lbsdiflsp0 - Mathbox for Thierry Arnoux

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Theorem lbsdiflsp0 33662

Description: The linear spans of two disjunct independent sets only have a trivial intersection. This can be seen as the opposite direction of lindsun 33661. (Contributed by Thierry Arnoux, 17-May-2023.)

**Hypotheses**
Ref	Expression
lbsdiflsp0.j	⊢ 𝐽 = (LBasis‘𝑊)
lbsdiflsp0.n	⊢ 𝑁 = (LSpan‘𝑊)
lbsdiflsp0.1	⊢ 0 = (0_g‘𝑊)

**Assertion**
Ref	Expression
lbsdiflsp0	⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽 ∧ 𝑉 ⊆ 𝐵) → ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉)) = { 0 })

Proof of Theorem lbsdiflsp0 Dummy variables 𝑎 𝑏 𝑐 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp-4r 783	. . . . . . . . 9 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣))))
2		fveq2 6830	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (𝑎‘𝑢) = (𝑎‘𝑣))
3		id 22	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → 𝑢 = 𝑣)
4	2, 3	oveq12d 7372	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢) = ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣))
5	4	cbvmptv 5199	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑉 ↦ ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢)) = (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣))
6	5	oveq2i 7365	. . . . . . . . 9 ⊢ (𝑊 Σ_g (𝑢 ∈ 𝑉 ↦ ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))
7	1, 6	eqtr4di 2786	. . . . . . . 8 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝑥 = (𝑊 Σ_g (𝑢 ∈ 𝑉 ↦ ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢))))
8		simp-4r 783	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → 𝑎 finSupp (0_g‘(Scalar‘𝑊)))
9		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → 𝑏 finSupp (0_g‘(Scalar‘𝑊)))
10		simp-8l 790	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → 𝑊 ∈ LVec)
11		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝐵 ∈ 𝐽)
12	11	ad6antr 736	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → 𝐵 ∈ 𝐽)
13		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝑉 ⊆ 𝐵)
14	13	ad6antr 736	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → 𝑉 ⊆ 𝐵)
15		simp-5r 785	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉))
16		fvexd 6845	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (Base‘(Scalar‘𝑊)) ∈ V)
17	11, 13	ssexd 5266	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝑉 ∈ V)
18	16, 17	elmapd 8772	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉) ↔ 𝑎:𝑉⟶(Base‘(Scalar‘𝑊))))
19	18	biimpa 476	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) → 𝑎:𝑉⟶(Base‘(Scalar‘𝑊)))
20	10, 12, 14, 15, 19	syl1111anc 840	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → 𝑎:𝑉⟶(Base‘(Scalar‘𝑊)))
21		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉)))
22		lveclmod 21044	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
23	22	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝑊 ∈ LMod)
24		eqid 2733	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (Base‘𝑊) = (Base‘𝑊)
25		lbsdiflsp0.j	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 𝐽 = (LBasis‘𝑊)
26	24, 25	lbsss 21015	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ⊆ (Base‘𝑊))
27	26	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝐵 ⊆ (Base‘𝑊))
28	27	ssdifssd 4096	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝐵 ∖ 𝑉) ⊆ (Base‘𝑊))
29		lbsdiflsp0.1	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 0 = (0_g‘𝑊)
30		lbsdiflsp0.n	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 𝑁 = (LSpan‘𝑊)
31	29, 24, 30	0ellsp 33343	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑊 ∈ LMod ∧ (𝐵 ∖ 𝑉) ⊆ (Base‘𝑊)) → 0 ∈ (𝑁‘(𝐵 ∖ 𝑉)))
32	23, 28, 31	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 0 ∈ (𝑁‘(𝐵 ∖ 𝑉)))
33	32	elfvexd 6866	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝐵 ∖ 𝑉) ∈ V)
34	16, 33	elmapd 8772	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉)) ↔ 𝑏:(𝐵 ∖ 𝑉)⟶(Base‘(Scalar‘𝑊))))
35	34	biimpa 476	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) → 𝑏:(𝐵 ∖ 𝑉)⟶(Base‘(Scalar‘𝑊)))
36	10, 12, 14, 21, 35	syl1111anc 840	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → 𝑏:(𝐵 ∖ 𝑉)⟶(Base‘(Scalar‘𝑊)))
37		disjdif 4421	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑉 ∩ (𝐵 ∖ 𝑉)) = ∅
38	37	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → (𝑉 ∩ (𝐵 ∖ 𝑉)) = ∅)
39	20, 36, 38	fun2d 6694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → (𝑎 ∪ 𝑏):(𝑉 ∪ (𝐵 ∖ 𝑉))⟶(Base‘(Scalar‘𝑊)))
40		undif 4431	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑉 ⊆ 𝐵 ↔ (𝑉 ∪ (𝐵 ∖ 𝑉)) = 𝐵)
41	14, 40	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → (𝑉 ∪ (𝐵 ∖ 𝑉)) = 𝐵)
42	41	feq2d 6642	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → ((𝑎 ∪ 𝑏):(𝑉 ∪ (𝐵 ∖ 𝑉))⟶(Base‘(Scalar‘𝑊)) ↔ (𝑎 ∪ 𝑏):𝐵⟶(Base‘(Scalar‘𝑊))))
43	39, 42	mpbid 232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → (𝑎 ∪ 𝑏):𝐵⟶(Base‘(Scalar‘𝑊)))
44	43	ffund 6662	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → Fun (𝑎 ∪ 𝑏))
45	44	fsuppunbi 9282	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → ((𝑎 ∪ 𝑏) finSupp (0_g‘(Scalar‘𝑊)) ↔ (𝑎 finSupp (0_g‘(Scalar‘𝑊)) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊)))))
46	8, 9, 45	mpbir2and 713	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) → (𝑎 ∪ 𝑏) finSupp (0_g‘(Scalar‘𝑊)))
47	46	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑎 ∪ 𝑏) finSupp (0_g‘(Scalar‘𝑊)))
48		eqid 2733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
49		lmodcmn 20847	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ CMnd)
50	22, 49	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ CMnd)
51	50	ad9antr 742	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝑊 ∈ CMnd)
52	11	ad7antr 738	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝐵 ∈ 𝐽)
53	23	ad8antr 740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) → 𝑊 ∈ LMod)
54		elmapfn 8797	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉) → 𝑎 Fn 𝑉)
55	54	ad6antlr 737	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝑎 Fn 𝑉)
56	55	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝑉) → 𝑎 Fn 𝑉)
57		elmapfn 8797	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉)) → 𝑏 Fn (𝐵 ∖ 𝑉))
58	57	ad3antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝑏 Fn (𝐵 ∖ 𝑉))
59	58	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝑉) → 𝑏 Fn (𝐵 ∖ 𝑉))
60	37	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝑉) → (𝑉 ∩ (𝐵 ∖ 𝑉)) = ∅)
61		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝑉) → 𝑢 ∈ 𝑉)
62		fvun1 6921	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 Fn 𝑉 ∧ 𝑏 Fn (𝐵 ∖ 𝑉) ∧ ((𝑉 ∩ (𝐵 ∖ 𝑉)) = ∅ ∧ 𝑢 ∈ 𝑉)) → ((𝑎 ∪ 𝑏)‘𝑢) = (𝑎‘𝑢))
63	56, 59, 60, 61, 62	syl112anc 1376	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝑉) → ((𝑎 ∪ 𝑏)‘𝑢) = (𝑎‘𝑢))
64	63	adantlr 715	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑢 ∈ 𝑉) → ((𝑎 ∪ 𝑏)‘𝑢) = (𝑎‘𝑢))
65	20	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑢 ∈ 𝑉) → 𝑎:𝑉⟶(Base‘(Scalar‘𝑊)))
66		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑢 ∈ 𝑉) → 𝑢 ∈ 𝑉)
67	65, 66	ffvelcdmd 7026	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑢 ∈ 𝑉) → (𝑎‘𝑢) ∈ (Base‘(Scalar‘𝑊)))
68	64, 67	eqeltrd 2833	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑢 ∈ 𝑉) → ((𝑎 ∪ 𝑏)‘𝑢) ∈ (Base‘(Scalar‘𝑊)))
69	55	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ (𝐵 ∖ 𝑉)) → 𝑎 Fn 𝑉)
70	58	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ (𝐵 ∖ 𝑉)) → 𝑏 Fn (𝐵 ∖ 𝑉))
71	37	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ (𝐵 ∖ 𝑉)) → (𝑉 ∩ (𝐵 ∖ 𝑉)) = ∅)
72		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ (𝐵 ∖ 𝑉)) → 𝑢 ∈ (𝐵 ∖ 𝑉))
73		fvun2 6922	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 Fn 𝑉 ∧ 𝑏 Fn (𝐵 ∖ 𝑉) ∧ ((𝑉 ∩ (𝐵 ∖ 𝑉)) = ∅ ∧ 𝑢 ∈ (𝐵 ∖ 𝑉))) → ((𝑎 ∪ 𝑏)‘𝑢) = (𝑏‘𝑢))
74	69, 70, 71, 72, 73	syl112anc 1376	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ (𝐵 ∖ 𝑉)) → ((𝑎 ∪ 𝑏)‘𝑢) = (𝑏‘𝑢))
75	74	adantlr 715	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑢 ∈ (𝐵 ∖ 𝑉)) → ((𝑎 ∪ 𝑏)‘𝑢) = (𝑏‘𝑢))
76	36	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑢 ∈ (𝐵 ∖ 𝑉)) → 𝑏:(𝐵 ∖ 𝑉)⟶(Base‘(Scalar‘𝑊)))
77		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑢 ∈ (𝐵 ∖ 𝑉)) → 𝑢 ∈ (𝐵 ∖ 𝑉))
78	76, 77	ffvelcdmd 7026	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑢 ∈ (𝐵 ∖ 𝑉)) → (𝑏‘𝑢) ∈ (Base‘(Scalar‘𝑊)))
79	75, 78	eqeltrd 2833	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑢 ∈ (𝐵 ∖ 𝑉)) → ((𝑎 ∪ 𝑏)‘𝑢) ∈ (Base‘(Scalar‘𝑊)))
80		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐵)
81	40	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑉 ⊆ 𝐵 → (𝑉 ∪ (𝐵 ∖ 𝑉)) = 𝐵)
82	81	ad8antlr 741	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑉 ∪ (𝐵 ∖ 𝑉)) = 𝐵)
83	82	eqcomd 2739	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝐵 = (𝑉 ∪ (𝐵 ∖ 𝑉)))
84	83	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) → 𝐵 = (𝑉 ∪ (𝐵 ∖ 𝑉)))
85	80, 84	eleqtrd 2835	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ (𝑉 ∪ (𝐵 ∖ 𝑉)))
86		elun 4102	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 ∈ (𝑉 ∪ (𝐵 ∖ 𝑉)) ↔ (𝑢 ∈ 𝑉 ∨ 𝑢 ∈ (𝐵 ∖ 𝑉)))
87	85, 86	sylib 218	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) → (𝑢 ∈ 𝑉 ∨ 𝑢 ∈ (𝐵 ∖ 𝑉)))
88	68, 79, 87	mpjaodan 960	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) → ((𝑎 ∪ 𝑏)‘𝑢) ∈ (Base‘(Scalar‘𝑊)))
89	27	ad8antr 740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) → 𝐵 ⊆ (Base‘𝑊))
90	89, 80	sseldd 3931	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ (Base‘𝑊))
91		eqid 2733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
92		eqid 2733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
93		eqid 2733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
94	24, 91, 92, 93	lmodvscl 20815	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑊 ∈ LMod ∧ ((𝑎 ∪ 𝑏)‘𝑢) ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑢 ∈ (Base‘𝑊)) → (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢) ∈ (Base‘𝑊))
95	53, 88, 90, 94	syl3anc 1373	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝐵) → (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢) ∈ (Base‘𝑊))
96		simp-9l 792	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝑊 ∈ LVec)
97	96, 22	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝑊 ∈ LMod)
98		eqidd 2734	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (Scalar‘𝑊) = (Scalar‘𝑊))
99		eqid 2733	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
100	43	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑎 ∪ 𝑏):𝐵⟶(Base‘(Scalar‘𝑊)))
101	100	feqmptd 6898	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑎 ∪ 𝑏) = (𝑢 ∈ 𝐵 ↦ ((𝑎 ∪ 𝑏)‘𝑢)))
102	101, 47	eqbrtrrd 5119	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑢 ∈ 𝐵 ↦ ((𝑎 ∪ 𝑏)‘𝑢)) finSupp (0_g‘(Scalar‘𝑊)))
103	52, 97, 98, 24, 88, 90, 29, 99, 92, 102	mptscmfsupp0 20864	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑢 ∈ 𝐵 ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢)) finSupp 0 )
104	37	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑉 ∩ (𝐵 ∖ 𝑉)) = ∅)
105	24, 29, 48, 51, 52, 95, 103, 104, 83	gsumsplit2 19845	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑊 Σ_g (𝑢 ∈ 𝐵 ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = ((𝑊 Σ_g (𝑢 ∈ 𝑉 ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢)))(+_g‘𝑊)(𝑊 Σ_g (𝑢 ∈ (𝐵 ∖ 𝑉) ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢)))))
106	63	oveq1d 7369	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝑉) → (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢) = ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢))
107	106	mpteq2dva 5188	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑢 ∈ 𝑉 ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢)) = (𝑢 ∈ 𝑉 ↦ ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢)))
108	107	oveq2d 7370	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑊 Σ_g (𝑢 ∈ 𝑉 ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = (𝑊 Σ_g (𝑢 ∈ 𝑉 ↦ ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢))))
109	74	oveq1d 7369	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ (𝐵 ∖ 𝑉)) → (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢) = ((𝑏‘𝑢)( ·_𝑠 ‘𝑊)𝑢))
110	109	mpteq2dva 5188	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑢 ∈ (𝐵 ∖ 𝑉) ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢)) = (𝑢 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑢)( ·_𝑠 ‘𝑊)𝑢)))
111	110	oveq2d 7370	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑊 Σ_g (𝑢 ∈ (𝐵 ∖ 𝑉) ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = (𝑊 Σ_g (𝑢 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑢)( ·_𝑠 ‘𝑊)𝑢))))
112	108, 111	oveq12d 7372	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → ((𝑊 Σ_g (𝑢 ∈ 𝑉 ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢)))(+_g‘𝑊)(𝑊 Σ_g (𝑢 ∈ (𝐵 ∖ 𝑉) ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢)))) = ((𝑊 Σ_g (𝑢 ∈ 𝑉 ↦ ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢)))(+_g‘𝑊)(𝑊 Σ_g (𝑢 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑢)( ·_𝑠 ‘𝑊)𝑢)))))
113		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣))))
114		fveq2 6830	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑢 = 𝑣 → (𝑏‘𝑢) = (𝑏‘𝑣))
115	114, 3	oveq12d 7372	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = 𝑣 → ((𝑏‘𝑢)( ·_𝑠 ‘𝑊)𝑢) = ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣))
116	115	cbvmptv 5199	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑢)( ·_𝑠 ‘𝑊)𝑢)) = (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣))
117	116	oveq2i 7365	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑊 Σ_g (𝑢 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))
118	113, 117	eqtr4di 2786	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑢 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑢)( ·_𝑠 ‘𝑊)𝑢))))
119	7, 118	oveq12d 7372	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑥(+_g‘𝑊)((inv_g‘𝑊)‘𝑥)) = ((𝑊 Σ_g (𝑢 ∈ 𝑉 ↦ ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢)))(+_g‘𝑊)(𝑊 Σ_g (𝑢 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑢)( ·_𝑠 ‘𝑊)𝑢)))))
120		lmodgrp 20804	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
121	96, 22, 120	3syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝑊 ∈ Grp)
122	13, 27	sstrd 3941	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝑉 ⊆ (Base‘𝑊))
123	24, 30	lspssv 20920	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑊 ∈ LMod ∧ 𝑉 ⊆ (Base‘𝑊)) → (𝑁‘𝑉) ⊆ (Base‘𝑊))
124	23, 122, 123	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝑁‘𝑉) ⊆ (Base‘𝑊))
125	124	ad7antr 738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑁‘𝑉) ⊆ (Base‘𝑊))
126		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉)))
127	126	elin2d 4154	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → 𝑥 ∈ (𝑁‘𝑉))
128	127	ad6antr 736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝑥 ∈ (𝑁‘𝑉))
129	125, 128	sseldd 3931	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝑥 ∈ (Base‘𝑊))
130		eqid 2733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (inv_g‘𝑊) = (inv_g‘𝑊)
131	24, 48, 29, 130	grprinv 18907	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑊 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥(+_g‘𝑊)((inv_g‘𝑊)‘𝑥)) = 0 )
132	121, 129, 131	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑥(+_g‘𝑊)((inv_g‘𝑊)‘𝑥)) = 0 )
133	112, 119, 132	3eqtr2d 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → ((𝑊 Σ_g (𝑢 ∈ 𝑉 ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢)))(+_g‘𝑊)(𝑊 Σ_g (𝑢 ∈ (𝐵 ∖ 𝑉) ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢)))) = 0 )
134	105, 133	eqtrd 2768	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑊 Σ_g (𝑢 ∈ 𝐵 ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = 0 )
135		breq1 5098	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → (𝑐 finSupp (0_g‘(Scalar‘𝑊)) ↔ (𝑎 ∪ 𝑏) finSupp (0_g‘(Scalar‘𝑊))))
136		fveq1 6829	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → (𝑐‘𝑢) = ((𝑎 ∪ 𝑏)‘𝑢))
137	136	oveq1d 7369	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → ((𝑐‘𝑢)( ·_𝑠 ‘𝑊)𝑢) = (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢))
138	137	mpteq2dv 5189	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → (𝑢 ∈ 𝐵 ↦ ((𝑐‘𝑢)( ·_𝑠 ‘𝑊)𝑢)) = (𝑢 ∈ 𝐵 ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢)))
139	138	oveq2d 7370	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → (𝑊 Σ_g (𝑢 ∈ 𝐵 ↦ ((𝑐‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = (𝑊 Σ_g (𝑢 ∈ 𝐵 ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢))))
140	139	eqeq1d 2735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → ((𝑊 Σ_g (𝑢 ∈ 𝐵 ↦ ((𝑐‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = 0 ↔ (𝑊 Σ_g (𝑢 ∈ 𝐵 ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = 0 ))
141	135, 140	anbi12d 632	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → ((𝑐 finSupp (0_g‘(Scalar‘𝑊)) ∧ (𝑊 Σ_g (𝑢 ∈ 𝐵 ↦ ((𝑐‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = 0 ) ↔ ((𝑎 ∪ 𝑏) finSupp (0_g‘(Scalar‘𝑊)) ∧ (𝑊 Σ_g (𝑢 ∈ 𝐵 ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = 0 )))
142		eqeq1 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → (𝑐 = (𝐵 × {(0_g‘(Scalar‘𝑊))}) ↔ (𝑎 ∪ 𝑏) = (𝐵 × {(0_g‘(Scalar‘𝑊))})))
143	141, 142	imbi12d 344	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → (((𝑐 finSupp (0_g‘(Scalar‘𝑊)) ∧ (𝑊 Σ_g (𝑢 ∈ 𝐵 ↦ ((𝑐‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = 0 ) → 𝑐 = (𝐵 × {(0_g‘(Scalar‘𝑊))})) ↔ (((𝑎 ∪ 𝑏) finSupp (0_g‘(Scalar‘𝑊)) ∧ (𝑊 Σ_g (𝑢 ∈ 𝐵 ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = 0 ) → (𝑎 ∪ 𝑏) = (𝐵 × {(0_g‘(Scalar‘𝑊))}))))
144	25	lbslinds 21774	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐽 ⊆ (LIndS‘𝑊)
145	144, 11	sselid 3928	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝐵 ∈ (LIndS‘𝑊))
146	24, 93, 91, 92, 29, 99	islinds5 33341	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ⊆ (Base‘𝑊)) → (𝐵 ∈ (LIndS‘𝑊) ↔ ∀𝑐 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝐵)((𝑐 finSupp (0_g‘(Scalar‘𝑊)) ∧ (𝑊 Σ_g (𝑢 ∈ 𝐵 ↦ ((𝑐‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = 0 ) → 𝑐 = (𝐵 × {(0_g‘(Scalar‘𝑊))}))))
147	146	biimpa 476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ⊆ (Base‘𝑊)) ∧ 𝐵 ∈ (LIndS‘𝑊)) → ∀𝑐 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝐵)((𝑐 finSupp (0_g‘(Scalar‘𝑊)) ∧ (𝑊 Σ_g (𝑢 ∈ 𝐵 ↦ ((𝑐‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = 0 ) → 𝑐 = (𝐵 × {(0_g‘(Scalar‘𝑊))})))
148	23, 27, 145, 147	syl21anc 837	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → ∀𝑐 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝐵)((𝑐 finSupp (0_g‘(Scalar‘𝑊)) ∧ (𝑊 Σ_g (𝑢 ∈ 𝐵 ↦ ((𝑐‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = 0 ) → 𝑐 = (𝐵 × {(0_g‘(Scalar‘𝑊))})))
149	148	ad7antr 738	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → ∀𝑐 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝐵)((𝑐 finSupp (0_g‘(Scalar‘𝑊)) ∧ (𝑊 Σ_g (𝑢 ∈ 𝐵 ↦ ((𝑐‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = 0 ) → 𝑐 = (𝐵 × {(0_g‘(Scalar‘𝑊))})))
150		fvexd 6845	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (Base‘(Scalar‘𝑊)) ∈ V)
151	150, 52	elmapd 8772	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → ((𝑎 ∪ 𝑏) ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝐵) ↔ (𝑎 ∪ 𝑏):𝐵⟶(Base‘(Scalar‘𝑊))))
152	100, 151	mpbird 257	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑎 ∪ 𝑏) ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝐵))
153	143, 149, 152	rspcdva 3574	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (((𝑎 ∪ 𝑏) finSupp (0_g‘(Scalar‘𝑊)) ∧ (𝑊 Σ_g (𝑢 ∈ 𝐵 ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = 0 ) → (𝑎 ∪ 𝑏) = (𝐵 × {(0_g‘(Scalar‘𝑊))})))
154	47, 134, 153	mp2and 699	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑎 ∪ 𝑏) = (𝐵 × {(0_g‘(Scalar‘𝑊))}))
155	154	reseq1d 5933	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → ((𝑎 ∪ 𝑏) ↾ 𝑉) = ((𝐵 × {(0_g‘(Scalar‘𝑊))}) ↾ 𝑉))
156		fnunres1 6600	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 Fn 𝑉 ∧ 𝑏 Fn (𝐵 ∖ 𝑉) ∧ (𝑉 ∩ (𝐵 ∖ 𝑉)) = ∅) → ((𝑎 ∪ 𝑏) ↾ 𝑉) = 𝑎)
157	55, 58, 104, 156	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → ((𝑎 ∪ 𝑏) ↾ 𝑉) = 𝑎)
158		xpssres 5973	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ⊆ 𝐵 → ((𝐵 × {(0_g‘(Scalar‘𝑊))}) ↾ 𝑉) = (𝑉 × {(0_g‘(Scalar‘𝑊))}))
159	158	ad8antlr 741	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → ((𝐵 × {(0_g‘(Scalar‘𝑊))}) ↾ 𝑉) = (𝑉 × {(0_g‘(Scalar‘𝑊))}))
160	155, 157, 159	3eqtr3d 2776	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝑎 = (𝑉 × {(0_g‘(Scalar‘𝑊))}))
161	160	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝑉) → 𝑎 = (𝑉 × {(0_g‘(Scalar‘𝑊))}))
162	161	fveq1d 6832	. . . . . . . . . . . . 13 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝑉) → (𝑎‘𝑢) = ((𝑉 × {(0_g‘(Scalar‘𝑊))})‘𝑢))
163		fvex 6843	. . . . . . . . . . . . . . 15 ⊢ (0_g‘(Scalar‘𝑊)) ∈ V
164	163	fvconst2 7146	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ 𝑉 → ((𝑉 × {(0_g‘(Scalar‘𝑊))})‘𝑢) = (0_g‘(Scalar‘𝑊)))
165	61, 164	syl 17	. . . . . . . . . . . . 13 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝑉) → ((𝑉 × {(0_g‘(Scalar‘𝑊))})‘𝑢) = (0_g‘(Scalar‘𝑊)))
166	162, 165	eqtrd 2768	. . . . . . . . . . . 12 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝑉) → (𝑎‘𝑢) = (0_g‘(Scalar‘𝑊)))
167	166	oveq1d 7369	. . . . . . . . . . 11 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝑉) → ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢) = ((0_g‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑢))
168	122	ad8antr 740	. . . . . . . . . . . . 13 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝑉) → 𝑉 ⊆ (Base‘𝑊))
169	168, 61	sseldd 3931	. . . . . . . . . . . 12 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝑉) → 𝑢 ∈ (Base‘𝑊))
170	24, 91, 92, 99, 29	lmod0vs 20832	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑢 ∈ (Base‘𝑊)) → ((0_g‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑢) = 0 )
171	97, 169, 170	syl2an2r 685	. . . . . . . . . . 11 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝑉) → ((0_g‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑢) = 0 )
172	167, 171	eqtrd 2768	. . . . . . . . . 10 ⊢ (((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑢 ∈ 𝑉) → ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢) = 0 )
173	172	mpteq2dva 5188	. . . . . . . . 9 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑢 ∈ 𝑉 ↦ ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢)) = (𝑢 ∈ 𝑉 ↦ 0 ))
174	173	oveq2d 7370	. . . . . . . 8 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑊 Σ_g (𝑢 ∈ 𝑉 ↦ ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢))) = (𝑊 Σ_g (𝑢 ∈ 𝑉 ↦ 0 )))
175		cmnmnd 19713	. . . . . . . . . 10 ⊢ (𝑊 ∈ CMnd → 𝑊 ∈ Mnd)
176	51, 175	syl 17	. . . . . . . . 9 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝑊 ∈ Mnd)
177	128	elfvexd 6866	. . . . . . . . 9 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝑉 ∈ V)
178	29	gsumz 18748	. . . . . . . . 9 ⊢ ((𝑊 ∈ Mnd ∧ 𝑉 ∈ V) → (𝑊 Σ_g (𝑢 ∈ 𝑉 ↦ 0 )) = 0 )
179	176, 177, 178	syl2anc 584	. . . . . . . 8 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → (𝑊 Σ_g (𝑢 ∈ 𝑉 ↦ 0 )) = 0 )
180	7, 174, 179	3eqtrd 2772	. . . . . . 7 ⊢ ((((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ 𝑏 finSupp (0_g‘(Scalar‘𝑊))) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝑥 = 0 )
181	180	anasss 466	. . . . . 6 ⊢ (((((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))) ∧ (𝑏 finSupp (0_g‘(Scalar‘𝑊)) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣))))) → 𝑥 = 0 )
182		eqid 2733	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
183	24, 182, 30	lspcl 20913	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ (𝐵 ∖ 𝑉) ⊆ (Base‘𝑊)) → (𝑁‘(𝐵 ∖ 𝑉)) ∈ (LSubSp‘𝑊))
184	23, 28, 183	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝑁‘(𝐵 ∖ 𝑉)) ∈ (LSubSp‘𝑊))
185	184	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → (𝑁‘(𝐵 ∖ 𝑉)) ∈ (LSubSp‘𝑊))
186	182	lsssubg 20894	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘(𝐵 ∖ 𝑉)) ∈ (LSubSp‘𝑊)) → (𝑁‘(𝐵 ∖ 𝑉)) ∈ (SubGrp‘𝑊))
187	23, 185, 186	syl2an2r 685	. . . . . . . . 9 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → (𝑁‘(𝐵 ∖ 𝑉)) ∈ (SubGrp‘𝑊))
188	126	elin1d 4153	. . . . . . . . 9 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → 𝑥 ∈ (𝑁‘(𝐵 ∖ 𝑉)))
189	130	subginvcl 19052	. . . . . . . . 9 ⊢ (((𝑁‘(𝐵 ∖ 𝑉)) ∈ (SubGrp‘𝑊) ∧ 𝑥 ∈ (𝑁‘(𝐵 ∖ 𝑉))) → ((inv_g‘𝑊)‘𝑥) ∈ (𝑁‘(𝐵 ∖ 𝑉)))
190	187, 188, 189	syl2anc 584	. . . . . . . 8 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → ((inv_g‘𝑊)‘𝑥) ∈ (𝑁‘(𝐵 ∖ 𝑉)))
191	30, 24, 93, 91, 99, 92, 23, 28	ellspds 33342	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (((inv_g‘𝑊)‘𝑥) ∈ (𝑁‘(𝐵 ∖ 𝑉)) ↔ ∃𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))(𝑏 finSupp (0_g‘(Scalar‘𝑊)) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣))))))
192	191	biimpa 476	. . . . . . . 8 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ ((inv_g‘𝑊)‘𝑥) ∈ (𝑁‘(𝐵 ∖ 𝑉))) → ∃𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))(𝑏 finSupp (0_g‘(Scalar‘𝑊)) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))))
193	190, 192	syldan 591	. . . . . . 7 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → ∃𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))(𝑏 finSupp (0_g‘(Scalar‘𝑊)) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))))
194	193	ad3antrrr 730	. . . . . 6 ⊢ (((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → ∃𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉))(𝑏 finSupp (0_g‘(Scalar‘𝑊)) ∧ ((inv_g‘𝑊)‘𝑥) = (𝑊 Σ_g (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))))
195	181, 194	r19.29a 3141	. . . . 5 ⊢ (((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑊))) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))) → 𝑥 = 0 )
196	195	anasss 466	. . . 4 ⊢ ((((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)) ∧ (𝑎 finSupp (0_g‘(Scalar‘𝑊)) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣))))) → 𝑥 = 0 )
197	30, 24, 93, 91, 99, 92, 23, 122	ellspds 33342	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝑥 ∈ (𝑁‘𝑉) ↔ ∃𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)(𝑎 finSupp (0_g‘(Scalar‘𝑊)) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣))))))
198	197	biimpa 476	. . . . 5 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑁‘𝑉)) → ∃𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)(𝑎 finSupp (0_g‘(Scalar‘𝑊)) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))))
199	127, 198	syldan 591	. . . 4 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → ∃𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉)(𝑎 finSupp (0_g‘(Scalar‘𝑊)) ∧ 𝑥 = (𝑊 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣)))))
200	196, 199	r19.29a 3141	. . 3 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → 𝑥 = 0 )
201	29, 24, 30	0ellsp 33343	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑉 ⊆ (Base‘𝑊)) → 0 ∈ (𝑁‘𝑉))
202	23, 122, 201	syl2anc 584	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 0 ∈ (𝑁‘𝑉))
203	32, 202	elind 4149	. . 3 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 0 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉)))
204	200, 203	eqsnd 4783	. 2 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉)) = { 0 })
205	204	3impa 1109	1 ⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽 ∧ 𝑉 ⊆ 𝐵) → ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉)) = { 0 })

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∀wral 3048 ∃wrex 3057 Vcvv 3437 ∖ cdif 3895 ∪ cun 3896 ∩ cin 3897 ⊆ wss 3898 ∅c0 4282 {csn 4577 class class class wbr 5095 ↦ cmpt 5176 × cxp 5619 ↾ cres 5623 Fn wfn 6483 ⟶wf 6484 ‘cfv 6488 (class class class)co 7354 ↑_m cmap 8758 finSupp cfsupp 9254 Basecbs 17124 +_gcplusg 17165 Scalarcsca 17168 ·_𝑠 cvsca 17169 0_gc0g 17347 Σ_g cgsu 17348 Mndcmnd 18646 Grpcgrp 18850 inv_gcminusg 18851 SubGrpcsubg 19037 CMndccmn 19696 LModclmod 20797 LSubSpclss 20868 LSpanclspn 20908 LBasisclbs 21012 LVecclvec 21040 LIndSclinds 21746

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 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-cnex 11071 ax-resscn 11072 ax-1cn 11073 ax-icn 11074 ax-addcl 11075 ax-addrcl 11076 ax-mulcl 11077 ax-mulrcl 11078 ax-mulcom 11079 ax-addass 11080 ax-mulass 11081 ax-distr 11082 ax-i2m1 11083 ax-1ne0 11084 ax-1rid 11085 ax-rnegex 11086 ax-rrecex 11087 ax-cnre 11088 ax-pre-lttri 11089 ax-pre-lttrn 11090 ax-pre-ltadd 11091 ax-pre-mulgt0 11092

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 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7311 df-ov 7357 df-oprab 7358 df-mpo 7359 df-of 7618 df-om 7805 df-1st 7929 df-2nd 7930 df-supp 8099 df-frecs 8219 df-wrecs 8250 df-recs 8299 df-rdg 8337 df-1o 8393 df-2o 8394 df-er 8630 df-map 8760 df-ixp 8830 df-en 8878 df-dom 8879 df-sdom 8880 df-fin 8881 df-fsupp 9255 df-sup 9335 df-oi 9405 df-card 9841 df-pnf 11157 df-mnf 11158 df-xr 11159 df-ltxr 11160 df-le 11161 df-sub 11355 df-neg 11356 df-nn 12135 df-2 12197 df-3 12198 df-4 12199 df-5 12200 df-6 12201 df-7 12202 df-8 12203 df-9 12204 df-n0 12391 df-z 12478 df-dec 12597 df-uz 12741 df-fz 13412 df-fzo 13559 df-seq 13913 df-hash 14242 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17125 df-ress 17146 df-plusg 17178 df-mulr 17179 df-sca 17181 df-vsca 17182 df-ip 17183 df-tset 17184 df-ple 17185 df-ds 17187 df-hom 17189 df-cco 17190 df-0g 17349 df-gsum 17350 df-prds 17355 df-pws 17357 df-mre 17492 df-mrc 17493 df-acs 17495 df-mgm 18552 df-sgrp 18631 df-mnd 18647 df-mhm 18695 df-submnd 18696 df-grp 18853 df-minusg 18854 df-sbg 18855 df-mulg 18985 df-subg 19040 df-ghm 19129 df-cntz 19233 df-cmn 19698 df-abl 19699 df-mgp 20063 df-rng 20075 df-ur 20104 df-ring 20157 df-nzr 20432 df-subrg 20489 df-lmod 20799 df-lss 20869 df-lsp 20909 df-lmhm 20960 df-lbs 21013 df-lvec 21041 df-sra 21111 df-rgmod 21112 df-dsmm 21673 df-frlm 21688 df-uvc 21724 df-lindf 21747 df-linds 21748

This theorem is referenced by: dimkerim 33663

W3C validator