Theorem dimkerim 34145

Description: Given a linear map 𝐹 between vector spaces 𝑉 and 𝑈, the dimension of the vector space 𝑉 is the sum of the dimension of 𝐹 's kernel and the dimension of 𝐹's image. Second part of theorem 5.3 in [Lang] p. 141 This can also be described as the Rank-nullity theorem, (dim‘𝐼) being the rank of 𝐹 (the dimension of its image), and (dim‘𝐾) its nullity (the dimension of its kernel). (Contributed by Thierry Arnoux, 17-May-2023.)

Hypotheses

Ref	Expression
dimkerim.0	⊢ 0 = (0g‘𝑈)
dimkerim.k	⊢ 𝐾 = (𝑉 ↾s (◡𝐹 “ { 0 }))
dimkerim.i	⊢ 𝐼 = (𝑈 ↾s ran 𝐹)

Assertion

Ref	Expression
dimkerim	⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))

Proof of Theorem dimkerim

Dummy variables 𝑏 𝑢 𝑣 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dimkerim.0	. . . . 5 ⊢ 0 = (0g‘𝑈)
2		dimkerim.k	. . . . 5 ⊢ 𝐾 = (𝑉 ↾s (◡𝐹 “ { 0 }))
3	1, 2	kerlmhm 34138	. . . 4 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐾 ∈ LVec)
4		eqid 2762	. . . . 5 ⊢ (LBasis‘𝐾) = (LBasis‘𝐾)
5	4	lbsex 21358	. . . 4 ⊢ (𝐾 ∈ LVec → (LBasis‘𝐾) ≠ ∅)
6	3, 5	syl 18	. . 3 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (LBasis‘𝐾) ≠ ∅)
7		n0 4303	. . 3 ⊢ ((LBasis‘𝐾) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (LBasis‘𝐾))
8	6, 7	sylib 221	. 2 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → ∃𝑤 𝑤 ∈ (LBasis‘𝐾))
9		simpllr 788	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ∈ (LBasis‘𝐾))
10		vex 3457	. . . . . . 7 ⊢ 𝑏 ∈ V
11	10	difexi 5299	. . . . . 6 ⊢ (𝑏 ∖ 𝑤) ∈ V
12	11	a1i 11	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ∈ V)
13		disjdif 4429	. . . . . 6 ⊢ (𝑤 ∩ (𝑏 ∖ 𝑤)) = ∅
14	13	a1i 11	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑤 ∩ (𝑏 ∖ 𝑤)) = ∅)
15		hashunx 14454	. . . . 5 ⊢ ((𝑤 ∈ (LBasis‘𝐾) ∧ (𝑏 ∖ 𝑤) ∈ V ∧ (𝑤 ∩ (𝑏 ∖ 𝑤)) = ∅) → (♯‘(𝑤 ∪ (𝑏 ∖ 𝑤))) = ((♯‘𝑤) +𝑒 (♯‘(𝑏 ∖ 𝑤))))
16	9, 12, 14, 15	syl3anc 1398	. . . 4 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (♯‘(𝑤 ∪ (𝑏 ∖ 𝑤))) = ((♯‘𝑤) +𝑒 (♯‘(𝑏 ∖ 𝑤))))
17		simp-4l 795	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑉 ∈ LVec)
18		simpr 490	. . . . . . 7 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ⊆ 𝑏)
19		undif 4441	. . . . . . 7 ⊢ (𝑤 ⊆ 𝑏 ↔ (𝑤 ∪ (𝑏 ∖ 𝑤)) = 𝑏)
20	18, 19	sylib 221	. . . . . 6 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑤 ∪ (𝑏 ∖ 𝑤)) = 𝑏)
21		simplr 781	. . . . . 6 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑏 ∈ (LBasis‘𝑉))
22	20, 21	eqeltrd 2862	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑤 ∪ (𝑏 ∖ 𝑤)) ∈ (LBasis‘𝑉))
23		eqid 2762	. . . . . 6 ⊢ (LBasis‘𝑉) = (LBasis‘𝑉)
24	23	dimval 34119	. . . . 5 ⊢ ((𝑉 ∈ LVec ∧ (𝑤 ∪ (𝑏 ∖ 𝑤)) ∈ (LBasis‘𝑉)) → (dim‘𝑉) = (♯‘(𝑤 ∪ (𝑏 ∖ 𝑤))))
25	17, 22, 24	syl2anc 596	. . . 4 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (dim‘𝑉) = (♯‘(𝑤 ∪ (𝑏 ∖ 𝑤))))
26	3	ad3antrrr 743	. . . . . 6 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐾 ∈ LVec)
27	4	dimval 34119	. . . . . 6 ⊢ ((𝐾 ∈ LVec ∧ 𝑤 ∈ (LBasis‘𝐾)) → (dim‘𝐾) = (♯‘𝑤))
28	26, 9, 27	syl2anc 596	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (dim‘𝐾) = (♯‘𝑤))
29		dimkerim.i	. . . . . . . . 9 ⊢ 𝐼 = (𝑈 ↾s ran 𝐹)
30	29	imlmhm 34139	. . . . . . . 8 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐼 ∈ LVec)
31	30	ad3antrrr 743	. . . . . . 7 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐼 ∈ LVec)
32		simp-4r 796	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹 ∈ (𝑉 LMHom 𝑈))
33		lmhmlmod2 21222	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝑈 ∈ LMod)
34	32, 33	syl 18	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑈 ∈ LMod)
35		lmhmrnlss 21240	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → ran 𝐹 ∈ (LSubSp‘𝑈))
36	32, 35	syl 18	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ran 𝐹 ∈ (LSubSp‘𝑈))
37		df-ima 5672	. . . . . . . . . . 11 ⊢ (𝐹 “ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
38		imassrn 6071	. . . . . . . . . . . 12 ⊢ (𝐹 “ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ⊆ ran 𝐹
39	38	a1i 11	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹 “ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ⊆ ran 𝐹)
40	37, 39	eqsstrrid 3973	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ⊆ ran 𝐹)
41		lveclmod 21296	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ LVec → 𝑉 ∈ LMod)
42	41	ad4antr 745	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑉 ∈ LMod)
43	23	lbslinds 22052	. . . . . . . . . . . . . . 15 ⊢ (LBasis‘𝑉) ⊆ (LIndS‘𝑉)
44	43, 21	sselid 3932	. . . . . . . . . . . . . 14 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑏 ∈ (LIndS‘𝑉))
45		difssd 4087	. . . . . . . . . . . . . 14 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ⊆ 𝑏)
46		lindsss 22043	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ LMod ∧ 𝑏 ∈ (LIndS‘𝑉) ∧ (𝑏 ∖ 𝑤) ⊆ 𝑏) → (𝑏 ∖ 𝑤) ∈ (LIndS‘𝑉))
47	42, 44, 45, 46	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ∈ (LIndS‘𝑉))
48		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑉) = (Base‘𝑉)
49	48	linds1 22029	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∖ 𝑤) ∈ (LIndS‘𝑉) → (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉))
50	47, 49	syl 18	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉))
51		eqid 2762	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝑉) = (LSubSp‘𝑉)
52		eqid 2762	. . . . . . . . . . . . 13 ⊢ (LSpan‘𝑉) = (LSpan‘𝑉)
53	48, 51, 52	lspcl 21166	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ LMod ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉))
54	42, 50, 53	syl2anc 596	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉))
55		eqid 2762	. . . . . . . . . . . 12 ⊢ (𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = (𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
56	51, 55	reslmhm 21242	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝑉 LMHom 𝑈) ∧ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉)) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) LMHom 𝑈))
57	32, 54, 56	syl2anc 596	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) LMHom 𝑈))
58		eqid 2762	. . . . . . . . . . . 12 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
59	29, 58	reslmhm2b 21244	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ ran 𝐹 ∈ (LSubSp‘𝑈) ∧ ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ⊆ ran 𝐹) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) LMHom 𝑈) ↔ (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) LMHom 𝐼)))
60	59	biimpa 482	. . . . . . . . . 10 ⊢ (((𝑈 ∈ LMod ∧ ran 𝐹 ∈ (LSubSp‘𝑈) ∧ ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ⊆ ran 𝐹) ∧ (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) LMHom 𝑈)) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) LMHom 𝐼))
61	34, 36, 40, 57, 60	syl31anc 1400	. . . . . . . . 9 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) LMHom 𝐼))
62		lmghm 21221	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
63	62	ad4antlr 746	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
64	48, 23	lbsss 21267	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ (LBasis‘𝑉) → 𝑏 ⊆ (Base‘𝑉))
65	21, 64	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑏 ⊆ (Base‘𝑉))
66	45, 65	sstrd 3944	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉))
67	42, 66, 53	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉))
68	51	lsssubg 21147	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉)) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (SubGrp‘𝑉))
69	42, 67, 68	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (SubGrp‘𝑉))
70	55	resghm 19365	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ (𝑉 GrpHom 𝑈) ∧ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (SubGrp‘𝑉)) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) GrpHom 𝑈))
71	63, 69, 70	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) GrpHom 𝑈))
72		eqid 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Base‘𝑈) = (Base‘𝑈)
73	48, 72	lmhmf 21224	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
74	73	ad4antlr 746	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
75	74	ffnd 6707	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹 Fn (Base‘𝑉))
76	48, 52	lspssv 21173	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑉 ∈ LMod ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ⊆ (Base‘𝑉))
77	42, 66, 76	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ⊆ (Base‘𝑉))
78	75, 77	fnssresd 6660	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) Fn ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
79		fniniseg 7056	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) Fn ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) → (𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) ↔ (𝑥 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘𝑥) = 0 )))
80	79	biimpa 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) Fn ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∧ 𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 })) → (𝑥 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘𝑥) = 0 ))
81	78, 80	sylan 592	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 })) → (𝑥 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘𝑥) = 0 ))
82	81	simpld 500	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 })) → 𝑥 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
83	75	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 })) → 𝐹 Fn (Base‘𝑉))
84	77	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 })) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ⊆ (Base‘𝑉))
85	84, 82	sseldd 3935	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 })) → 𝑥 ∈ (Base‘𝑉))
86	82	fvresd 6902	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 })) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘𝑥) = (𝐹‘𝑥))
87	81	simprd 501	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 })) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘𝑥) = 0 )
88	86, 87	eqtr3d 2799	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 })) → (𝐹‘𝑥) = 0 )
89		fniniseg 7056	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 Fn (Base‘𝑉) → (𝑥 ∈ (◡𝐹 “ { 0 }) ↔ (𝑥 ∈ (Base‘𝑉) ∧ (𝐹‘𝑥) = 0 )))
90	89	biimpar 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Base‘𝑉) ∧ (𝑥 ∈ (Base‘𝑉) ∧ (𝐹‘𝑥) = 0 )) → 𝑥 ∈ (◡𝐹 “ { 0 }))
91	83, 85, 88, 90	syl12anc 850	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 })) → 𝑥 ∈ (◡𝐹 “ { 0 }))
92	82, 91	elind 4149	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 })) → 𝑥 ∈ (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∩ (◡𝐹 “ { 0 })))
93		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LBasis‘𝐾))
94		eqid 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Base‘𝐾) = (Base‘𝐾)
95		eqid 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾)
96	94, 4, 95	lbssp 21269	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑤 ∈ (LBasis‘𝐾) → ((LSpan‘𝐾)‘𝑤) = (Base‘𝐾))
97	93, 96	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝐾)‘𝑤) = (Base‘𝐾))
98	41	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑉 ∈ LMod)
99		eqid 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (◡𝐹 “ { 0 }) = (◡𝐹 “ { 0 })
100	99, 1, 51	lmhmkerlss 21241	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉))
101	100	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉))
102	94, 4	lbsss 21267	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 ∈ (LBasis‘𝐾) → 𝑤 ⊆ (Base‘𝐾))
103	93, 102	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ⊆ (Base‘𝐾))
104		cnvimass 6082	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (◡𝐹 “ { 0 }) ⊆ dom 𝐹
105	104, 73	fssdm 6726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (◡𝐹 “ { 0 }) ⊆ (Base‘𝑉))
106	2, 48	ressbas2 17336	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((◡𝐹 “ { 0 }) ⊆ (Base‘𝑉) → (◡𝐹 “ { 0 }) = (Base‘𝐾))
107	105, 106	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (◡𝐹 “ { 0 }) = (Base‘𝐾))
108	107	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (◡𝐹 “ { 0 }) = (Base‘𝐾))
109	103, 108	sseqtrrd 3971	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ⊆ (◡𝐹 “ { 0 }))
110	2, 52, 95, 51	lsslsp 21205	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑉 ∈ LMod ∧ (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (◡𝐹 “ { 0 })) → ((LSpan‘𝐾)‘𝑤) = ((LSpan‘𝑉)‘𝑤))
111	110	eqcomd 2768	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑉 ∈ LMod ∧ (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (◡𝐹 “ { 0 })) → ((LSpan‘𝑉)‘𝑤) = ((LSpan‘𝐾)‘𝑤))
112	98, 101, 109, 111	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝑉)‘𝑤) = ((LSpan‘𝐾)‘𝑤))
113	97, 112, 108	3eqtr4d 2807	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝑉)‘𝑤) = (◡𝐹 “ { 0 }))
114	113	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑤) = (◡𝐹 “ { 0 }))
115	114	ineq2d 4169	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∩ ((LSpan‘𝑉)‘𝑤)) = (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∩ (◡𝐹 “ { 0 })))
116		eqid 2762	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0g‘𝑉) = (0g‘𝑉)
117	23, 52, 116	lbsdiflsp0 34144	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑉 ∈ LVec ∧ 𝑏 ∈ (LBasis‘𝑉) ∧ 𝑤 ⊆ 𝑏) → (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∩ ((LSpan‘𝑉)‘𝑤)) = {(0g‘𝑉)})
118	117	ad5ant145 1396	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∩ ((LSpan‘𝑉)‘𝑤)) = {(0g‘𝑉)})
119	115, 118	eqtr3d 2799	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∩ (◡𝐹 “ { 0 })) = {(0g‘𝑉)})
120	119	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 })) → (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∩ (◡𝐹 “ { 0 })) = {(0g‘𝑉)})
121	92, 120	eleqtrd 2864	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 })) → 𝑥 ∈ {(0g‘𝑉)})
122	121	ex 418	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) → 𝑥 ∈ {(0g‘𝑉)}))
123	122	ssrdv 3940	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) ⊆ {(0g‘𝑉)})
124	116, 48, 52	0ellsp 33812	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 ∈ LMod ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → (0g‘𝑉) ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
125	42, 66, 124	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (0g‘𝑉) ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
126		fvexd 6897	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) ∈ V)
127	125	fvresd 6902	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) = (𝐹‘(0g‘𝑉)))
128	116, 1	ghmid 19355	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 ∈ (𝑉 GrpHom 𝑈) → (𝐹‘(0g‘𝑉)) = 0 )
129	62, 128	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (𝐹‘(0g‘𝑉)) = 0 )
130	129	ad4antlr 746	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹‘(0g‘𝑉)) = 0 )
131	127, 130	eqtrd 2797	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) = 0 )
132		elsng 4601	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) ∈ V → (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) ∈ { 0 } ↔ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) = 0 ))
133	132	biimpar 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) ∈ V ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) = 0 ) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) ∈ { 0 })
134	126, 131, 133	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) ∈ { 0 })
135	78, 125, 134	elpreimad 7055	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (0g‘𝑉) ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }))
136	135	snssd 4750	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → {(0g‘𝑉)} ⊆ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }))
137	123, 136	eqssd 3951	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) = {(0g‘𝑉)})
138		lmodgrp 21057	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑉 ∈ LMod → 𝑉 ∈ Grp)
139		grpmnd 19070	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑉 ∈ Grp → 𝑉 ∈ Mnd)
140	42, 138, 139	3syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑉 ∈ Mnd)
141	55, 48, 116	ress0g 18873	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑉 ∈ Mnd ∧ (0g‘𝑉) ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∧ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ⊆ (Base‘𝑉)) → (0g‘𝑉) = (0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))))
142	140, 125, 77, 141	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (0g‘𝑉) = (0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))))
143	142	sneqd 4599	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → {(0g‘𝑉)} = {(0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))})
144	137, 143	eqtrd 2797	. . . . . . . . . . . . . . 15 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) = {(0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))})
145		eqid 2762	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
146		eqid 2762	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
147	145, 72, 146, 1	kerf1ghm 19380	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) GrpHom 𝑈) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):(Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))–1-1→(Base‘𝑈) ↔ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) = {(0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))}))
148	147	biimpar 483	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) GrpHom 𝑈) ∧ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) = {(0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))}) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):(Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))–1-1→(Base‘𝑈))
149	71, 144, 148	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):(Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))–1-1→(Base‘𝑈))
150		eqidd 2763	. . . . . . . . . . . . . . 15 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
151	55, 48	ressbas2 17336	. . . . . . . . . . . . . . . 16 ⊢ (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ⊆ (Base‘𝑉) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) = (Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))))
152	77, 151	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) = (Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))))
153		eqidd 2763	. . . . . . . . . . . . . . 15 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (Base‘𝑈) = (Base‘𝑈))
154	150, 152, 153	f1eq123d 6813	. . . . . . . . . . . . . 14 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))–1-1→(Base‘𝑈) ↔ (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):(Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))–1-1→(Base‘𝑈)))
155	149, 154	mpbird 260	. . . . . . . . . . . . 13 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))–1-1→(Base‘𝑈))
156		f1ssr 6783	. . . . . . . . . . . . 13 ⊢ (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))–1-1→(Base‘𝑈) ∧ ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ⊆ ran 𝐹) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))–1-1→ran 𝐹)
157	155, 40, 156	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))–1-1→ran 𝐹)
158		f1f1orn 6833	. . . . . . . . . . . 12 ⊢ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))–1-1→ran 𝐹 → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))–1-1-onto→ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
159	157, 158	syl 18	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))–1-1-onto→ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
160		simp-4r 796	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → (𝐹‘𝑥) = 𝑦)
161	75	ad6antr 749	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → 𝐹 Fn (Base‘𝑉))
162		simpllr 788	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → 𝑢 ∈ ((LSpan‘𝑉)‘𝑤))
163	113	ad8antr 753	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → ((LSpan‘𝑉)‘𝑤) = (◡𝐹 “ { 0 }))
164	162, 163	eleqtrd 2864	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → 𝑢 ∈ (◡𝐹 “ { 0 }))
165		fniniseg 7056	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 Fn (Base‘𝑉) → (𝑢 ∈ (◡𝐹 “ { 0 }) ↔ (𝑢 ∈ (Base‘𝑉) ∧ (𝐹‘𝑢) = 0 )))
166	165	simplbda 505	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 Fn (Base‘𝑉) ∧ 𝑢 ∈ (◡𝐹 “ { 0 })) → (𝐹‘𝑢) = 0 )
167	161, 164, 166	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → (𝐹‘𝑢) = 0 )
168	167	oveq1d 7432	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → ((𝐹‘𝑢)(+g‘𝑈)(𝐹‘𝑣)) = ( 0 (+g‘𝑈)(𝐹‘𝑣)))
169		simpr 490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → 𝑥 = (𝑢(+g‘𝑉)𝑣))
170	169	fveq2d 6886	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → (𝐹‘𝑥) = (𝐹‘(𝑢(+g‘𝑉)𝑣)))
171	63	ad6antr 749	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
172	48, 52	lspss 21174	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑉 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑉) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ ((LSpan‘𝑉)‘𝑏))
173	42, 65, 18, 172	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ ((LSpan‘𝑉)‘𝑏))
174	48, 23, 52	lbssp 21269	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 ∈ (LBasis‘𝑉) → ((LSpan‘𝑉)‘𝑏) = (Base‘𝑉))
175	21, 174	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑏) = (Base‘𝑉))
176	173, 175	sseqtrd 3970	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ (Base‘𝑉))
177	176	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) → ((LSpan‘𝑉)‘𝑤) ⊆ (Base‘𝑉))
178	177	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → ((LSpan‘𝑉)‘𝑤) ⊆ (Base‘𝑉))
179	178, 162	sseldd 3935	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → 𝑢 ∈ (Base‘𝑉))
180	77	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ⊆ (Base‘𝑉))
181	180	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ⊆ (Base‘𝑉))
182		simplr 781	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
183	181, 182	sseldd 3935	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → 𝑣 ∈ (Base‘𝑉))
184		eqid 2762	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (+g‘𝑉) = (+g‘𝑉)
185		eqid 2762	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (+g‘𝑈) = (+g‘𝑈)
186	48, 184, 185	ghmlin 19354	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 ∈ (𝑉 GrpHom 𝑈) ∧ 𝑢 ∈ (Base‘𝑉) ∧ 𝑣 ∈ (Base‘𝑉)) → (𝐹‘(𝑢(+g‘𝑉)𝑣)) = ((𝐹‘𝑢)(+g‘𝑈)(𝐹‘𝑣)))
187	171, 179, 183, 186	syl3anc 1398	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → (𝐹‘(𝑢(+g‘𝑉)𝑣)) = ((𝐹‘𝑢)(+g‘𝑈)(𝐹‘𝑣)))
188	170, 187	eqtr2d 2798	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → ((𝐹‘𝑢)(+g‘𝑈)(𝐹‘𝑣)) = (𝐹‘𝑥))
189		lmhmlvec2 34137	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec)
190	189	lvecgrpd 21298	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ Grp)
191	190	ad9antr 755	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → 𝑈 ∈ Grp)
192	74	ad6antr 749	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
193	192, 183	ffvelcdmd 7082	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → (𝐹‘𝑣) ∈ (Base‘𝑈))
194	72, 185, 1, 191, 193	grplidd 19099	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → ( 0 (+g‘𝑈)(𝐹‘𝑣)) = (𝐹‘𝑣))
195	168, 188, 194	3eqtr3d 2805	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → (𝐹‘𝑥) = (𝐹‘𝑣))
196	160, 195	eqtr3d 2799	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → 𝑦 = (𝐹‘𝑣))
197	161, 183, 182	fnfvimad 7237	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → (𝐹‘𝑣) ∈ (𝐹 “ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
198	196, 197	eqeltrd 2862	. . . . . . . . . . . . . . . 16 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → 𝑦 ∈ (𝐹 “ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
199		simp-7l 801	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) → 𝑉 ∈ LVec)
200		simplr 781	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) → 𝑥 ∈ (Base‘𝑉))
201	109	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ⊆ (◡𝐹 “ { 0 }))
202	105	ad4antlr 746	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (◡𝐹 “ { 0 }) ⊆ (Base‘𝑉))
203	201, 202	sstrd 3944	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ⊆ (Base‘𝑉))
204		eqid 2762	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (LSSum‘𝑉) = (LSSum‘𝑉)
205	48, 52, 204	lsmsp2 21277	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑉 ∈ LMod ∧ 𝑤 ⊆ (Base‘𝑉) ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → (((LSpan‘𝑉)‘𝑤)(LSSum‘𝑉)((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = ((LSpan‘𝑉)‘(𝑤 ∪ (𝑏 ∖ 𝑤))))
206	42, 203, 66, 205	syl3anc 1398	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (((LSpan‘𝑉)‘𝑤)(LSSum‘𝑉)((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = ((LSpan‘𝑉)‘(𝑤 ∪ (𝑏 ∖ 𝑤))))
207	20	fveq2d 6886	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘(𝑤 ∪ (𝑏 ∖ 𝑤))) = ((LSpan‘𝑉)‘𝑏))
208	206, 207, 175	3eqtrrd 2802	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (Base‘𝑉) = (((LSpan‘𝑉)‘𝑤)(LSSum‘𝑉)((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
209	208	ad3antrrr 743	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) → (Base‘𝑉) = (((LSpan‘𝑉)‘𝑤)(LSSum‘𝑉)((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
210	200, 209	eleqtrd 2864	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) → 𝑥 ∈ (((LSpan‘𝑉)‘𝑤)(LSSum‘𝑉)((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
211	48, 184, 204	lsmelvalx 19773	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑉 ∈ LVec ∧ ((LSpan‘𝑉)‘𝑤) ⊆ (Base‘𝑉) ∧ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ⊆ (Base‘𝑉)) → (𝑥 ∈ (((LSpan‘𝑉)‘𝑤)(LSSum‘𝑉)((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ↔ ∃𝑢 ∈ ((LSpan‘𝑉)‘𝑤)∃𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))𝑥 = (𝑢(+g‘𝑉)𝑣)))
212	211	biimpa 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑉 ∈ LVec ∧ ((LSpan‘𝑉)‘𝑤) ⊆ (Base‘𝑉) ∧ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ⊆ (Base‘𝑉)) ∧ 𝑥 ∈ (((LSpan‘𝑉)‘𝑤)(LSSum‘𝑉)((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) → ∃𝑢 ∈ ((LSpan‘𝑉)‘𝑤)∃𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))𝑥 = (𝑢(+g‘𝑉)𝑣))
213	199, 177, 180, 210, 212	syl31anc 1400	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) → ∃𝑢 ∈ ((LSpan‘𝑉)‘𝑤)∃𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))𝑥 = (𝑢(+g‘𝑉)𝑣))
214	198, 213	r19.29vva 3224	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ (𝐹 “ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
215		fvelrnb 6942	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn (Base‘𝑉) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ (Base‘𝑉)(𝐹‘𝑥) = 𝑦))
216	215	biimpa 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn (Base‘𝑉) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ (Base‘𝑉)(𝐹‘𝑥) = 𝑦)
217	75, 216	sylan 592	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ (Base‘𝑉)(𝐹‘𝑥) = 𝑦)
218	214, 217	r19.29a 3172	. . . . . . . . . . . . . 14 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ (𝐹 “ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
219	39, 218	eqelssd 3955	. . . . . . . . . . . . 13 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹 “ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = ran 𝐹)
220	37, 219	eqtr3id 2811	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = ran 𝐹)
221	220	f1oeq3d 6818	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))–1-1-onto→ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ↔ (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))–1-1-onto→ran 𝐹))
222	159, 221	mpbid 235	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))–1-1-onto→ran 𝐹)
223	42, 50, 76	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ⊆ (Base‘𝑉))
224	223, 151	syl 18	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) = (Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))))
225		frn 6714	. . . . . . . . . . . 12 ⊢ (𝐹:(Base‘𝑉)⟶(Base‘𝑈) → ran 𝐹 ⊆ (Base‘𝑈))
226	29, 72	ressbas2 17336	. . . . . . . . . . . 12 ⊢ (ran 𝐹 ⊆ (Base‘𝑈) → ran 𝐹 = (Base‘𝐼))
227	32, 73, 225, 226	4syl 20	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ran 𝐹 = (Base‘𝐼))
228	150, 224, 227	f1oeq123d 6815	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))–1-1-onto→ran 𝐹 ↔ (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):(Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))–1-1-onto→(Base‘𝐼)))
229	222, 228	mpbid 235	. . . . . . . . 9 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):(Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))–1-1-onto→(Base‘𝐼))
230		eqid 2762	. . . . . . . . . 10 ⊢ (Base‘𝐼) = (Base‘𝐼)
231	145, 230	islmim 21252	. . . . . . . . 9 ⊢ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) LMIso 𝐼) ↔ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) LMHom 𝐼) ∧ (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):(Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))–1-1-onto→(Base‘𝐼)))
232	61, 229, 231	sylanbrc 595	. . . . . . . 8 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) LMIso 𝐼))
233	48, 52	lspssid 21175	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ LMod ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → (𝑏 ∖ 𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
234	42, 50, 233	syl2anc 596	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
235	51, 55	lsslinds 22050	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉) ∧ (𝑏 ∖ 𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) → ((𝑏 ∖ 𝑤) ∈ (LIndS‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) ↔ (𝑏 ∖ 𝑤) ∈ (LIndS‘𝑉)))
236	235	biimpar 483	. . . . . . . . . 10 ⊢ (((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉) ∧ (𝑏 ∖ 𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ (𝑏 ∖ 𝑤) ∈ (LIndS‘𝑉)) → (𝑏 ∖ 𝑤) ∈ (LIndS‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))))
237	42, 67, 234, 47, 236	syl31anc 1400	. . . . . . . . 9 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ∈ (LIndS‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))))
238		eqid 2762	. . . . . . . . . . . . 13 ⊢ (LSpan‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (LSpan‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
239	55, 52, 238, 51	lsslsp 21205	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉) ∧ (𝑏 ∖ 𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) → ((LSpan‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))‘(𝑏 ∖ 𝑤)) = ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
240	239	eqcomd 2768	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉) ∧ (𝑏 ∖ 𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) = ((LSpan‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))‘(𝑏 ∖ 𝑤)))
241	42, 54, 234, 240	syl3anc 1398	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) = ((LSpan‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))‘(𝑏 ∖ 𝑤)))
242	241, 224	eqtr3d 2799	. . . . . . . . 9 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))‘(𝑏 ∖ 𝑤)) = (Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))))
243		eqid 2762	. . . . . . . . . 10 ⊢ (LBasis‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (LBasis‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
244	145, 243, 238	islbs4 22051	. . . . . . . . 9 ⊢ ((𝑏 ∖ 𝑤) ∈ (LBasis‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) ↔ ((𝑏 ∖ 𝑤) ∈ (LIndS‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) ∧ ((LSpan‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))‘(𝑏 ∖ 𝑤)) = (Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))))
245	237, 242, 244	sylanbrc 595	. . . . . . . 8 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ∈ (LBasis‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))))
246		eqid 2762	. . . . . . . . 9 ⊢ (LBasis‘𝐼) = (LBasis‘𝐼)
247	243, 246	lmimlbs 22055	. . . . . . . 8 ⊢ (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) LMIso 𝐼) ∧ (𝑏 ∖ 𝑤) ∈ (LBasis‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤)) ∈ (LBasis‘𝐼))
248	232, 245, 247	syl2anc 596	. . . . . . 7 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤)) ∈ (LBasis‘𝐼))
249	246	dimval 34119	. . . . . . 7 ⊢ ((𝐼 ∈ LVec ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤)) ∈ (LBasis‘𝐼)) → (dim‘𝐼) = (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤))))
250	31, 248, 249	syl2anc 596	. . . . . 6 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (dim‘𝐼) = (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤))))
251		f1imaeng 9024	. . . . . . . 8 ⊢ (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))–1-1→ran 𝐹 ∧ (𝑏 ∖ 𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∧ (𝑏 ∖ 𝑤) ∈ (LIndS‘𝑉)) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤)) ≈ (𝑏 ∖ 𝑤))
252		hasheni 14416	. . . . . . . 8 ⊢ (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤)) ≈ (𝑏 ∖ 𝑤) → (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤))) = (♯‘(𝑏 ∖ 𝑤)))
253	251, 252	syl 18	. . . . . . 7 ⊢ (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))–1-1→ran 𝐹 ∧ (𝑏 ∖ 𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∧ (𝑏 ∖ 𝑤) ∈ (LIndS‘𝑉)) → (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤))) = (♯‘(𝑏 ∖ 𝑤)))
254	157, 234, 47, 253	syl3anc 1398	. . . . . 6 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤))) = (♯‘(𝑏 ∖ 𝑤)))
255	250, 254	eqtrd 2797	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (dim‘𝐼) = (♯‘(𝑏 ∖ 𝑤)))
256	28, 255	oveq12d 7435	. . . 4 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((dim‘𝐾) +𝑒 (dim‘𝐼)) = ((♯‘𝑤) +𝑒 (♯‘(𝑏 ∖ 𝑤))))
257	16, 25, 256	3eqtr4d 2807	. . 3 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))
258	4	lbslinds 22052	. . . . . 6 ⊢ (LBasis‘𝐾) ⊆ (LIndS‘𝐾)
259	258, 93	sselid 3932	. . . . 5 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LIndS‘𝐾))
260	51, 2	lsslinds 22050	. . . . . 6 ⊢ ((𝑉 ∈ LMod ∧ (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (◡𝐹 “ { 0 })) → (𝑤 ∈ (LIndS‘𝐾) ↔ 𝑤 ∈ (LIndS‘𝑉)))
261	260	biimpa 482	. . . . 5 ⊢ (((𝑉 ∈ LMod ∧ (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (◡𝐹 “ { 0 })) ∧ 𝑤 ∈ (LIndS‘𝐾)) → 𝑤 ∈ (LIndS‘𝑉))
262	98, 101, 109, 259, 261	syl31anc 1400	. . . 4 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LIndS‘𝑉))
263	23	islinds4 22054	. . . . 5 ⊢ (𝑉 ∈ LVec → (𝑤 ∈ (LIndS‘𝑉) ↔ ∃𝑏 ∈ (LBasis‘𝑉)𝑤 ⊆ 𝑏))
264	263	ad2antrr 739	. . . 4 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (𝑤 ∈ (LIndS‘𝑉) ↔ ∃𝑏 ∈ (LBasis‘𝑉)𝑤 ⊆ 𝑏))
265	262, 264	mpbid 235	. . 3 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ∃𝑏 ∈ (LBasis‘𝑉)𝑤 ⊆ 𝑏)
266	257, 265	r19.29a 3172	. 2 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))
267	8, 266	exlimddv 1968	1 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2957  ∃wrex 3088  Vcvv 3453   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  {csn 4587   class class class wbr 5107  ^◡ccnv 5658  ran crn 5660   ↾ cres 5661   “ cima 5662   Fn wfn 6532  ⟶wf 6533  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7417   ≈ cen 8953   +_𝑒 cxad 13165  ♯chash 14398  Basecbs 17307   ↾_s cress 17328  +_gcplusg 17348  0_gc0g 17530  Mndcmnd 18842  Grpcgrp 19063  SubGrpcsubg 19249   GrpHom cghm 19346  LSSumclsm 19767  LModclmod 21050  LSubSpclss 21121  LSpanclspn 21161   LMHom clmhm 21209   LMIso clmim 21210  LBasisclbs 21264  LVecclvec 21292  LIndSclinds 22024  dimcldim 34117

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-reg 9568  ax-inf2 9624  ax-ac2 10469  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  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 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-of 7682  df-rpss 7728  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8163  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-oadd 8463  df-er 8700  df-map 8832  df-ixp 8909  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-fsupp 9336  df-sup 9416  df-oi 9486  df-r1 9750  df-rank 9751  df-scott 9872  df-dju 9910  df-card 9948  df-acn 9951  df-ac 10123  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-7 12336  df-8 12337  df-9 12338  df-n0 12533  df-xnn0 12606  df-z 12620  df-dec 12741  df-uz 12892  df-xadd 13168  df-fz 13566  df-fzo 13714  df-seq 14070  df-hash 14399  df-struct 17245  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-plusg 17361  df-mulr 17362  df-sca 17364  df-vsca 17365  df-ip 17366  df-tset 17367  df-ple 17368  df-ocomp 17369  df-ds 17370  df-hom 17372  df-cco 17373  df-0g 17532  df-gsum 17533  df-prds 17538  df-pws 17540  df-mre 17676  df-mrc 17677  df-mri 17678  df-acs 17679  df-proset 18388  df-drs 18389  df-poset 18407  df-ipo 18622  df-mgm 18736  df-sgrp 18827  df-mnd 18843  df-mhm 18897  df-submnd 18898  df-grp 19066  df-minusg 19067  df-sbg 19068  df-mulg 19197  df-subg 19252  df-ghm 19347  df-cntz 19450  df-lsm 19769  df-cmn 19915  df-abl 19916  df-mgp 20280  df-rng 20294  df-ur 20327  df-ring 20380  df-oppr 20484  df-dvdsr 20504  df-unit 20505  df-invr 20535  df-nzr 20679  df-subrg 20738  df-drng 20898  df-lmod 21052  df-lss 21122  df-lsp 21162  df-lmhm 21212  df-lmim 21213  df-lbs 21265  df-lvec 21293  df-sra 21363  df-rgmod 21364  df-dsmm 21951  df-frlm 21966  df-uvc 22002  df-lindf 22025  df-linds 22026  df-dim 34118

This theorem is used by:  qusdimsum  34146  lvecendof1f1o  34151