Theorem dimkerim 33672

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 33665	. . . 4 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐾 ∈ LVec)
4		eqid 2736	. . . . 5 ⊢ (LBasis‘𝐾) = (LBasis‘𝐾)
5	4	lbsex 21131	. . . 4 ⊢ (𝐾 ∈ LVec → (LBasis‘𝐾) ≠ ∅)
6	3, 5	syl 17	. . 3 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (LBasis‘𝐾) ≠ ∅)
7		n0 4333	. . 3 ⊢ ((LBasis‘𝐾) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (LBasis‘𝐾))
8	6, 7	sylib 218	. 2 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → ∃𝑤 𝑤 ∈ (LBasis‘𝐾))
9		simpllr 775	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ∈ (LBasis‘𝐾))
10		vex 3468	. . . . . . 7 ⊢ 𝑏 ∈ V
11	10	difexi 5305	. . . . . 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 4452	. . . . . 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 14409	. . . . 5 ⊢ ((𝑤 ∈ (LBasis‘𝐾) ∧ (𝑏 ∖ 𝑤) ∈ V ∧ (𝑤 ∩ (𝑏 ∖ 𝑤)) = ∅) → (♯‘(𝑤 ∪ (𝑏 ∖ 𝑤))) = ((♯‘𝑤) +𝑒 (♯‘(𝑏 ∖ 𝑤))))
16	9, 12, 14, 15	syl3anc 1373	. . . 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 782	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑉 ∈ LVec)
18		simpr 484	. . . . . . 7 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ⊆ 𝑏)
19		undif 4462	. . . . . . 7 ⊢ (𝑤 ⊆ 𝑏 ↔ (𝑤 ∪ (𝑏 ∖ 𝑤)) = 𝑏)
20	18, 19	sylib 218	. . . . . 6 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑤 ∪ (𝑏 ∖ 𝑤)) = 𝑏)
21		simplr 768	. . . . . 6 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑏 ∈ (LBasis‘𝑉))
22	20, 21	eqeltrd 2835	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑤 ∪ (𝑏 ∖ 𝑤)) ∈ (LBasis‘𝑉))
23		eqid 2736	. . . . . 6 ⊢ (LBasis‘𝑉) = (LBasis‘𝑉)
24	23	dimval 33645	. . . . 5 ⊢ ((𝑉 ∈ LVec ∧ (𝑤 ∪ (𝑏 ∖ 𝑤)) ∈ (LBasis‘𝑉)) → (dim‘𝑉) = (♯‘(𝑤 ∪ (𝑏 ∖ 𝑤))))
25	17, 22, 24	syl2anc 584	. . . 4 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (dim‘𝑉) = (♯‘(𝑤 ∪ (𝑏 ∖ 𝑤))))
26	3	ad3antrrr 730	. . . . . 6 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐾 ∈ LVec)
27	4	dimval 33645	. . . . . 6 ⊢ ((𝐾 ∈ LVec ∧ 𝑤 ∈ (LBasis‘𝐾)) → (dim‘𝐾) = (♯‘𝑤))
28	26, 9, 27	syl2anc 584	. . . . 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 33666	. . . . . . . 8 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐼 ∈ LVec)
31	30	ad3antrrr 730	. . . . . . 7 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐼 ∈ LVec)
32		simp-4r 783	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹 ∈ (𝑉 LMHom 𝑈))
33		lmhmlmod2 20995	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝑈 ∈ LMod)
34	32, 33	syl 17	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑈 ∈ LMod)
35		lmhmrnlss 21013	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → ran 𝐹 ∈ (LSubSp‘𝑈))
36	32, 35	syl 17	. . . . . . . . . 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 6063	. . . . . . . . . . . 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 4003	. . . . . . . . . 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 21069	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ LVec → 𝑉 ∈ LMod)
42	41	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑉 ∈ LMod)
43	23	lbslinds 21798	. . . . . . . . . . . . . . 15 ⊢ (LBasis‘𝑉) ⊆ (LIndS‘𝑉)
44	43, 21	sselid 3961	. . . . . . . . . . . . . 14 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑏 ∈ (LIndS‘𝑉))
45		difssd 4117	. . . . . . . . . . . . . 14 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ⊆ 𝑏)
46		lindsss 21789	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ LMod ∧ 𝑏 ∈ (LIndS‘𝑉) ∧ (𝑏 ∖ 𝑤) ⊆ 𝑏) → (𝑏 ∖ 𝑤) ∈ (LIndS‘𝑉))
47	42, 44, 45, 46	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ∈ (LIndS‘𝑉))
48		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑉) = (Base‘𝑉)
49	48	linds1 21775	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∖ 𝑤) ∈ (LIndS‘𝑉) → (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉))
50	47, 49	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉))
51		eqid 2736	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝑉) = (LSubSp‘𝑉)
52		eqid 2736	. . . . . . . . . . . . 13 ⊢ (LSpan‘𝑉) = (LSpan‘𝑉)
53	48, 51, 52	lspcl 20938	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ LMod ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉))
54	42, 50, 53	syl2anc 584	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉))
55		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = (𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
56	51, 55	reslmhm 21015	. . . . . . . . . . 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 584	. . . . . . . . . 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 2736	. . . . . . . . . . . 12 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
59	29, 58	reslmhm2b 21017	. . . . . . . . . . 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 476	. . . . . . . . . 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 1375	. . . . . . . . 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 20994	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
63	62	ad4antlr 733	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
64	48, 23	lbsss 21040	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ (LBasis‘𝑉) → 𝑏 ⊆ (Base‘𝑉))
65	21, 64	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑏 ⊆ (Base‘𝑉))
66	45, 65	sstrd 3974	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉))
67	42, 66, 53	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉))
68	51	lsssubg 20919	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉)) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (SubGrp‘𝑉))
69	42, 67, 68	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (SubGrp‘𝑉))
70	55	resghm 19220	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Base‘𝑈) = (Base‘𝑈)
73	48, 72	lmhmf 20997	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
74	73	ad4antlr 733	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
75	74	ffnd 6712	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹 Fn (Base‘𝑉))
76	48, 52	lspssv 20945	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑉 ∈ LMod ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ⊆ (Base‘𝑉))
77	42, 66, 76	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . 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 6667	. . . . . . . . . . . . . . . . . . . . . . 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 7055	. . . . . . . . . . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . . . . . . . . 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 580	. . . . . . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . 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 3964	. . . . . . . . . . . . . . . . . . . . . 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 6901	. . . . . . . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . . . . . 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 7055	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 Fn (Base‘𝑉) → (𝑥 ∈ (◡𝐹 “ { 0 }) ↔ (𝑥 ∈ (Base‘𝑉) ∧ (𝐹‘𝑥) = 0 )))
90	89	biimpar 477	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Base‘𝑉) ∧ (𝑥 ∈ (Base‘𝑉) ∧ (𝐹‘𝑥) = 0 )) → 𝑥 ∈ (◡𝐹 “ { 0 }))
91	83, 85, 88, 90	syl12anc 836	. . . . . . . . . . . . . . . . . . . . 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 4180	. . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LBasis‘𝐾))
94		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Base‘𝐾) = (Base‘𝐾)
95		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾)
96	94, 4, 95	lbssp 21042	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑤 ∈ (LBasis‘𝐾) → ((LSpan‘𝐾)‘𝑤) = (Base‘𝐾))
97	93, 96	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝐾)‘𝑤) = (Base‘𝐾))
98	41	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑉 ∈ LMod)
99		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (◡𝐹 “ { 0 }) = (◡𝐹 “ { 0 })
100	99, 1, 51	lmhmkerlss 21014	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉))
101	100	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉))
102	94, 4	lbsss 21040	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 ∈ (LBasis‘𝐾) → 𝑤 ⊆ (Base‘𝐾))
103	93, 102	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ⊆ (Base‘𝐾))
104		cnvimass 6074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (◡𝐹 “ { 0 }) ⊆ dom 𝐹
105	104, 73	fssdm 6730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (◡𝐹 “ { 0 }) ⊆ (Base‘𝑉))
106	2, 48	ressbas2 17264	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((◡𝐹 “ { 0 }) ⊆ (Base‘𝑉) → (◡𝐹 “ { 0 }) = (Base‘𝐾))
107	105, 106	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (◡𝐹 “ { 0 }) = (Base‘𝐾))
108	107	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (◡𝐹 “ { 0 }) = (Base‘𝐾))
109	103, 108	sseqtrrd 4001	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ⊆ (◡𝐹 “ { 0 }))
110	2, 52, 95, 51	lsslsp 20977	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑉 ∈ LMod ∧ (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (◡𝐹 “ { 0 })) → ((LSpan‘𝐾)‘𝑤) = ((LSpan‘𝑉)‘𝑤))
111	110	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑉 ∈ LMod ∧ (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (◡𝐹 “ { 0 })) → ((LSpan‘𝑉)‘𝑤) = ((LSpan‘𝐾)‘𝑤))
112	98, 101, 109, 111	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝑉)‘𝑤) = ((LSpan‘𝐾)‘𝑤))
113	97, 112, 108	3eqtr4d 2781	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝑉)‘𝑤) = (◡𝐹 “ { 0 }))
114	113	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑤) = (◡𝐹 “ { 0 }))
115	114	ineq2d 4200	. . . . . . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0g‘𝑉) = (0g‘𝑉)
117	23, 52, 116	lbsdiflsp0 33671	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑉 ∈ LVec ∧ 𝑏 ∈ (LBasis‘𝑉) ∧ 𝑤 ⊆ 𝑏) → (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∩ ((LSpan‘𝑉)‘𝑤)) = {(0g‘𝑉)})
118	117	ad5ant145 1371	. . . . . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . 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 3969	. . . . . . . . . . . . . . . . 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 33389	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 ∈ LMod ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → (0g‘𝑉) ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
125	42, 66, 124	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (0g‘𝑉) ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
126		fvexd 6896	. . . . . . . . . . . . . . . . . . . 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 6901	. . . . . . . . . . . . . . . . . . . . 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 19210	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 ∈ (𝑉 GrpHom 𝑈) → (𝐹‘(0g‘𝑉)) = 0 )
129	62, 128	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (𝐹‘(0g‘𝑉)) = 0 )
130	129	ad4antlr 733	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹‘(0g‘𝑉)) = 0 )
131	127, 130	eqtrd 2771	. . . . . . . . . . . . . . . . . . . 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 4620	. . . . . . . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . 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 7054	. . . . . . . . . . . . . . . . . 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 4790	. . . . . . . . . . . . . . . . 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 3981	. . . . . . . . . . . . . . . 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 20829	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑉 ∈ LMod → 𝑉 ∈ Grp)
139		grpmnd 18928	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑉 ∈ Grp → 𝑉 ∈ Mnd)
140	42, 138, 139	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑉 ∈ Mnd)
141	55, 48, 116	ress0g 18745	. . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . 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 4618	. . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
146		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
147	145, 72, 146, 1	kerf1ghm 19235	. . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . 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 17264	. . . . . . . . . . . . . . . 16 ⊢ (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ⊆ (Base‘𝑉) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) = (Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))))
152	77, 151	syl 17	. . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . 15 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (Base‘𝑈) = (Base‘𝑈))
154	150, 152, 153	f1eq123d 6815	. . . . . . . . . . . . . 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 257	. . . . . . . . . . . . 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 6785	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 6834	. . . . . . . . . . . 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 17	. . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . . . . . . 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 7055	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 Fn (Base‘𝑉) → (𝑢 ∈ (◡𝐹 “ { 0 }) ↔ (𝑢 ∈ (Base‘𝑉) ∧ (𝐹‘𝑢) = 0 )))
166	165	simplbda 499	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 Fn (Base‘𝑉) ∧ 𝑢 ∈ (◡𝐹 “ { 0 })) → (𝐹‘𝑢) = 0 )
167	161, 164, 166	syl2anc 584	. . . . . . . . . . . . . . . . . . . 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 7425	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . 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 20946	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑉 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑉) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ ((LSpan‘𝑉)‘𝑏))
173	42, 65, 18, 172	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . 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 21042	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 ∈ (LBasis‘𝑉) → ((LSpan‘𝑉)‘𝑏) = (Base‘𝑉))
175	21, 174	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑏) = (Base‘𝑉))
176	173, 175	sseqtrd 4000	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ (Base‘𝑉))
177	176	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . 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 3964	. . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . . . . 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 3964	. . . . . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (+g‘𝑉) = (+g‘𝑉)
185		eqid 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (+g‘𝑈) = (+g‘𝑈)
186	48, 184, 185	ghmlin 19209	. . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . 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 33664	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec)
190	189	lvecgrpd 21071	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ Grp)
191	190	ad9antr 742	. . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . 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 7080	. . . . . . . . . . . . . . . . . . . 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 18957	. . . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . 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 7231	. . . . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ⊆ (◡𝐹 “ { 0 }))
202	105	ad4antlr 733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (◡𝐹 “ { 0 }) ⊆ (Base‘𝑉))
203	201, 202	sstrd 3974	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ⊆ (Base‘𝑉))
204		eqid 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (LSSum‘𝑉) = (LSSum‘𝑉)
205	48, 52, 204	lsmsp2 21050	. . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . . 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 19626	. . . . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . . 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 3205	. . . . . . . . . . . . . . 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 6944	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn (Base‘𝑉) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ (Base‘𝑉)(𝐹‘𝑥) = 𝑦))
216	215	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn (Base‘𝑉) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ (Base‘𝑉)(𝐹‘𝑥) = 𝑦)
217	75, 216	sylan 580	. . . . . . . . . . . . . . 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 3149	. . . . . . . . . . . . . 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 3985	. . . . . . . . . . . . 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 2785	. . . . . . . . . . . 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 6820	. . . . . . . . . . 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 232	. . . . . . . . . 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 584	. . . . . . . . . . . 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 17	. . . . . . . . . . 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 6718	. . . . . . . . . . . 12 ⊢ (𝐹:(Base‘𝑉)⟶(Base‘𝑈) → ran 𝐹 ⊆ (Base‘𝑈))
226	29, 72	ressbas2 17264	. . . . . . . . . . . 12 ⊢ (ran 𝐹 ⊆ (Base‘𝑈) → ran 𝐹 = (Base‘𝐼))
227	32, 73, 225, 226	4syl 19	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ran 𝐹 = (Base‘𝐼))
228	150, 224, 227	f1oeq123d 6817	. . . . . . . . . 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 232	. . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ (Base‘𝐼) = (Base‘𝐼)
231	145, 230	islmim 21025	. . . . . . . . 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 583	. . . . . . . 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 20947	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ LMod ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → (𝑏 ∖ 𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
234	42, 50, 233	syl2anc 584	. . . . . . . . . 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 21796	. . . . . . . . . . 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 477	. . . . . . . . . 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 1375	. . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (LSpan‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (LSpan‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
239	55, 52, 238, 51	lsslsp 20977	. . . . . . . . . . . 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 2742	. . . . . . . . . . 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 1373	. . . . . . . . . 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 2773	. . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ (LBasis‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (LBasis‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
244	145, 243, 238	islbs4 21797	. . . . . . . . 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 583	. . . . . . . 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 2736	. . . . . . . . 9 ⊢ (LBasis‘𝐼) = (LBasis‘𝐼)
247	243, 246	lmimlbs 21801	. . . . . . . 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 584	. . . . . . 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 33645	. . . . . . 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 584	. . . . . 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 9033	. . . . . . . 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 14371	. . . . . . . 8 ⊢ (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤)) ≈ (𝑏 ∖ 𝑤) → (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤))) = (♯‘(𝑏 ∖ 𝑤)))
253	251, 252	syl 17	. . . . . . 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 1373	. . . . . 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 2771	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (dim‘𝐼) = (♯‘(𝑏 ∖ 𝑤)))
256	28, 255	oveq12d 7428	. . . 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 2781	. . 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 21798	. . . . . 6 ⊢ (LBasis‘𝐾) ⊆ (LIndS‘𝐾)
259	258, 93	sselid 3961	. . . . 5 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LIndS‘𝐾))
260	51, 2	lsslinds 21796	. . . . . 6 ⊢ ((𝑉 ∈ LMod ∧ (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (◡𝐹 “ { 0 })) → (𝑤 ∈ (LIndS‘𝐾) ↔ 𝑤 ∈ (LIndS‘𝑉)))
261	260	biimpa 476	. . . . 5 ⊢ (((𝑉 ∈ LMod ∧ (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (◡𝐹 “ { 0 })) ∧ 𝑤 ∈ (LIndS‘𝐾)) → 𝑤 ∈ (LIndS‘𝑉))
262	98, 101, 109, 259, 261	syl31anc 1375	. . . 4 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LIndS‘𝑉))
263	23	islinds4 21800	. . . . 5 ⊢ (𝑉 ∈ LVec → (𝑤 ∈ (LIndS‘𝑉) ↔ ∃𝑏 ∈ (LBasis‘𝑉)𝑤 ⊆ 𝑏))
264	263	ad2antrr 726	. . . 4 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (𝑤 ∈ (LIndS‘𝑉) ↔ ∃𝑏 ∈ (LBasis‘𝑉)𝑤 ⊆ 𝑏))
265	262, 264	mpbid 232	. . 3 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ∃𝑏 ∈ (LBasis‘𝑉)𝑤 ⊆ 𝑏)
266	257, 265	r19.29a 3149	. 2 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))
267	8, 266	exlimddv 1935	1 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2933  ∃wrex 3061  Vcvv 3464   ∖ cdif 3928   ∪ cun 3929   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  {csn 4606   class class class wbr 5124  ^◡ccnv 5658  ran crn 5660   ↾ cres 5661   “ cima 5662   Fn wfn 6531  ⟶wf 6532  –1-1→wf1 6533  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410   ≈ cen 8961   +_𝑒 cxad 13131  ♯chash 14353  Basecbs 17233   ↾_s cress 17256  +_gcplusg 17276  0_gc0g 17458  Mndcmnd 18717  Grpcgrp 18921  SubGrpcsubg 19108   GrpHom cghm 19200  LSSumclsm 19620  LModclmod 20822  LSubSpclss 20893  LSpanclspn 20933   LMHom clmhm 20982   LMIso clmim 20983  LBasisclbs 21037  LVecclvec 21065  LIndSclinds 21770  dimcldim 33643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-reg 9611  ax-inf2 9660  ax-ac2 10482  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  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 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7676  df-rpss 7722  df-om 7867  df-1st 7993  df-2nd 7994  df-supp 8165  df-tpos 8230  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-er 8724  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9379  df-sup 9459  df-oi 9529  df-r1 9783  df-rank 9784  df-dju 9920  df-card 9958  df-acn 9961  df-ac 10135  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-xnn0 12580  df-z 12594  df-dec 12714  df-uz 12858  df-xadd 13134  df-fz 13530  df-fzo 13677  df-seq 14025  df-hash 14354  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  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-ocomp 17297  df-ds 17298  df-hom 17300  df-cco 17301  df-0g 17460  df-gsum 17461  df-prds 17466  df-pws 17468  df-mre 17603  df-mrc 17604  df-mri 17605  df-acs 17606  df-proset 18311  df-drs 18312  df-poset 18330  df-ipo 18543  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-mhm 18766  df-submnd 18767  df-grp 18924  df-minusg 18925  df-sbg 18926  df-mulg 19056  df-subg 19111  df-ghm 19201  df-cntz 19305  df-lsm 19622  df-cmn 19768  df-abl 19769  df-mgp 20106  df-rng 20118  df-ur 20147  df-ring 20200  df-oppr 20302  df-dvdsr 20322  df-unit 20323  df-invr 20353  df-nzr 20478  df-subrg 20535  df-drng 20696  df-lmod 20824  df-lss 20894  df-lsp 20934  df-lmhm 20985  df-lmim 20986  df-lbs 21038  df-lvec 21066  df-sra 21136  df-rgmod 21137  df-dsmm 21697  df-frlm 21712  df-uvc 21748  df-lindf 21771  df-linds 21772  df-dim 33644

This theorem is referenced by:  qusdimsum  33673  lvecendof1f1o  33678