dimkerim - Mathbox for Thierry Arnoux

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Theorem dimkerim 31708

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 = (0_g‘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 = (0_g‘𝑈)
2		dimkerim.k	. . . . 5 ⊢ 𝐾 = (𝑉 ↾_s (^◡𝐹 “ { 0 }))
3	1, 2	kerlmhm 31703	. . . 4 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐾 ∈ LVec)
4		eqid 2738	. . . . 5 ⊢ (LBasis‘𝐾) = (LBasis‘𝐾)
5	4	lbsex 20427	. . . 4 ⊢ (𝐾 ∈ LVec → (LBasis‘𝐾) ≠ ∅)
6	3, 5	syl 17	. . 3 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (LBasis‘𝐾) ≠ ∅)
7		n0 4280	. . 3 ⊢ ((LBasis‘𝐾) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (LBasis‘𝐾))
8	6, 7	sylib 217	. 2 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → ∃𝑤 𝑤 ∈ (LBasis‘𝐾))
9		simpllr 773	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ∈ (LBasis‘𝐾))
10		vex 3436	. . . . . . 7 ⊢ 𝑏 ∈ V
11	10	difexi 5252	. . . . . 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 4405	. . . . . 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 14101	. . . . 5 ⊢ ((𝑤 ∈ (LBasis‘𝐾) ∧ (𝑏 ∖ 𝑤) ∈ V ∧ (𝑤 ∩ (𝑏 ∖ 𝑤)) = ∅) → (♯‘(𝑤 ∪ (𝑏 ∖ 𝑤))) = ((♯‘𝑤) +_𝑒 (♯‘(𝑏 ∖ 𝑤))))
16	9, 12, 14, 15	syl3anc 1370	. . . 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 780	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑉 ∈ LVec)
18		simpr 485	. . . . . . 7 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ⊆ 𝑏)
19		undif 4415	. . . . . . 7 ⊢ (𝑤 ⊆ 𝑏 ↔ (𝑤 ∪ (𝑏 ∖ 𝑤)) = 𝑏)
20	18, 19	sylib 217	. . . . . 6 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑤 ∪ (𝑏 ∖ 𝑤)) = 𝑏)
21		simplr 766	. . . . . 6 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑏 ∈ (LBasis‘𝑉))
22	20, 21	eqeltrd 2839	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑤 ∪ (𝑏 ∖ 𝑤)) ∈ (LBasis‘𝑉))
23		eqid 2738	. . . . . 6 ⊢ (LBasis‘𝑉) = (LBasis‘𝑉)
24	23	dimval 31686	. . . . 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 727	. . . . . 6 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐾 ∈ LVec)
27	4	dimval 31686	. . . . . 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 31704	. . . . . . . 8 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐼 ∈ LVec)
31	30	ad3antrrr 727	. . . . . . 7 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐼 ∈ LVec)
32		simp-4r 781	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹 ∈ (𝑉 LMHom 𝑈))
33		lmhmlmod2 20294	. . . . . . . . . . 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 20312	. . . . . . . . . . 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 5602	. . . . . . . . . . 11 ⊢ (𝐹 “ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
38		imassrn 5980	. . . . . . . . . . . 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 3970	. . . . . . . . . 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 20368	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ LVec → 𝑉 ∈ LMod)
42	41	ad4antr 729	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑉 ∈ LMod)
43	23	lbslinds 21040	. . . . . . . . . . . . . . 15 ⊢ (LBasis‘𝑉) ⊆ (LIndS‘𝑉)
44	43, 21	sselid 3919	. . . . . . . . . . . . . 14 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑏 ∈ (LIndS‘𝑉))
45		difssd 4067	. . . . . . . . . . . . . 14 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ⊆ 𝑏)
46		lindsss 21031	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ LMod ∧ 𝑏 ∈ (LIndS‘𝑉) ∧ (𝑏 ∖ 𝑤) ⊆ 𝑏) → (𝑏 ∖ 𝑤) ∈ (LIndS‘𝑉))
47	42, 44, 45, 46	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ∈ (LIndS‘𝑉))
48		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑉) = (Base‘𝑉)
49	48	linds1 21017	. . . . . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝑉) = (LSubSp‘𝑉)
52		eqid 2738	. . . . . . . . . . . . 13 ⊢ (LSpan‘𝑉) = (LSpan‘𝑉)
53	48, 51, 52	lspcl 20238	. . . . . . . . . . . 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 2738	. . . . . . . . . . . 12 ⊢ (𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = (𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
56	51, 55	reslmhm 20314	. . . . . . . . . . 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 2738	. . . . . . . . . . . 12 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
59	29, 58	reslmhm2b 20316	. . . . . . . . . . 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 477	. . . . . . . . . 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 1372	. . . . . . . . 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 20293	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
63	62	ad4antlr 730	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
64	48, 23	lbsss 20339	. . . . . . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . . . . . 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 20219	. . . . . . . . . . . . . . . . 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 18850	. . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Base‘𝑈) = (Base‘𝑈)
73	48, 72	lmhmf 20296	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
74	73	ad4antlr 730	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
75	74	ffnd 6601	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹 Fn (Base‘𝑉))
76	48, 52	lspssv 20245	. . . . . . . . . . . . . . . . . . . . . . . . 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		fnssres 6555	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹 Fn (Base‘𝑉) ∧ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ⊆ (Base‘𝑉)) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) Fn ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
79	75, 77, 78	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 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)))
80		fniniseg 6937	. . . . . . . . . . . . . . . . . . . . . . . 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 )))
81	80	biimpa 477	. . . . . . . . . . . . . . . . . . . . . . 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 ))
82	79, 81	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 ))
83	82	simpld 495	. . . . . . . . . . . . . . . . . . . . 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)))
84	75	adantr 481	. . . . . . . . . . . . . . . . . . . . . 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))
85	77	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 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))
86	85, 83	sseldd 3922	. . . . . . . . . . . . . . . . . . . . . 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))
87	83	fvresd 6794	. . . . . . . . . . . . . . . . . . . . . . 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))
88	82	simprd 496	. . . . . . . . . . . . . . . . . . . . . . 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 )
89	87, 88	eqtr3d 2780	. . . . . . . . . . . . . . . . . . . . . 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 )
90		fniniseg 6937	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 Fn (Base‘𝑉) → (𝑥 ∈ (^◡𝐹 “ { 0 }) ↔ (𝑥 ∈ (Base‘𝑉) ∧ (𝐹‘𝑥) = 0 )))
91	90	biimpar 478	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Base‘𝑉) ∧ (𝑥 ∈ (Base‘𝑉) ∧ (𝐹‘𝑥) = 0 )) → 𝑥 ∈ (^◡𝐹 “ { 0 }))
92	84, 86, 89, 91	syl12anc 834	. . . . . . . . . . . . . . . . . . . . 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 }))
93	83, 92	elind 4128	. . . . . . . . . . . . . . . . . . . 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 })))
94		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LBasis‘𝐾))
95		eqid 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Base‘𝐾) = (Base‘𝐾)
96		eqid 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾)
97	95, 4, 96	lbssp 20341	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑤 ∈ (LBasis‘𝐾) → ((LSpan‘𝐾)‘𝑤) = (Base‘𝐾))
98	94, 97	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝐾)‘𝑤) = (Base‘𝐾))
99	41	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑉 ∈ LMod)
100		eqid 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (^◡𝐹 “ { 0 }) = (^◡𝐹 “ { 0 })
101	100, 1, 51	lmhmkerlss 20313	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (^◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉))
102	101	ad2antlr 724	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (^◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉))
103	95, 4	lbsss 20339	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 ∈ (LBasis‘𝐾) → 𝑤 ⊆ (Base‘𝐾))
104	94, 103	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ⊆ (Base‘𝐾))
105		cnvimass 5989	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (^◡𝐹 “ { 0 }) ⊆ dom 𝐹
106	105, 73	fssdm 6620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (^◡𝐹 “ { 0 }) ⊆ (Base‘𝑉))
107	2, 48	ressbas2 16949	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((^◡𝐹 “ { 0 }) ⊆ (Base‘𝑉) → (^◡𝐹 “ { 0 }) = (Base‘𝐾))
108	106, 107	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (^◡𝐹 “ { 0 }) = (Base‘𝐾))
109	108	ad2antlr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (^◡𝐹 “ { 0 }) = (Base‘𝐾))
110	104, 109	sseqtrrd 3962	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ⊆ (^◡𝐹 “ { 0 }))
111	2, 52, 96, 51	lsslsp 20277	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑉 ∈ LMod ∧ (^◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (^◡𝐹 “ { 0 })) → ((LSpan‘𝑉)‘𝑤) = ((LSpan‘𝐾)‘𝑤))
112	99, 102, 110, 111	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝑉)‘𝑤) = ((LSpan‘𝐾)‘𝑤))
113	98, 112, 109	3eqtr4d 2788	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝑉)‘𝑤) = (^◡𝐹 “ { 0 }))
114	113	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑤) = (^◡𝐹 “ { 0 }))
115	114	ineq2d 4146	. . . . . . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0_g‘𝑉) = (0_g‘𝑉)
117	23, 52, 116	lbsdiflsp0 31707	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑉 ∈ LVec ∧ 𝑏 ∈ (LBasis‘𝑉) ∧ 𝑤 ⊆ 𝑏) → (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∩ ((LSpan‘𝑉)‘𝑤)) = {(0_g‘𝑉)})
118	117	ad5ant145 1368	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∩ ((LSpan‘𝑉)‘𝑤)) = {(0_g‘𝑉)})
119	115, 118	eqtr3d 2780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∩ (^◡𝐹 “ { 0 })) = {(0_g‘𝑉)})
120	119	adantr 481	. . . . . . . . . . . . . . . . . . . 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 })) = {(0_g‘𝑉)})
121	93, 120	eleqtrd 2841	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑥 ∈ (^◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 })) → 𝑥 ∈ {(0_g‘𝑉)})
122	121	ex 413	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑥 ∈ (^◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) → 𝑥 ∈ {(0_g‘𝑉)}))
123	122	ssrdv 3927	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (^◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) ⊆ {(0_g‘𝑉)})
124	116, 48, 52	0ellsp 31565	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 ∈ LMod ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → (0_g‘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) → (0_g‘𝑉) ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
126		fvexd 6789	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0_g‘𝑉)) ∈ V)
127	125	fvresd 6794	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0_g‘𝑉)) = (𝐹‘(0_g‘𝑉)))
128	116, 1	ghmid 18840	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 ∈ (𝑉 GrpHom 𝑈) → (𝐹‘(0_g‘𝑉)) = 0 )
129	62, 128	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (𝐹‘(0_g‘𝑉)) = 0 )
130	129	ad4antlr 730	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹‘(0_g‘𝑉)) = 0 )
131	127, 130	eqtrd 2778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0_g‘𝑉)) = 0 )
132		elsng 4575	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0_g‘𝑉)) ∈ V → (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0_g‘𝑉)) ∈ { 0 } ↔ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0_g‘𝑉)) = 0 ))
133	132	biimpar 478	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0_g‘𝑉)) ∈ V ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0_g‘𝑉)) = 0 ) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0_g‘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)))‘(0_g‘𝑉)) ∈ { 0 })
135	79, 125, 134	elpreimad 6936	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (0_g‘𝑉) ∈ (^◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }))
136	135	snssd 4742	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → {(0_g‘𝑉)} ⊆ (^◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }))
137	123, 136	eqssd 3938	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (^◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) = {(0_g‘𝑉)})
138		lmodgrp 20130	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑉 ∈ LMod → 𝑉 ∈ Grp)
139		grpmnd 18584	. . . . . . . . . . . . . . . . . . 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 18413	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑉 ∈ Mnd ∧ (0_g‘𝑉) ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∧ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ⊆ (Base‘𝑉)) → (0_g‘𝑉) = (0_g‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))))
142	140, 125, 77, 141	syl3anc 1370	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (0_g‘𝑉) = (0_g‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))))
143	142	sneqd 4573	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → {(0_g‘𝑉)} = {(0_g‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))})
144	137, 143	eqtrd 2778	. . . . . . . . . . . . . . 15 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (^◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) = {(0_g‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))})
145		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (Base‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
146		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (0_g‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
147	145, 72, 146, 1	kerf1ghm 19987	. . . . . . . . . . . . . . . 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 }) = {(0_g‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))}))
148	147	biimpar 478	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) GrpHom 𝑈) ∧ (^◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) = {(0_g‘(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 2739	. . . . . . . . . . . . . . 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 16949	. . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . 15 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (Base‘𝑈) = (Base‘𝑈))
154	150, 152, 153	f1eq123d 6708	. . . . . . . . . . . . . 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 256	. . . . . . . . . . . . 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 6677	. . . . . . . . . . . . 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 6727	. . . . . . . . . . . 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 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) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+_g‘𝑉)𝑣)) → (𝐹‘𝑥) = 𝑦)
161	75	ad6antr 733	. . . . . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . . . . . . . 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 6937	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 Fn (Base‘𝑉) → (𝑢 ∈ (^◡𝐹 “ { 0 }) ↔ (𝑢 ∈ (Base‘𝑉) ∧ (𝐹‘𝑢) = 0 )))
166	165	simplbda 500	. . . . . . . . . . . . . . . . . . . . 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 7290	. . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . 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 6778	. . . . . . . . . . . . . . . . . . . 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 733	. . . . . . . . . . . . . . . . . . . . 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 20246	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑉 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑉) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ ((LSpan‘𝑉)‘𝑏))
173	42, 65, 18, 172	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . 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 20341	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3961	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ (Base‘𝑉))
177	176	ad3antrrr 727	. . . . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (+_g‘𝑉) = (+_g‘𝑉)
185		eqid 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
186	48, 184, 185	ghmlin 18839	. . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . . . 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 31702	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec)
190		lveclmod 20368	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑈 ∈ LVec → 𝑈 ∈ LMod)
191		lmodgrp 20130	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Grp)
192	189, 190, 191	3syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ Grp)
193	192	ad9antr 739	. . . . . . . . . . . . . . . . . . . 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)
194	74	ad6antr 733	. . . . . . . . . . . . . . . . . . . . 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))
195	194, 183	ffvelrnd 6962	. . . . . . . . . . . . . . . . . . . 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))
196	72, 185, 1	grplid 18609	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑈 ∈ Grp ∧ (𝐹‘𝑣) ∈ (Base‘𝑈)) → ( 0 (+_g‘𝑈)(𝐹‘𝑣)) = (𝐹‘𝑣))
197	193, 195, 196	syl2anc 584	. . . . . . . . . . . . . . . . . . 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))
198	168, 188, 197	3eqtr3d 2786	. . . . . . . . . . . . . . . . . 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))
199	160, 198	eqtr3d 2780	. . . . . . . . . . . . . . . . 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))
200	161, 183, 182	fnfvimad 7110	. . . . . . . . . . . . . . . . 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))))
201	199, 200	eqeltrd 2839	. . . . . . . . . . . . . . . 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))))
202		simp-7l 786	. . . . . . . . . . . . . . . . 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)
203		simplr 766	. . . . . . . . . . . . . . . . . 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))
204	110	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ⊆ (^◡𝐹 “ { 0 }))
205	106	ad4antlr 730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (^◡𝐹 “ { 0 }) ⊆ (Base‘𝑉))
206	204, 205	sstrd 3931	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ⊆ (Base‘𝑉))
207		eqid 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (LSSum‘𝑉) = (LSSum‘𝑉)
208	48, 52, 207	lsmsp2 20349	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑉 ∈ LMod ∧ 𝑤 ⊆ (Base‘𝑉) ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → (((LSpan‘𝑉)‘𝑤)(LSSum‘𝑉)((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = ((LSpan‘𝑉)‘(𝑤 ∪ (𝑏 ∖ 𝑤))))
209	42, 206, 66, 208	syl3anc 1370	. . . . . . . . . . . . . . . . . . . 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))))
210	20	fveq2d 6778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘(𝑤 ∪ (𝑏 ∖ 𝑤))) = ((LSpan‘𝑉)‘𝑏))
211	209, 210, 175	3eqtrrd 2783	. . . . . . . . . . . . . . . . . . 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))))
212	211	ad3antrrr 727	. . . . . . . . . . . . . . . . . 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))))
213	203, 212	eleqtrd 2841	. . . . . . . . . . . . . . . . 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))))
214	48, 184, 207	lsmelvalx 19245	. . . . . . . . . . . . . . . . . 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)))
215	214	biimpa 477	. . . . . . . . . . . . . . . . 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))
216	202, 177, 180, 213, 215	syl31anc 1372	. . . . . . . . . . . . . . . 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))
217	201, 216	r19.29vva 3266	. . . . . . . . . . . . . . 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))))
218		fvelrnb 6830	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn (Base‘𝑉) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ (Base‘𝑉)(𝐹‘𝑥) = 𝑦))
219	218	biimpa 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn (Base‘𝑉) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ (Base‘𝑉)(𝐹‘𝑥) = 𝑦)
220	75, 219	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)
221	217, 220	r19.29a 3218	. . . . . . . . . . . . . 14 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ (𝐹 “ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
222	39, 221	eqelssd 3942	. . . . . . . . . . . . 13 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹 “ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = ran 𝐹)
223	37, 222	eqtr3id 2792	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = ran 𝐹)
224	223	f1oeq3d 6713	. . . . . . . . . . 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))
225	159, 224	mpbid 231	. . . . . . . . . 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)
226	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))
227	226, 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)))))
228		frn 6607	. . . . . . . . . . . . 13 ⊢ (𝐹:(Base‘𝑉)⟶(Base‘𝑈) → ran 𝐹 ⊆ (Base‘𝑈))
229	29, 72	ressbas2 16949	. . . . . . . . . . . . 13 ⊢ (ran 𝐹 ⊆ (Base‘𝑈) → ran 𝐹 = (Base‘𝐼))
230	73, 228, 229	3syl 18	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → ran 𝐹 = (Base‘𝐼))
231	32, 230	syl 17	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ran 𝐹 = (Base‘𝐼))
232	150, 227, 231	f1oeq123d 6710	. . . . . . . . . 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)))
233	225, 232	mpbid 231	. . . . . . . . 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))
234		eqid 2738	. . . . . . . . . 10 ⊢ (Base‘𝐼) = (Base‘𝐼)
235	145, 234	islmim 20324	. . . . . . . . 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)))
236	61, 233, 235	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))
237	48, 52	lspssid 20247	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ LMod ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → (𝑏 ∖ 𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
238	42, 50, 237	syl2anc 584	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
239	51, 55	lsslinds 21038	. . . . . . . . . . 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)))
240	239	biimpar 478	. . . . . . . . . 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)))))
241	42, 67, 238, 47, 240	syl31anc 1372	. . . . . . . . 9 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ∈ (LIndS‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))))
242		eqid 2738	. . . . . . . . . . . 12 ⊢ (LSpan‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (LSpan‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
243	55, 52, 242, 51	lsslsp 20277	. . . . . . . . . . 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)))
244	42, 54, 238, 243	syl3anc 1370	. . . . . . . . . 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)))
245	244, 227	eqtr3d 2780	. . . . . . . . 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)))))
246		eqid 2738	. . . . . . . . . 10 ⊢ (LBasis‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (LBasis‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
247	145, 246, 242	islbs4 21039	. . . . . . . . 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))))))
248	241, 245, 247	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)))))
249		eqid 2738	. . . . . . . . 9 ⊢ (LBasis‘𝐼) = (LBasis‘𝐼)
250	246, 249	lmimlbs 21043	. . . . . . . 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))
251	236, 248, 250	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))
252	249	dimval 31686	. . . . . . 7 ⊢ ((𝐼 ∈ LVec ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤)) ∈ (LBasis‘𝐼)) → (dim‘𝐼) = (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤))))
253	31, 251, 252	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))))
254		f1imaeng 8800	. . . . . . . 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))
255		hasheni 14062	. . . . . . . 8 ⊢ (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤)) ≈ (𝑏 ∖ 𝑤) → (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤))) = (♯‘(𝑏 ∖ 𝑤)))
256	254, 255	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)))
257	157, 238, 47, 256	syl3anc 1370	. . . . . 6 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (♯‘((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ (𝑏 ∖ 𝑤))) = (♯‘(𝑏 ∖ 𝑤)))
258	253, 257	eqtrd 2778	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (dim‘𝐼) = (♯‘(𝑏 ∖ 𝑤)))
259	28, 258	oveq12d 7293	. . . 4 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((dim‘𝐾) +_𝑒 (dim‘𝐼)) = ((♯‘𝑤) +_𝑒 (♯‘(𝑏 ∖ 𝑤))))
260	16, 25, 259	3eqtr4d 2788	. . 3 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (dim‘𝑉) = ((dim‘𝐾) +_𝑒 (dim‘𝐼)))
261	4	lbslinds 21040	. . . . . 6 ⊢ (LBasis‘𝐾) ⊆ (LIndS‘𝐾)
262	261, 94	sselid 3919	. . . . 5 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LIndS‘𝐾))
263	51, 2	lsslinds 21038	. . . . . 6 ⊢ ((𝑉 ∈ LMod ∧ (^◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (^◡𝐹 “ { 0 })) → (𝑤 ∈ (LIndS‘𝐾) ↔ 𝑤 ∈ (LIndS‘𝑉)))
264	263	biimpa 477	. . . . 5 ⊢ (((𝑉 ∈ LMod ∧ (^◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (^◡𝐹 “ { 0 })) ∧ 𝑤 ∈ (LIndS‘𝐾)) → 𝑤 ∈ (LIndS‘𝑉))
265	99, 102, 110, 262, 264	syl31anc 1372	. . . 4 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LIndS‘𝑉))
266	23	islinds4 21042	. . . . 5 ⊢ (𝑉 ∈ LVec → (𝑤 ∈ (LIndS‘𝑉) ↔ ∃𝑏 ∈ (LBasis‘𝑉)𝑤 ⊆ 𝑏))
267	266	ad2antrr 723	. . . 4 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (𝑤 ∈ (LIndS‘𝑉) ↔ ∃𝑏 ∈ (LBasis‘𝑉)𝑤 ⊆ 𝑏))
268	265, 267	mpbid 231	. . 3 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ∃𝑏 ∈ (LBasis‘𝑉)𝑤 ⊆ 𝑏)
269	260, 268	r19.29a 3218	. 2 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (dim‘𝑉) = ((dim‘𝐾) +_𝑒 (dim‘𝐼)))
270	8, 269	exlimddv 1938	1 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (dim‘𝑉) = ((dim‘𝐾) +_𝑒 (dim‘𝐼)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 Vcvv 3432 ∖ cdif 3884 ∪ cun 3885 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 {csn 4561 class class class wbr 5074 ^◡ccnv 5588 ran crn 5590 ↾ cres 5591 “ cima 5592 Fn wfn 6428 ⟶wf 6429 –1-1→wf1 6430 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 ≈ cen 8730 +_𝑒 cxad 12846 ♯chash 14044 Basecbs 16912 ↾_s cress 16941 +_gcplusg 16962 0_gc0g 17150 Mndcmnd 18385 Grpcgrp 18577 SubGrpcsubg 18749 GrpHom cghm 18831 LSSumclsm 19239 LModclmod 20123 LSubSpclss 20193 LSpanclspn 20233 LMHom clmhm 20281 LMIso clmim 20282 LBasisclbs 20336 LVecclvec 20364 LIndSclinds 21012 dimcldim 31684

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-reg 9351 ax-inf2 9399 ax-ac2 10219 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-rpss 7576 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-oi 9269 df-r1 9522 df-rank 9523 df-dju 9659 df-card 9697 df-acn 9700 df-ac 9872 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12438 df-uz 12583 df-xadd 12849 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ocomp 16983 df-ds 16984 df-hom 16986 df-cco 16987 df-0g 17152 df-gsum 17153 df-prds 17158 df-pws 17160 df-mre 17295 df-mrc 17296 df-mri 17297 df-acs 17298 df-proset 18013 df-drs 18014 df-poset 18031 df-ipo 18246 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-ghm 18832 df-cntz 18923 df-lsm 19241 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-drng 19993 df-subrg 20022 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lmhm 20284 df-lmim 20285 df-lbs 20337 df-lvec 20365 df-sra 20434 df-rgmod 20435 df-nzr 20529 df-dsmm 20939 df-frlm 20954 df-uvc 20990 df-lindf 21013 df-linds 21014 df-dim 31685

This theorem is referenced by: qusdimsum 31709

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