 19842	. . . . . . . 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 20827	. . . . . . . 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 1373	. . . . . . 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 2734	. . . . . . . 8 ⊢ (invg‘𝑀) = (invg‘𝑀)
124	8, 54, 13, 123	grpinvid2 18920	. . . . . . 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 1373	. . . . . 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 20855	. . . . . . . 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 2736	. . . . . . 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 1196	. . . . . . . . 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 48662	. . . . . . . . . 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 2741	. . . . . . . . . 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 1375	. . . . . . . 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 1117	. 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 2769	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 1086   = wceq 1541   ∈ wcel 2113  Vcvv 3438   ∖ cdif 3896   ⊆ wss 3899  𝒫 cpw 4552  {csn 4578   class class class wbr 5096   ↦ cmpt 5177   ↾ cres 5624  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356   ↑_m cmap 8761   finSupp cfsupp 9262  Basecbs 17134  +_gcplusg 17175  ._rcmulr 17176  Scalarcsca 17178   ·_𝑠 cvsca 17179  0_gc0g 17357   Σ_g cgsu 17358  Grpcgrp 18861  inv_gcminusg 18862  CMndccmn 19707  Unitcui 20289  inv_rcinvr 20321  LModclmod 20809   linC clinc 48592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-map 8763  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-fsupp 9263  df-oi 9413  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-3 12207  df-n0 12400  df-z 12487  df-uz 12750  df-fz 13422  df-fzo 13569  df-seq 13923  df-hash 14252  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-mulr 17189  df-0g 17359  df-gsum 17360  df-mre 17503  df-mrc 17504  df-acs 17506  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-mhm 18706  df-submnd 18707  df-grp 18864  df-minusg 18865  df-mulg 18996  df-ghm 19140  df-cntz 19244  df-cmn 19709  df-abl 19710  df-mgp 20074  df-rng 20086  df-ur 20115  df-ring 20168  df-oppr 20271  df-dvdsr 20291  df-unit 20292  df-invr 20322  df-lmod 20811  df-linc 48594

This theorem is referenced by:  lincreslvec3  48670