dimkerim - Mathbox for Thierry Arnoux

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Theorem dimkerim 33453

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 33446	. . . 4 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐾 ∈ LVec)
4		eqid 2725	. . . . 5 ⊢ (LBasis‘𝐾) = (LBasis‘𝐾)
5	4	lbsex 21065	. . . 4 ⊢ (𝐾 ∈ LVec → (LBasis‘𝐾) ≠ ∅)
6	3, 5	syl 17	. . 3 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (LBasis‘𝐾) ≠ ∅)
7		n0 4346	. . 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 774	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ∈ (LBasis‘𝐾))
10		vex 3465	. . . . . . 7 ⊢ 𝑏 ∈ V
11	10	difexi 5331	. . . . . 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 4473	. . . . . 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 14381	. . . . 5 ⊢ ((𝑤 ∈ (LBasis‘𝐾) ∧ (𝑏 ∖ 𝑤) ∈ V ∧ (𝑤 ∩ (𝑏 ∖ 𝑤)) = ∅) → (♯‘(𝑤 ∪ (𝑏 ∖ 𝑤))) = ((♯‘𝑤) +_𝑒 (♯‘(𝑏 ∖ 𝑤))))
16	9, 12, 14, 15	syl3anc 1368	. . . 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 781	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑉 ∈ LVec)
18		simpr 483	. . . . . . 7 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ⊆ 𝑏)
19		undif 4483	. . . . . . 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 767	. . . . . 6 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑏 ∈ (LBasis‘𝑉))
22	20, 21	eqeltrd 2825	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑤 ∪ (𝑏 ∖ 𝑤)) ∈ (LBasis‘𝑉))
23		eqid 2725	. . . . . 6 ⊢ (LBasis‘𝑉) = (LBasis‘𝑉)
24	23	dimval 33426	. . . . 5 ⊢ ((𝑉 ∈ LVec ∧ (𝑤 ∪ (𝑏 ∖ 𝑤)) ∈ (LBasis‘𝑉)) → (dim‘𝑉) = (♯‘(𝑤 ∪ (𝑏 ∖ 𝑤))))
25	17, 22, 24	syl2anc 582	. . . 4 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (dim‘𝑉) = (♯‘(𝑤 ∪ (𝑏 ∖ 𝑤))))
26	3	ad3antrrr 728	. . . . . 6 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐾 ∈ LVec)
27	4	dimval 33426	. . . . . 6 ⊢ ((𝐾 ∈ LVec ∧ 𝑤 ∈ (LBasis‘𝐾)) → (dim‘𝐾) = (♯‘𝑤))
28	26, 9, 27	syl2anc 582	. . . . 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 33447	. . . . . . . 8 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐼 ∈ LVec)
31	30	ad3antrrr 728	. . . . . . 7 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐼 ∈ LVec)
32		simp-4r 782	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹 ∈ (𝑉 LMHom 𝑈))
33		lmhmlmod2 20929	. . . . . . . . . . 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 20947	. . . . . . . . . . 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 5691	. . . . . . . . . . 11 ⊢ (𝐹 “ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
38		imassrn 6075	. . . . . . . . . . . 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 4026	. . . . . . . . . 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 21003	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ LVec → 𝑉 ∈ LMod)
42	41	ad4antr 730	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑉 ∈ LMod)
43	23	lbslinds 21784	. . . . . . . . . . . . . . 15 ⊢ (LBasis‘𝑉) ⊆ (LIndS‘𝑉)
44	43, 21	sselid 3974	. . . . . . . . . . . . . 14 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑏 ∈ (LIndS‘𝑉))
45		difssd 4129	. . . . . . . . . . . . . 14 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ⊆ 𝑏)
46		lindsss 21775	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ LMod ∧ 𝑏 ∈ (LIndS‘𝑉) ∧ (𝑏 ∖ 𝑤) ⊆ 𝑏) → (𝑏 ∖ 𝑤) ∈ (LIndS‘𝑉))
47	42, 44, 45, 46	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ∈ (LIndS‘𝑉))
48		eqid 2725	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑉) = (Base‘𝑉)
49	48	linds1 21761	. . . . . . . . . . . . 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 2725	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝑉) = (LSubSp‘𝑉)
52		eqid 2725	. . . . . . . . . . . . 13 ⊢ (LSpan‘𝑉) = (LSpan‘𝑉)
53	48, 51, 52	lspcl 20872	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ LMod ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉))
54	42, 50, 53	syl2anc 582	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉))
55		eqid 2725	. . . . . . . . . . . 12 ⊢ (𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = (𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
56	51, 55	reslmhm 20949	. . . . . . . . . . 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 582	. . . . . . . . . 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 2725	. . . . . . . . . . . 12 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
59	29, 58	reslmhm2b 20951	. . . . . . . . . . 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 475	. . . . . . . . . 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 1370	. . . . . . . . 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 20928	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
63	62	ad4antlr 731	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
64	48, 23	lbsss 20974	. . . . . . . . . . . . . . . . . . . 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 3987	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉))
67	42, 66, 53	syl2anc 582	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉))
68	51	lsssubg 20853	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑉 ∈ LMod ∧ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (LSubSp‘𝑉)) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (SubGrp‘𝑉))
69	42, 67, 68	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∈ (SubGrp‘𝑉))
70	55	resghm 19195	. . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . 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 2725	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Base‘𝑈) = (Base‘𝑈)
73	48, 72	lmhmf 20931	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
74	73	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
75	74	ffnd 6724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹 Fn (Base‘𝑉))
76	48, 52	lspssv 20879	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑉 ∈ LMod ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ⊆ (Base‘𝑉))
77	42, 66, 76	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . . 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 6680	. . . . . . . . . . . . . . . . . . . . . . 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 7068	. . . . . . . . . . . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . . . . . . . . . . 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 578	. . . . . . . . . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . 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 3977	. . . . . . . . . . . . . . . . . . . . . 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 6916	. . . . . . . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . . . . 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 7068	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 Fn (Base‘𝑉) → (𝑥 ∈ (^◡𝐹 “ { 0 }) ↔ (𝑥 ∈ (Base‘𝑉) ∧ (𝐹‘𝑥) = 0 )))
90	89	biimpar 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Base‘𝑉) ∧ (𝑥 ∈ (Base‘𝑉) ∧ (𝐹‘𝑥) = 0 )) → 𝑥 ∈ (^◡𝐹 “ { 0 }))
91	83, 85, 88, 90	syl12anc 835	. . . . . . . . . . . . . . . . . . . . 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 4192	. . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LBasis‘𝐾))
94		eqid 2725	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Base‘𝐾) = (Base‘𝐾)
95		eqid 2725	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾)
96	94, 4, 95	lbssp 20976	. . . . . . . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑉 ∈ LMod)
99		eqid 2725	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (^◡𝐹 “ { 0 }) = (^◡𝐹 “ { 0 })
100	99, 1, 51	lmhmkerlss 20948	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (^◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉))
101	100	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (^◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉))
102	94, 4	lbsss 20974	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6086	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (^◡𝐹 “ { 0 }) ⊆ dom 𝐹
105	104, 73	fssdm 6742	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (^◡𝐹 “ { 0 }) ⊆ (Base‘𝑉))
106	2, 48	ressbas2 17221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (^◡𝐹 “ { 0 }) = (Base‘𝐾))
109	103, 108	sseqtrrd 4018	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ⊆ (^◡𝐹 “ { 0 }))
110	2, 52, 95, 51	lsslsp 20911	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑉 ∈ LMod ∧ (^◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (^◡𝐹 “ { 0 })) → ((LSpan‘𝐾)‘𝑤) = ((LSpan‘𝑉)‘𝑤))
111	110	eqcomd 2731	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑉 ∈ LMod ∧ (^◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (^◡𝐹 “ { 0 })) → ((LSpan‘𝑉)‘𝑤) = ((LSpan‘𝐾)‘𝑤))
112	98, 101, 109, 111	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝑉)‘𝑤) = ((LSpan‘𝐾)‘𝑤))
113	97, 112, 108	3eqtr4d 2775	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝑉)‘𝑤) = (^◡𝐹 “ { 0 }))
114	113	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑤) = (^◡𝐹 “ { 0 }))
115	114	ineq2d 4210	. . . . . . . . . . . . . . . . . . . . . 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 2725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0_g‘𝑉) = (0_g‘𝑉)
117	23, 52, 116	lbsdiflsp0 33452	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑉 ∈ LVec ∧ 𝑏 ∈ (LBasis‘𝑉) ∧ 𝑤 ⊆ 𝑏) → (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∩ ((LSpan‘𝑉)‘𝑤)) = {(0_g‘𝑉)})
118	117	ad5ant145 1366	. . . . . . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . 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	92, 120	eleqtrd 2827	. . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . 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 3982	. . . . . . . . . . . . . . . . 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 33180	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 ∈ LMod ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → (0_g‘𝑉) ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
125	42, 66, 124	syl2anc 582	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (0_g‘𝑉) ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
126		fvexd 6911	. . . . . . . . . . . . . . . . . . . 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 6916	. . . . . . . . . . . . . . . . . . . . 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 19185	. . . . . . . . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹‘(0_g‘𝑉)) = 0 )
131	127, 130	eqtrd 2765	. . . . . . . . . . . . . . . . . . . 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 4644	. . . . . . . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . . 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	78, 125, 134	elpreimad 7067	. . . . . . . . . . . . . . . . . 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 4814	. . . . . . . . . . . . . . . . 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 3994	. . . . . . . . . . . . . . . 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 20762	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑉 ∈ LMod → 𝑉 ∈ Grp)
139		grpmnd 18905	. . . . . . . . . . . . . . . . . . 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 18725	. . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . 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 4642	. . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . 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 2725	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (Base‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
146		eqid 2725	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (0_g‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
147	145, 72, 146, 1	kerf1ghm 19210	. . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . 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 2726	. . . . . . . . . . . . . . 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 17221	. . . . . . . . . . . . . . . 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 2726	. . . . . . . . . . . . . . 15 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (Base‘𝑈) = (Base‘𝑈))
154	150, 152, 153	f1eq123d 6830	. . . . . . . . . . . . . 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 6799	. . . . . . . . . . . . 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 582	. . . . . . . . . . . 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 6849	. . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . . . . . . . . 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 7068	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 Fn (Base‘𝑉) → (𝑢 ∈ (^◡𝐹 “ { 0 }) ↔ (𝑢 ∈ (Base‘𝑉) ∧ (𝐹‘𝑢) = 0 )))
166	165	simplbda 498	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 Fn (Base‘𝑉) ∧ 𝑢 ∈ (^◡𝐹 “ { 0 })) → (𝐹‘𝑢) = 0 )
167	161, 164, 166	syl2anc 582	. . . . . . . . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . 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 6900	. . . . . . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . . . . . 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 20880	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑉 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑉) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ ((LSpan‘𝑉)‘𝑏))
173	42, 65, 18, 172	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . . . . 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 20976	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4017	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ (Base‘𝑉))
177	176	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . . . . 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 3977	. . . . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . . . . 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 3977	. . . . . . . . . . . . . . . . . . . . 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 2725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (+_g‘𝑉) = (+_g‘𝑉)
185		eqid 2725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
186	48, 184, 185	ghmlin 19184	. . . . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . 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 33445	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec)
190	189	lvecgrpd 21005	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ Grp)
191	190	ad9antr 740	. . . . . . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . . . . . 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 7094	. . . . . . . . . . . . . . . . . . . 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 18934	. . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . 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 7246	. . . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ⊆ (^◡𝐹 “ { 0 }))
202	105	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (^◡𝐹 “ { 0 }) ⊆ (Base‘𝑉))
203	201, 202	sstrd 3987	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ⊆ (Base‘𝑉))
204		eqid 2725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (LSSum‘𝑉) = (LSSum‘𝑉)
205	48, 52, 204	lsmsp2 20984	. . . . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . . . 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 6900	. . . . . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . . . . 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 19607	. . . . . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . 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 3203	. . . . . . . . . . . . . . 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 6958	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn (Base‘𝑉) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ (Base‘𝑉)(𝐹‘𝑥) = 𝑦))
216	215	biimpa 475	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn (Base‘𝑉) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ (Base‘𝑉)(𝐹‘𝑥) = 𝑦)
217	75, 216	sylan 578	. . . . . . . . . . . . . . 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 3151	. . . . . . . . . . . . . 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 3998	. . . . . . . . . . . . 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 2779	. . . . . . . . . . . 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 6835	. . . . . . . . . . 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 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)
223	42, 50, 76	syl2anc 582	. . . . . . . . . . . 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 6730	. . . . . . . . . . . 12 ⊢ (𝐹:(Base‘𝑉)⟶(Base‘𝑈) → ran 𝐹 ⊆ (Base‘𝑈))
226	29, 72	ressbas2 17221	. . . . . . . . . . . 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 6832	. . . . . . . . . 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 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))
230		eqid 2725	. . . . . . . . . 10 ⊢ (Base‘𝐼) = (Base‘𝐼)
231	145, 230	islmim 20959	. . . . . . . . 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 581	. . . . . . . 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 20881	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ LMod ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → (𝑏 ∖ 𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
234	42, 50, 233	syl2anc 582	. . . . . . . . . 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 21782	. . . . . . . . . . 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 476	. . . . . . . . . 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 1370	. . . . . . . . 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 2725	. . . . . . . . . . . . 13 ⊢ (LSpan‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (LSpan‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
239	55, 52, 238, 51	lsslsp 20911	. . . . . . . . . . . 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 2731	. . . . . . . . . . 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 1368	. . . . . . . . . 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 2767	. . . . . . . . 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 2725	. . . . . . . . . 10 ⊢ (LBasis‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (LBasis‘(𝑉 ↾_s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
244	145, 243, 238	islbs4 21783	. . . . . . . . 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 581	. . . . . . . 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 2725	. . . . . . . . 9 ⊢ (LBasis‘𝐼) = (LBasis‘𝐼)
247	243, 246	lmimlbs 21787	. . . . . . . 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 582	. . . . . . 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 33426	. . . . . . 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 582	. . . . . 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 9035	. . . . . . . 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 14343	. . . . . . . 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 1368	. . . . . 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 2765	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (dim‘𝐼) = (♯‘(𝑏 ∖ 𝑤)))
256	28, 255	oveq12d 7437	. . . 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 2775	. . 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 21784	. . . . . 6 ⊢ (LBasis‘𝐾) ⊆ (LIndS‘𝐾)
259	258, 93	sselid 3974	. . . . 5 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LIndS‘𝐾))
260	51, 2	lsslinds 21782	. . . . . 6 ⊢ ((𝑉 ∈ LMod ∧ (^◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (^◡𝐹 “ { 0 })) → (𝑤 ∈ (LIndS‘𝐾) ↔ 𝑤 ∈ (LIndS‘𝑉)))
261	260	biimpa 475	. . . . 5 ⊢ (((𝑉 ∈ LMod ∧ (^◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (^◡𝐹 “ { 0 })) ∧ 𝑤 ∈ (LIndS‘𝐾)) → 𝑤 ∈ (LIndS‘𝑉))
262	98, 101, 109, 259, 261	syl31anc 1370	. . . 4 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LIndS‘𝑉))
263	23	islinds4 21786	. . . . 5 ⊢ (𝑉 ∈ LVec → (𝑤 ∈ (LIndS‘𝑉) ↔ ∃𝑏 ∈ (LBasis‘𝑉)𝑤 ⊆ 𝑏))
264	263	ad2antrr 724	. . . 4 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (𝑤 ∈ (LIndS‘𝑉) ↔ ∃𝑏 ∈ (LBasis‘𝑉)𝑤 ⊆ 𝑏))
265	262, 264	mpbid 231	. . 3 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ∃𝑏 ∈ (LBasis‘𝑉)𝑤 ⊆ 𝑏)
266	257, 265	r19.29a 3151	. 2 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (dim‘𝑉) = ((dim‘𝐾) +_𝑒 (dim‘𝐼)))
267	8, 266	exlimddv 1930	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 394 ∧ w3a 1084 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2929 ∃wrex 3059 Vcvv 3461 ∖ cdif 3941 ∪ cun 3942 ∩ cin 3943 ⊆ wss 3944 ∅c0 4322 {csn 4630 class class class wbr 5149 ^◡ccnv 5677 ran crn 5679 ↾ cres 5680 “ cima 5681 Fn wfn 6544 ⟶wf 6545 –1-1→wf1 6546 –1-1-onto→wf1o 6548 ‘cfv 6549 (class class class)co 7419 ≈ cen 8961 +_𝑒 cxad 13125 ♯chash 14325 Basecbs 17183 ↾_s cress 17212 +_gcplusg 17236 0_gc0g 17424 Mndcmnd 18697 Grpcgrp 18898 SubGrpcsubg 19083 GrpHom cghm 19175 LSSumclsm 19601 LModclmod 20755 LSubSpclss 20827 LSpanclspn 20867 LMHom clmhm 20916 LMIso clmim 20917 LBasisclbs 20971 LVecclvec 20999 LIndSclinds 21756 dimcldim 33424

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-reg 9617 ax-inf2 9666 ax-ac2 10488 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-rpss 7729 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-sup 9467 df-oi 9535 df-r1 9789 df-rank 9790 df-dju 9926 df-card 9964 df-acn 9967 df-ac 10141 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 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 12506 df-xnn0 12578 df-z 12592 df-dec 12711 df-uz 12856 df-xadd 13128 df-fz 13520 df-fzo 13663 df-seq 14003 df-hash 14326 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ocomp 17257 df-ds 17258 df-hom 17260 df-cco 17261 df-0g 17426 df-gsum 17427 df-prds 17432 df-pws 17434 df-mre 17569 df-mrc 17570 df-mri 17571 df-acs 17572 df-proset 18290 df-drs 18291 df-poset 18308 df-ipo 18523 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18743 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19032 df-subg 19086 df-ghm 19176 df-cntz 19280 df-lsm 19603 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-nzr 20464 df-subrg 20520 df-drng 20638 df-lmod 20757 df-lss 20828 df-lsp 20868 df-lmhm 20919 df-lmim 20920 df-lbs 20972 df-lvec 21000 df-sra 21070 df-rgmod 21071 df-dsmm 21683 df-frlm 21698 df-uvc 21734 df-lindf 21757 df-linds 21758 df-dim 33425

This theorem is referenced by: qusdimsum 33454

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