lbsdiflsp0 - Mathbox for Thierry Arnoux

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Theorem lbsdiflsp0 33883

Description: The linear spans of two disjunct independent sets only have a trivial intersection. This can be seen as the opposite direction of lindsun 33882. (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 793	. . . . . . . . 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 6861	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (𝑎‘𝑢) = (𝑎‘𝑣))
3		id 22	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → 𝑢 = 𝑣)
4	2, 3	oveq12d 7408	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢) = ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣))
5	4	cbvmptv 5201	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑉 ↦ ((𝑎‘𝑢)( ·_𝑠 ‘𝑊)𝑢)) = (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣)( ·_𝑠 ‘𝑊)𝑣))
6	5	oveq2i 7401	. . . . . . . . 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 2814	. . . . . . . 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 793	. . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . 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 800	. . . . . . . . . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝐵 ∈ 𝐽)
12	11	ad6antr 746	. . . . . . . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝑉 ⊆ 𝐵)
14	13	ad6antr 746	. . . . . . . . . . . . . . . . . . . . . . . . 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 795	. . . . . . . . . . . . . . . . . . . . . . . . 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 6876	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (Base‘(Scalar‘𝑊)) ∈ V)
17	11, 13	ssexd 5277	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝑉 ∈ V)
18	16, 17	elmapd 8814	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉) ↔ 𝑎:𝑉⟶(Base‘(Scalar‘𝑊))))
19	18	biimpa 480	. . . . . . . . . . . . . . . . . . . . . . . . 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 851	. . . . . . . . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . . . . . . . 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 21160	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
23	22	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝑊 ∈ LMod)
24		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (Base‘𝑊) = (Base‘𝑊)
25		lbsdiflsp0.j	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 𝐽 = (LBasis‘𝑊)
26	24, 25	lbsss 21131	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ⊆ (Base‘𝑊))
27	26	ad2antlr 737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝐵 ⊆ (Base‘𝑊))
28	27	ssdifssd 4098	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 33515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑊 ∈ LMod ∧ (𝐵 ∖ 𝑉) ⊆ (Base‘𝑊)) → 0 ∈ (𝑁‘(𝐵 ∖ 𝑉)))
32	23, 28, 31	syl2anc 593	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 0 ∈ (𝑁‘(𝐵 ∖ 𝑉)))
33	32	elfvexd 6897	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝐵 ∖ 𝑉) ∈ V)
34	16, 33	elmapd 8814	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉)) ↔ 𝑏:(𝐵 ∖ 𝑉)⟶(Base‘(Scalar‘𝑊))))
35	34	biimpa 480	. . . . . . . . . . . . . . . . . . . . . . . . 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 851	. . . . . . . . . . . . . . . . . . . . . . . 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 4423	. . . . . . . . . . . . . . . . . . . . . . . . 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 6722	. . . . . . . . . . . . . . . . . . . . . . 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 4433	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑉 ⊆ 𝐵 ↔ (𝑉 ∪ (𝐵 ∖ 𝑉)) = 𝐵)
41	14, 40	sylib 220	. . . . . . . . . . . . . . . . . . . . . . . 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 6669	. . . . . . . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . . . . . . 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 6690	. . . . . . . . . . . . . . . . . . . . 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 9328	. . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
49		lmodcmn 20964	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ CMnd)
50	22, 49	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ CMnd)
51	50	ad9antr 752	. . . . . . . . . . . . . . . . . . . 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 748	. . . . . . . . . . . . . . . . . . . 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 750	. . . . . . . . . . . . . . . . . . . . 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 8839	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 ∈ ((Base‘(Scalar‘𝑊)) ↑_m 𝑉) → 𝑎 Fn 𝑉)
55	54	ad6antlr 747	. . . . . . . . . . . . . . . . . . . . . . . . . 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 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) → 𝑎 Fn 𝑉)
57		elmapfn 8839	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑏 ∈ ((Base‘(Scalar‘𝑊)) ↑_m (𝐵 ∖ 𝑉)) → 𝑏 Fn (𝐵 ∖ 𝑉))
58	57	ad3antlr 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)))) → 𝑏 Fn (𝐵 ∖ 𝑉))
59	58	adantr 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) → 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 488	. . . . . . . . . . . . . . . . . . . . . . . . 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 6952	. . . . . . . . . . . . . . . . . . . . . . . . 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 1392	. . . . . . . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . . . 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 7060	. . . . . . . . . . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . . . . . . . . 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 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)) → 𝑎 Fn 𝑉)
70	58	adantr 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)) → 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 488	. . . . . . . . . . . . . . . . . . . . . . . . 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 6953	. . . . . . . . . . . . . . . . . . . . . . . . 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 1392	. . . . . . . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . . . 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 7060	. . . . . . . . . . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑉 ⊆ 𝐵 → (𝑉 ∪ (𝐵 ∖ 𝑉)) = 𝐵)
82	81	ad8antlr 751	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . . . . . . . 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 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) → 𝐵 = (𝑉 ∪ (𝐵 ∖ 𝑉)))
85	80, 84	eleqtrd 2863	. . . . . . . . . . . . . . . . . . . . . . 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 4104	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 ∈ (𝑉 ∪ (𝐵 ∖ 𝑉)) ↔ (𝑢 ∈ 𝑉 ∨ 𝑢 ∈ (𝐵 ∖ 𝑉)))
87	85, 86	sylib 220	. . . . . . . . . . . . . . . . . . . . . 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 971	. . . . . . . . . . . . . . . . . . . . 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 750	. . . . . . . . . . . . . . . . . . . . . 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 3935	. . . . . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
92		eqid 2761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
93		eqid 2761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
94	24, 91, 92, 93	lmodvscl 20932	. . . . . . . . . . . . . . . . . . . . 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 1389	. . . . . . . . . . . . . . . . . . . 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 802	. . . . . . . . . . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
100	43	adantr 484	. . . . . . . . . . . . . . . . . . . . . . 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 6929	. . . . . . . . . . . . . . . . . . . . . 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 5121	. . . . . . . . . . . . . . . . . . . . 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 20981	. . . . . . . . . . . . . . . . . . . 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 19959	. . . . . . . . . . . . . . . . . . 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 7405	. . . . . . . . . . . . . . . . . . . . . . 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 5190	. . . . . . . . . . . . . . . . . . . . . 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 7406	. . . . . . . . . . . . . . . . . . . . 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 7405	. . . . . . . . . . . . . . . . . . . . . . 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 5190	. . . . . . . . . . . . . . . . . . . . . 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 7406	. . . . . . . . . . . . . . . . . . . . 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 7408	. . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑢 = 𝑣 → (𝑏‘𝑢) = (𝑏‘𝑣))
115	114, 3	oveq12d 7408	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = 𝑣 → ((𝑏‘𝑢)( ·_𝑠 ‘𝑊)𝑢) = ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣))
116	115	cbvmptv 5201	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑢)( ·_𝑠 ‘𝑊)𝑢)) = (𝑣 ∈ (𝐵 ∖ 𝑉) ↦ ((𝑏‘𝑣)( ·_𝑠 ‘𝑊)𝑣))
117	116	oveq2i 7401	. . . . . . . . . . . . . . . . . . . . . 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 2814	. . . . . . . . . . . . . . . . . . . . 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 7408	. . . . . . . . . . . . . . . . . . . 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 20921	. . . . . . . . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝑉 ⊆ (Base‘𝑊))
123	24, 30	lspssv 21037	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑊 ∈ LMod ∧ 𝑉 ⊆ (Base‘𝑊)) → (𝑁‘𝑉) ⊆ (Base‘𝑊))
124	23, 122, 123	syl2anc 593	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝑁‘𝑉) ⊆ (Base‘𝑊))
125	124	ad7antr 748	. . . . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉)))
127	126	elin2d 4155	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → 𝑥 ∈ (𝑁‘𝑉))
128	127	ad6antr 746	. . . . . . . . . . . . . . . . . . . . . 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 3935	. . . . . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (inv_g‘𝑊) = (inv_g‘𝑊)
131	24, 48, 29, 130	grprinv 19022	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑊 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥(+_g‘𝑊)((inv_g‘𝑊)‘𝑥)) = 0 )
132	121, 129, 131	syl2anc 593	. . . . . . . . . . . . . . . . . . . 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 2802	. . . . . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . . . . 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 5100	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → (𝑐 finSupp (0_g‘(Scalar‘𝑊)) ↔ (𝑎 ∪ 𝑏) finSupp (0_g‘(Scalar‘𝑊))))
136		fveq1 6860	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → (𝑐‘𝑢) = ((𝑎 ∪ 𝑏)‘𝑢))
137	136	oveq1d 7405	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → ((𝑐‘𝑢)( ·_𝑠 ‘𝑊)𝑢) = (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢))
138	137	mpteq2dv 5191	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → (𝑢 ∈ 𝐵 ↦ ((𝑐‘𝑢)( ·_𝑠 ‘𝑊)𝑢)) = (𝑢 ∈ 𝐵 ↦ (((𝑎 ∪ 𝑏)‘𝑢)( ·_𝑠 ‘𝑊)𝑢)))
139	138	oveq2d 7406	. . . . . . . . . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . . . . . 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 641	. . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = (𝑎 ∪ 𝑏) → (𝑐 = (𝐵 × {(0_g‘(Scalar‘𝑊))}) ↔ (𝑎 ∪ 𝑏) = (𝐵 × {(0_g‘(Scalar‘𝑊))})))
143	141, 142	imbi12d 346	. . . . . . . . . . . . . . . . . . 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 21872	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐽 ⊆ (LIndS‘𝑊)
145	144, 11	sselid 3932	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 𝐵 ∈ (LIndS‘𝑊))
146	24, 93, 91, 92, 29, 99	islinds5 33513	. . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . 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 848	. . . . . . . . . . . . . . . . . . . 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 748	. . . . . . . . . . . . . . . . . . 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 6876	. . . . . . . . . . . . . . . . . . . . 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 8814	. . . . . . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . . . . . 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 3581	. . . . . . . . . . . . . . . . . 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 709	. . . . . . . . . . . . . . . . 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 5960	. . . . . . . . . . . . . . . 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 6627	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 Fn 𝑉 ∧ 𝑏 Fn (𝐵 ∖ 𝑉) ∧ (𝑉 ∩ (𝐵 ∖ 𝑉)) = ∅) → ((𝑎 ∪ 𝑏) ↾ 𝑉) = 𝑎)
157	55, 58, 104, 156	syl3anc 1389	. . . . . . . . . . . . . . . 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 6000	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ⊆ 𝐵 → ((𝐵 × {(0_g‘(Scalar‘𝑊))}) ↾ 𝑉) = (𝑉 × {(0_g‘(Scalar‘𝑊))}))
159	158	ad8antlr 751	. . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 6863	. . . . . . . . . . . . 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 6874	. . . . . . . . . . . . . . 15 ⊢ (0_g‘(Scalar‘𝑊)) ∈ V
164	163	fvconst2 7182	. . . . . . . . . . . . . 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 2796	. . . . . . . . . . . 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 7405	. . . . . . . . . . 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 750	. . . . . . . . . . . . 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 3935	. . . . . . . . . . . 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 20949	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑢 ∈ (Base‘𝑊)) → ((0_g‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑢) = 0 )
171	97, 169, 170	syl2an2r 695	. . . . . . . . . . 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 2796	. . . . . . . . . 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 5190	. . . . . . . . 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 7406	. . . . . . . 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 19827	. . . . . . . . . 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 6897	. . . . . . . . 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 18860	. . . . . . . . 9 ⊢ ((𝑊 ∈ Mnd ∧ 𝑉 ∈ V) → (𝑊 Σ_g (𝑢 ∈ 𝑉 ↦ 0 )) = 0 )
179	176, 177, 178	syl2anc 593	. . . . . . . 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 2800	. . . . . . 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 470	. . . . . 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 2761	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
183	24, 182, 30	lspcl 21030	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ (𝐵 ∖ 𝑉) ⊆ (Base‘𝑊)) → (𝑁‘(𝐵 ∖ 𝑉)) ∈ (LSubSp‘𝑊))
184	23, 28, 183	syl2anc 593	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → (𝑁‘(𝐵 ∖ 𝑉)) ∈ (LSubSp‘𝑊))
185	184	adantr 484	. . . . . . . . . 10 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → (𝑁‘(𝐵 ∖ 𝑉)) ∈ (LSubSp‘𝑊))
186	182	lsssubg 21011	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘(𝐵 ∖ 𝑉)) ∈ (LSubSp‘𝑊)) → (𝑁‘(𝐵 ∖ 𝑉)) ∈ (SubGrp‘𝑊))
187	23, 185, 186	syl2an2r 695	. . . . . . . . 9 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → (𝑁‘(𝐵 ∖ 𝑉)) ∈ (SubGrp‘𝑊))
188	126	elin1d 4154	. . . . . . . . 9 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → 𝑥 ∈ (𝑁‘(𝐵 ∖ 𝑉)))
189	130	subginvcl 19167	. . . . . . . . 9 ⊢ (((𝑁‘(𝐵 ∖ 𝑉)) ∈ (SubGrp‘𝑊) ∧ 𝑥 ∈ (𝑁‘(𝐵 ∖ 𝑉))) → ((inv_g‘𝑊)‘𝑥) ∈ (𝑁‘(𝐵 ∖ 𝑉)))
190	187, 188, 189	syl2anc 593	. . . . . . . 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 33514	. . . . . . . . 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 480	. . . . . . . 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 600	. . . . . . 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 740	. . . . . 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 3169	. . . . 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 470	. . . 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 33514	. . . . . 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 480	. . . . 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 600	. . . 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 3169	. . 3 ⊢ ((((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) ∧ 𝑥 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉))) → 𝑥 = 0 )
201	29, 24, 30	0ellsp 33515	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑉 ⊆ (Base‘𝑊)) → 0 ∈ (𝑁‘𝑉))
202	23, 122, 201	syl2anc 593	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 0 ∈ (𝑁‘𝑉))
203	32, 202	elind 4150	. . 3 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → 0 ∈ ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉)))
204	200, 203	eqsnd 4785	. 2 ⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑉 ⊆ 𝐵) → ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉)) = { 0 })
205	204	3impa 1121	1 ⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽 ∧ 𝑉 ⊆ 𝐵) → ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉)) = { 0 })

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∨ wo 858 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ∀wral 3075 ∃wrex 3085 Vcvv 3453 ∖ cdif 3899 ∪ cun 3900 ∩ cin 3901 ⊆ wss 3902 ∅c0 4283 {csn 4579 class class class wbr 5097 ↦ cmpt 5178 × cxp 5641 ↾ cres 5645 Fn wfn 6510 ⟶wf 6511 ‘cfv 6515 (class class class)co 7390 ↑_m cmap 8801 finSupp cfsupp 9300 Basecbs 17235 +_gcplusg 17276 Scalarcsca 17279 ·_𝑠 cvsca 17280 0_gc0g 17458 Σ_g cgsu 17459 Mndcmnd 18758 Grpcgrp 18965 inv_gcminusg 18966 SubGrpcsubg 19152 CMndccmn 19810 LModclmod 20914 LSubSpclss 20985 LSpanclspn 21025 LBasisclbs 21128 LVecclvec 21156 LIndSclinds 21844

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7654 df-om 7841 df-1st 7964 df-2nd 7965 df-supp 8134 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-er 8671 df-map 8803 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9301 df-sup 9381 df-oi 9451 df-card 9890 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-nn 12204 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12475 df-z 12562 df-dec 12682 df-uz 12833 df-fz 13506 df-fzo 13653 df-seq 14008 df-hash 14337 df-struct 17173 df-sets 17190 df-slot 17208 df-ndx 17220 df-base 17236 df-ress 17257 df-plusg 17289 df-mulr 17290 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-hom 17300 df-cco 17301 df-0g 17460 df-gsum 17461 df-prds 17466 df-pws 17468 df-mre 17604 df-mrc 17605 df-acs 17607 df-mgm 18664 df-sgrp 18743 df-mnd 18759 df-mhm 18807 df-submnd 18808 df-grp 18968 df-minusg 18969 df-sbg 18970 df-mulg 19100 df-subg 19155 df-ghm 19244 df-cntz 19347 df-cmn 19812 df-abl 19813 df-mgp 20177 df-rng 20189 df-ur 20218 df-ring 20271 df-nzr 20549 df-subrg 20606 df-lmod 20916 df-lss 20986 df-lsp 21026 df-lmhm 21076 df-lbs 21129 df-lvec 21157 df-sra 21227 df-rgmod 21228 df-dsmm 21771 df-frlm 21786 df-uvc 21822 df-lindf 21845 df-linds 21846

This theorem is referenced by: dimkerim 33884

W3C validator