lbsdiflsp0 - Mathbox for Thierry Arnoux

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Theorem lbsdiflsp0 33380

Description: The linear spans of two disjunct independent sets only have a trivial intersection. This can be seen as the opposite direction of lindsun 33379. (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 782	. . . . . . . . 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 6891	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (𝑎‘𝑢) = (𝑎‘𝑣))
3		id 22	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → 𝑢 = 𝑣)
4	2, 3	oveq12d 7433	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢) = ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣))
5	4	cbvmptv 5256	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑉 ↦ ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢)) = (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣))
6	5	oveq2i 7426	. . . . . . . . 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 2783	. . . . . . . 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 782	. . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝐵 ∈ 𝐽)
12	11	ad6antr 734	. . . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝑉 ⊆ 𝐵)
14	13	ad6antr 734	. . . . . . . . . . . . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (Base‘(Scalar‘𝑊)) ∈ V)
17	11, 13	ssexd 5319	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝑉 ∈ V)
18	16, 17	elmapd 8855	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉) ↔ 𝑎:𝑉⟶(Base‘(Scalar‘𝑊))))
19	18	biimpa 475	. . . . . . . . . . . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . . . . . . . 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 20993	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
23	22	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝑊 ∈ LMod)
24		eqid 2725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (Base‘𝑊) = (Base‘𝑊)
25		lbsdiflsp0.j	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 𝐽 = (LBasis‘𝑊)
26	24, 25	lbsss 20964	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ⊆ (Base‘𝑊))
27	26	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝐵 ⊆ (Base‘𝑊))
28	27	ssdifssd 4135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 33127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑊 ∈ LMod ∧ (𝐵 ∖ 𝑉) ⊆ (Base‘𝑊)) → 0 ∈ (𝑁‘(𝐵 ∖ 𝑉)))
32	23, 28, 31	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 0 ∈ (𝑁‘(𝐵 ∖ 𝑉)))
33	32	elfvexd 6930	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝐵 ∖ 𝑉) ∈ V)
34	16, 33	elmapd 8855	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉)) ↔ 𝑏:(𝐵 ∖ 𝑉)⟶(Base‘(Scalar‘𝑊))))
35	34	biimpa 475	. . . . . . . . . . . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . . . . . . . . . . . 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 4467	. . . . . . . . . . . . . . . . . . . . . . . . 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 6755	. . . . . . . . . . . . . . . . . . . . . . 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 4477	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑉 ⊆ 𝐵 ↔ (𝑉 ∪ (𝐵 ∖ 𝑉)) = 𝐵)
41	14, 40	sylib 217	. . . . . . . . . . . . . . . . . . . . . . . 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 6702	. . . . . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . . . . 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 6720	. . . . . . . . . . . . . . . . . . . . 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 9410	. . . . . . . . . . . . . . . . . . . 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 711	. . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 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 2725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
49		lmodcmn 20795	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ CMnd)
50	22, 49	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ CMnd)
51	50	ad9antr 740	. . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . . . 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 8880	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉) → 𝑎 Fn 𝑉)
55	54	ad6antlr 735	. . . . . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . . 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 8880	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉)) → 𝑏 Fn (𝐵 ∖ 𝑉))
58	57	ad3antlr 729	. . . . . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . 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 6983	. . . . . . . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . 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 7089	. . . . . . . . . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . 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 6984	. . . . . . . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . 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 7089	. . . . . . . . . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑉 ⊆ 𝐵 → (𝑉 ∪ (𝐵 ∖ 𝑉)) = 𝐵)
82	81	ad8antlr 739	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . . . . . . . . . . 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 4141	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 ∈ (𝑉 ∪ (𝐵 ∖ 𝑉)) ↔ (𝑢 ∈ 𝑉 ∨ 𝑢 ∈ (𝐵 ∖ 𝑉)))
87	85, 86	sylib 217	. . . . . . . . . . . . . . . . . . . . . 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 956	. . . . . . . . . . . . . . . . . . . . 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 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)))) ∧ 𝑢 ∈ 𝐵) → 𝐵 ⊆ (Base‘𝑊))
90	89, 80	sseldd 3973	. . . . . . . . . . . . . . . . . . . . 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 2725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
92		eqid 2725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
93		eqid 2725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
94	24, 91, 92, 93	lmodvscl 20763	. . . . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . . . 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 791	. . . . . . . . . . . . . . . . . . . . . 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 2726	. . . . . . . . . . . . . . . . . . . . 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 2725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
100	43	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 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 6961	. . . . . . . . . . . . . . . . . . . . . 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 5167	. . . . . . . . . . . . . . . . . . . . 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 20812	. . . . . . . . . . . . . . . . . . . 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 19886	. . . . . . . . . . . . . . . . . . 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 7430	. . . . . . . . . . . . . . . . . . . . . . 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 5243	. . . . . . . . . . . . . . . . . . . . . 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 7431	. . . . . . . . . . . . . . . . . . . . 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 7430	. . . . . . . . . . . . . . . . . . . . . . 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 5243	. . . . . . . . . . . . . . . . . . . . . 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 7431	. . . . . . . . . . . . . . . . . . . . 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 7433	. . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . 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 6891	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑢 = 𝑣 → (𝑏‘𝑢) = (𝑏‘𝑣))
115	114, 3	oveq12d 7433	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = 𝑣 → ((𝑏‘𝑢)( ·_𝑠 ‘𝑊)𝑢) = ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣))
116	115	cbvmptv 5256	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑢)( ·_𝑠 ‘𝑊)𝑢)) = (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣))
117	116	oveq2i 7426	. . . . . . . . . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . . . . . . . . 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 7433	. . . . . . . . . . . . . . . . . . . 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 20752	. . . . . . . . . . . . . . . . . . . . . 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 3983	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝑉 ⊆ (Base‘𝑊))
123	24, 30	lspssv 20869	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑊 ∈ LMod ∧ 𝑉 ⊆ (Base‘𝑊)) → (𝑁‘𝑉) ⊆ (Base‘𝑊))
124	23, 122, 123	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝑁‘𝑉) ⊆ (Base‘𝑊))
125	124	ad7antr 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)))) → (𝑁‘𝑉) ⊆ (Base‘𝑊))
126		simpr 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉)))
127	126	elin2d 4193	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → 𝑥 ∈ (𝑁‘𝑉))
128	127	ad6antr 734	. . . . . . . . . . . . . . . . . . . . . 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 3973	. . . . . . . . . . . . . . . . . . . . 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 2725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (inv_g‘𝑊) = (inv_g‘𝑊)
131	24, 48, 29, 130	grprinv 18949	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑊 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥(+_g‘𝑊)((inv_g‘𝑊)‘𝑥)) = 0 )
132	121, 129, 131	syl2anc 582	. . . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . 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 5146	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → (𝑐 finSupp (0_g‘(Scalar‘𝑊)) ↔ (𝑎 ∪ 𝑏) finSupp (0_g‘(Scalar‘𝑊))))
136		fveq1 6890	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → (𝑐‘𝑢) = ((𝑎 ∪ 𝑏)‘𝑢))
137	136	oveq1d 7430	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → ((𝑐‘𝑢)( ·_𝑠 ‘𝑊)𝑢) = (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢))
138	137	mpteq2dv 5245	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → (𝑢 ∈ 𝐵 ↦ ((𝑐‘𝑢)( ·_𝑠 ‘𝑊)𝑢)) = (𝑢 ∈ 𝐵 ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢)))
139	138	oveq2d 7431	. . . . . . . . . . . . . . . . . . . . . 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 2727	. . . . . . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → (𝑐 = (𝐵 × {(0_g‘(Scalar‘𝑊))}) ↔ (𝑎 ∪ 𝑏) = (𝐵 × {(0_g‘(Scalar‘𝑊))})))
143	141, 142	imbi12d 343	. . . . . . . . . . . . . . . . . . 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 21769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐽 ⊆ (LIndS‘𝑊)
145	144, 11	sselid 3970	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝐵 ∈ (LIndS‘𝑊))
146	24, 93, 91, 92, 29, 99	islinds5 33125	. . . . . . . . . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . . . . . . . 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 8855	. . . . . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . . . . 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 3603	. . . . . . . . . . . . . . . . . 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 697	. . . . . . . . . . . . . . . . 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 5978	. . . . . . . . . . . . . . . 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 6660	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 Fn 𝑉 ∧ 𝑏 Fn (𝐵 ∖ 𝑉) ∧ (𝑉 ∩ (𝐵 ∖ 𝑉)) = ∅) → ((𝑎 ∪ 𝑏) ↾ 𝑉) = 𝑎)
157	55, 58, 104, 156	syl3anc 1368	. . . . . . . . . . . . . . . 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 6017	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ⊆ 𝐵 → ((𝐵 × {(0_g‘(Scalar‘𝑊))}) ↾ 𝑉) = (𝑉 × {(0_g‘(Scalar‘𝑊))}))
159	158	ad8antlr 739	. . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . 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 6893	. . . . . . . . . . . . 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 6904	. . . . . . . . . . . . . . 15 ⊢ (0_g‘(Scalar‘𝑊)) ∈ V
164	163	fvconst2 7211	. . . . . . . . . . . . . 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 2765	. . . . . . . . . . . 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 7430	. . . . . . . . . . 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 738	. . . . . . . . . . . . 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 3973	. . . . . . . . . . . 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 20780	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑢 ∈ (Base‘𝑊)) → ((0_g‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑢) = 0 )
171	97, 169, 170	syl2an2r 683	. . . . . . . . . . 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 2765	. . . . . . . . . 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 5243	. . . . . . . . 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 7431	. . . . . . . 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 19754	. . . . . . . . . 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 6930	. . . . . . . . 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 18790	. . . . . . . . 9 ⊢ ((𝑊 ∈ Mnd ∧ 𝑉 ∈ V) → (𝑊 Σ_g (𝑢 ∈ 𝑉 ↦ 0 )) = 0 )
179	176, 177, 178	syl2anc 582	. . . . . . . 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 2769	. . . . . . 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 465	. . . . . 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 2725	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
183	24, 182, 30	lspcl 20862	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ (𝐵 ∖ 𝑉) ⊆ (Base‘𝑊)) → (𝑁‘(𝐵 ∖ 𝑉)) ∈ (LSubSp‘𝑊))
184	23, 28, 183	syl2anc 582	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝑁‘(𝐵 ∖ 𝑉)) ∈ (LSubSp‘𝑊))
185	184	adantr 479	. . . . . . . . . 10 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → (𝑁‘(𝐵 ∖ 𝑉)) ∈ (LSubSp‘𝑊))
186	182	lsssubg 20843	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘(𝐵 ∖ 𝑉)) ∈ (LSubSp‘𝑊)) → (𝑁‘(𝐵 ∖ 𝑉)) ∈ (SubGrp‘𝑊))
187	23, 185, 186	syl2an2r 683	. . . . . . . . 9 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → (𝑁‘(𝐵 ∖ 𝑉)) ∈ (SubGrp‘𝑊))
188	126	elin1d 4192	. . . . . . . . 9 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → 𝑥 ∈ (𝑁‘(𝐵 ∖ 𝑉)))
189	130	subginvcl 19092	. . . . . . . . 9 ⊢ (((𝑁‘(𝐵 ∖ 𝑉)) ∈ (SubGrp‘𝑊) ∧ 𝑥 ∈ (𝑁‘(𝐵 ∖ 𝑉))) → ((inv_g‘𝑊)‘𝑥) ∈ (𝑁‘(𝐵 ∖ 𝑉)))
190	187, 188, 189	syl2anc 582	. . . . . . . 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 33126	. . . . . . . . 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 475	. . . . . . . 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 589	. . . . . . 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 728	. . . . . 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 3152	. . . . 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 465	. . . 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 33126	. . . . . 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 475	. . . . 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 589	. . . 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 3152	. . 3 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → 𝑥 = 0 )
201	29, 24, 30	0ellsp 33127	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑉 ⊆ (Base‘𝑊)) → 0 ∈ (𝑁‘𝑉))
202	23, 122, 201	syl2anc 582	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 0 ∈ (𝑁‘𝑉))
203	32, 202	elind 4188	. . 3 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 0 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉)))
204	200, 203	eqsnd 32368	. 2 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉)) = { 0 })
205	204	3impa 1107	1 ⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽 ∧ 𝑉 ⊆ 𝐵) → ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉)) = { 0 })

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ∃wrex 3060 Vcvv 3463 ∖ cdif 3937 ∪ cun 3938 ∩ cin 3939 ⊆ wss 3940 ∅c0 4318 {csn 4624 class class class wbr 5143 ↦ cmpt 5226 × cxp 5670 ↾ cres 5674 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7415 ↑_m cmap 8841 finSupp cfsupp 9383 Basecbs 17177 +_gcplusg 17230 Scalarcsca 17233 ·_𝑠 cvsca 17234 0_gc0g 17418 Σ_g cgsu 17419 Mndcmnd 18691 Grpcgrp 18892 inv_gcminusg 18893 SubGrpcsubg 19077 CMndccmn 19737 LModclmod 20745 LSubSpclss 20817 LSpanclspn 20857 LBasisclbs 20961 LVecclvec 20989 LIndSclinds 21741

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-sup 9463 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-fzo 13658 df-seq 13997 df-hash 14320 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-hom 17254 df-cco 17255 df-0g 17420 df-gsum 17421 df-prds 17426 df-pws 17428 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-ghm 19170 df-cntz 19270 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-nzr 20454 df-subrg 20510 df-lmod 20747 df-lss 20818 df-lsp 20858 df-lmhm 20909 df-lbs 20962 df-lvec 20990 df-sra 21060 df-rgmod 21061 df-dsmm 21668 df-frlm 21683 df-uvc 21719 df-lindf 21742 df-linds 21743

This theorem is referenced by: dimkerim 33381

W3C validator