Theorem dimkerim 33000

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

Hypotheses

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

Assertion

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

Proof of Theorem dimkerim

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

Step	Hyp	Ref	Expression
1		dimkerim.0	. . . . 5 ⊢ 0 = (0g‘𝑈)
2		dimkerim.k	. . . . 5 ⊢ 𝐾 = (𝑉 ↾s (◡𝐹 “ { 0 }))
3	1, 2	kerlmhm 32993	. . . 4 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐾 ∈ LVec)
4		eqid 2730	. . . . 5 ⊢ (LBasis‘𝐾) = (LBasis‘𝐾)
5	4	lbsex 20923	. . . 4 ⊢ (𝐾 ∈ LVec → (LBasis‘𝐾) ≠ ∅)
6	3, 5	syl 17	. . 3 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (LBasis‘𝐾) ≠ ∅)
7		n0 4345	. . 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 772	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ∈ (LBasis‘𝐾))
10		vex 3476	. . . . . . 7 ⊢ 𝑏 ∈ V
11	10	difexi 5327	. . . . . 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 4470	. . . . . 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 14350	. . . . 5 ⊢ ((𝑤 ∈ (LBasis‘𝐾) ∧ (𝑏 ∖ 𝑤) ∈ V ∧ (𝑤 ∩ (𝑏 ∖ 𝑤)) = ∅) → (♯‘(𝑤 ∪ (𝑏 ∖ 𝑤))) = ((♯‘𝑤) +𝑒 (♯‘(𝑏 ∖ 𝑤))))
16	9, 12, 14, 15	syl3anc 1369	. . . 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 779	. . . . 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 4480	. . . . . . 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 765	. . . . . 6 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑏 ∈ (LBasis‘𝑉))
22	20, 21	eqeltrd 2831	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑤 ∪ (𝑏 ∖ 𝑤)) ∈ (LBasis‘𝑉))
23		eqid 2730	. . . . . 6 ⊢ (LBasis‘𝑉) = (LBasis‘𝑉)
24	23	dimval 32973	. . . . 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 726	. . . . . 6 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐾 ∈ LVec)
27	4	dimval 32973	. . . . . 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 32994	. . . . . . . 8 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐼 ∈ LVec)
31	30	ad3antrrr 726	. . . . . . 7 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐼 ∈ LVec)
32		simp-4r 780	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹 ∈ (𝑉 LMHom 𝑈))
33		lmhmlmod2 20787	. . . . . . . . . . 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 20805	. . . . . . . . . . 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 5688	. . . . . . . . . . 11 ⊢ (𝐹 “ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = ran (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
38		imassrn 6069	. . . . . . . . . . . 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 4030	. . . . . . . . . 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 20861	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ LVec → 𝑉 ∈ LMod)
42	41	ad4antr 728	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑉 ∈ LMod)
43	23	lbslinds 21607	. . . . . . . . . . . . . . 15 ⊢ (LBasis‘𝑉) ⊆ (LIndS‘𝑉)
44	43, 21	sselid 3979	. . . . . . . . . . . . . 14 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑏 ∈ (LIndS‘𝑉))
45		difssd 4131	. . . . . . . . . . . . . 14 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ⊆ 𝑏)
46		lindsss 21598	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ LMod ∧ 𝑏 ∈ (LIndS‘𝑉) ∧ (𝑏 ∖ 𝑤) ⊆ 𝑏) → (𝑏 ∖ 𝑤) ∈ (LIndS‘𝑉))
47	42, 44, 45, 46	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑏 ∖ 𝑤) ∈ (LIndS‘𝑉))
48		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑉) = (Base‘𝑉)
49	48	linds1 21584	. . . . . . . . . . . . 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 2730	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝑉) = (LSubSp‘𝑉)
52		eqid 2730	. . . . . . . . . . . . 13 ⊢ (LSpan‘𝑉) = (LSpan‘𝑉)
53	48, 51, 52	lspcl 20731	. . . . . . . . . . . 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 2730	. . . . . . . . . . . 12 ⊢ (𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) = (𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
56	51, 55	reslmhm 20807	. . . . . . . . . . 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 2730	. . . . . . . . . . . 12 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
59	29, 58	reslmhm2b 20809	. . . . . . . . . . 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 1371	. . . . . . . . 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 20786	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
63	62	ad4antlr 729	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹 ∈ (𝑉 GrpHom 𝑈))
64	48, 23	lbsss 20832	. . . . . . . . . . . . . . . . . . . 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 3991	. . . . . . . . . . . . . . . . . 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 20712	. . . . . . . . . . . . . . . . 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 19146	. . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Base‘𝑈) = (Base‘𝑈)
73	48, 72	lmhmf 20789	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
74	73	ad4antlr 729	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹:(Base‘𝑉)⟶(Base‘𝑈))
75	74	ffnd 6717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝐹 Fn (Base‘𝑉))
76	48, 52	lspssv 20738	. . . . . . . . . . . . . . . . . . . . . . . . 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		fnssres 6672	. . . . . . . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . . . . . . 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 7060	. . . . . . . . . . . . . . . . . . . . . . . 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 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 ))
82	79, 81	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 ))
83	82	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)))
84	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))
85	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))
86	85, 83	sseldd 3982	. . . . . . . . . . . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . . . . . . . . . . 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 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 )
89	87, 88	eqtr3d 2772	. . . . . . . . . . . . . . . . . . . . . 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 7060	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 Fn (Base‘𝑉) → (𝑥 ∈ (◡𝐹 “ { 0 }) ↔ (𝑥 ∈ (Base‘𝑉) ∧ (𝐹‘𝑥) = 0 )))
91	90	biimpar 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Base‘𝑉) ∧ (𝑥 ∈ (Base‘𝑉) ∧ (𝐹‘𝑥) = 0 )) → 𝑥 ∈ (◡𝐹 “ { 0 }))
92	84, 86, 89, 91	syl12anc 833	. . . . . . . . . . . . . . . . . . . . 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 4193	. . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LBasis‘𝐾))
95		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Base‘𝐾) = (Base‘𝐾)
96		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾)
97	95, 4, 96	lbssp 20834	. . . . . . . . . . . . . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑉 ∈ LMod)
100		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (◡𝐹 “ { 0 }) = (◡𝐹 “ { 0 })
101	100, 1, 51	lmhmkerlss 20806	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉))
102	101	ad2antlr 723	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉))
103	95, 4	lbsss 20832	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6079	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (◡𝐹 “ { 0 }) ⊆ dom 𝐹
106	105, 73	fssdm 6736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (◡𝐹 “ { 0 }) ⊆ (Base‘𝑉))
107	2, 48	ressbas2 17186	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (◡𝐹 “ { 0 }) = (Base‘𝐾))
110	104, 109	sseqtrrd 4022	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ⊆ (◡𝐹 “ { 0 }))
111	2, 52, 96, 51	lsslsp 20770	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑉 ∈ LMod ∧ (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (◡𝐹 “ { 0 })) → ((LSpan‘𝑉)‘𝑤) = ((LSpan‘𝐾)‘𝑤))
112	99, 102, 110, 111	syl3anc 1369	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝑉)‘𝑤) = ((LSpan‘𝐾)‘𝑤))
113	98, 112, 109	3eqtr4d 2780	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → ((LSpan‘𝑉)‘𝑤) = (◡𝐹 “ { 0 }))
114	113	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑤) = (◡𝐹 “ { 0 }))
115	114	ineq2d 4211	. . . . . . . . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0g‘𝑉) = (0g‘𝑉)
117	23, 52, 116	lbsdiflsp0 32999	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑉 ∈ LVec ∧ 𝑏 ∈ (LBasis‘𝑉) ∧ 𝑤 ⊆ 𝑏) → (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∩ ((LSpan‘𝑉)‘𝑤)) = {(0g‘𝑉)})
118	117	ad5ant145 1367	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∩ ((LSpan‘𝑉)‘𝑤)) = {(0g‘𝑉)})
119	115, 118	eqtr3d 2772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∩ (◡𝐹 “ { 0 })) = {(0g‘𝑉)})
120	119	adantr 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 })) = {(0g‘𝑉)})
121	93, 120	eleqtrd 2833	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 })) → 𝑥 ∈ {(0g‘𝑉)})
122	121	ex 411	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝑥 ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) → 𝑥 ∈ {(0g‘𝑉)}))
123	122	ssrdv 3987	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) ⊆ {(0g‘𝑉)})
124	116, 48, 52	0ellsp 32756	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 ∈ LMod ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → (0g‘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) → (0g‘𝑉) ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
126		fvexd 6905	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) ∈ V)
127	125	fvresd 6910	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) = (𝐹‘(0g‘𝑉)))
128	116, 1	ghmid 19136	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 ∈ (𝑉 GrpHom 𝑈) → (𝐹‘(0g‘𝑉)) = 0 )
129	62, 128	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (𝐹‘(0g‘𝑉)) = 0 )
130	129	ad4antlr 729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (𝐹‘(0g‘𝑉)) = 0 )
131	127, 130	eqtrd 2770	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) = 0 )
132		elsng 4641	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) ∈ V → (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) ∈ { 0 } ↔ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) = 0 ))
133	132	biimpar 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) ∈ V ∧ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) = 0 ) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) ∈ { 0 })
134	126, 131, 133	syl2anc 582	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))‘(0g‘𝑉)) ∈ { 0 })
135	79, 125, 134	elpreimad 7059	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (0g‘𝑉) ∈ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }))
136	135	snssd 4811	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → {(0g‘𝑉)} ⊆ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }))
137	123, 136	eqssd 3998	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) = {(0g‘𝑉)})
138		lmodgrp 20621	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑉 ∈ LMod → 𝑉 ∈ Grp)
139		grpmnd 18862	. . . . . . . . . . . . . . . . . . 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 18687	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑉 ∈ Mnd ∧ (0g‘𝑉) ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ∧ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)) ⊆ (Base‘𝑉)) → (0g‘𝑉) = (0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))))
142	140, 125, 77, 141	syl3anc 1369	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (0g‘𝑉) = (0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))))
143	142	sneqd 4639	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → {(0g‘𝑉)} = {(0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))})
144	137, 143	eqtrd 2770	. . . . . . . . . . . . . . 15 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) = {(0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))})
145		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
146		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
147	145, 72, 146, 1	kerf1ghm 19161	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) GrpHom 𝑈) → ((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):(Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))–1-1→(Base‘𝑈) ↔ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) = {(0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))}))
148	147	biimpar 476	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∈ ((𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) GrpHom 𝑈) ∧ (◡(𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) “ { 0 }) = {(0g‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))}) → (𝐹 ↾ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))):(Base‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))–1-1→(Base‘𝑈))
149	71, 144, 148	syl2anc 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 2731	. . . . . . . . . . . . . . 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 17186	. . . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . 15 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (Base‘𝑈) = (Base‘𝑈))
154	150, 152, 153	f1eq123d 6824	. . . . . . . . . . . . . 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 6793	. . . . . . . . . . . . 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 6843	. . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . . . . . . . . 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 7060	. . . . . . . . . . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . . . . . 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 20739	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑉 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑉) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ ((LSpan‘𝑉)‘𝑏))
173	42, 65, 18, 172	syl3anc 1369	. . . . . . . . . . . . . . . . . . . . . . . . 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 20834	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4021	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → ((LSpan‘𝑉)‘𝑤) ⊆ (Base‘𝑉))
177	176	ad3antrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . 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 3982	. . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . . . . . . 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 3982	. . . . . . . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (+g‘𝑉) = (+g‘𝑉)
185		eqid 2730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (+g‘𝑈) = (+g‘𝑈)
186	48, 184, 185	ghmlin 19135	. . . . . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . . 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 32992	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec)
190		lveclmod 20861	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑈 ∈ LVec → 𝑈 ∈ LMod)
191		lmodgrp 20621	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Grp)
192	189, 190, 191	3syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ Grp)
193	192	ad9antr 738	. . . . . . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . . . . . 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	ffvelcdmd 7086	. . . . . . . . . . . . . . . . . . . 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 18888	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑈 ∈ Grp ∧ (𝐹‘𝑣) ∈ (Base‘𝑈)) → ( 0 (+g‘𝑈)(𝐹‘𝑣)) = (𝐹‘𝑣))
197	193, 195, 196	syl2anc 582	. . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . 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 7237	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑥 ∈ (Base‘𝑉)) ∧ (𝐹‘𝑥) = 𝑦) ∧ 𝑢 ∈ ((LSpan‘𝑉)‘𝑤)) ∧ 𝑣 ∈ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))) ∧ 𝑥 = (𝑢(+g‘𝑉)𝑣)) → (𝐹‘𝑣) ∈ (𝐹 “ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
201	199, 200	eqeltrd 2831	. . . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ⊆ (◡𝐹 “ { 0 }))
205	106	ad4antlr 729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (◡𝐹 “ { 0 }) ⊆ (Base‘𝑉))
206	204, 205	sstrd 3991	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → 𝑤 ⊆ (Base‘𝑉))
207		eqid 2730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (LSSum‘𝑉) = (LSSum‘𝑉)
208	48, 52, 207	lsmsp2 20842	. . . . . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . . . . 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 19549	. . . . . . . . . . . . . . . . . 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 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))
216	202, 177, 180, 213, 215	syl31anc 1371	. . . . . . . . . . . . . . . 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 3211	. . . . . . . . . . . . . . 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 6951	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn (Base‘𝑉) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ (Base‘𝑉)(𝐹‘𝑥) = 𝑦))
219	218	biimpa 475	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn (Base‘𝑉) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ (Base‘𝑉)(𝐹‘𝑥) = 𝑦)
220	75, 219	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)
221	217, 220	r19.29a 3160	. . . . . . . . . . . . . 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 4002	. . . . . . . . . . . . 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 2784	. . . . . . . . . . . 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 6829	. . . . . . . . . . 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 582	. . . . . . . . . . . 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 6723	. . . . . . . . . . . . 13 ⊢ (𝐹:(Base‘𝑉)⟶(Base‘𝑈) → ran 𝐹 ⊆ (Base‘𝑈))
229	29, 72	ressbas2 17186	. . . . . . . . . . . . 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 6826	. . . . . . . . . 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 2730	. . . . . . . . . 10 ⊢ (Base‘𝐼) = (Base‘𝐼)
235	145, 234	islmim 20817	. . . . . . . . 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 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))
237	48, 52	lspssid 20740	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ LMod ∧ (𝑏 ∖ 𝑤) ⊆ (Base‘𝑉)) → (𝑏 ∖ 𝑤) ⊆ ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))
238	42, 50, 237	syl2anc 582	. . . . . . . . . 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 21605	. . . . . . . . . . 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 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)))))
241	42, 67, 238, 47, 240	syl31anc 1371	. . . . . . . . 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 2730	. . . . . . . . . . . 12 ⊢ (LSpan‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (LSpan‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
243	55, 52, 242, 51	lsslsp 20770	. . . . . . . . . . 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 1369	. . . . . . . . . 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 2772	. . . . . . . . 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 2730	. . . . . . . . . 10 ⊢ (LBasis‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤)))) = (LBasis‘(𝑉 ↾s ((LSpan‘𝑉)‘(𝑏 ∖ 𝑤))))
247	145, 246, 242	islbs4 21606	. . . . . . . . 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 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)))))
249		eqid 2730	. . . . . . . . 9 ⊢ (LBasis‘𝐼) = (LBasis‘𝐼)
250	246, 249	lmimlbs 21610	. . . . . . . 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 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))
252	249	dimval 32973	. . . . . . 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 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))))
254		f1imaeng 9012	. . . . . . . 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 14312	. . . . . . . 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 1369	. . . . . 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 2770	. . . . 5 ⊢ (((((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑤 ⊆ 𝑏) → (dim‘𝐼) = (♯‘(𝑏 ∖ 𝑤)))
259	28, 258	oveq12d 7429	. . . 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 2780	. . 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 21607	. . . . . 6 ⊢ (LBasis‘𝐾) ⊆ (LIndS‘𝐾)
262	261, 94	sselid 3979	. . . . 5 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LIndS‘𝐾))
263	51, 2	lsslinds 21605	. . . . . 6 ⊢ ((𝑉 ∈ LMod ∧ (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (◡𝐹 “ { 0 })) → (𝑤 ∈ (LIndS‘𝐾) ↔ 𝑤 ∈ (LIndS‘𝑉)))
264	263	biimpa 475	. . . . 5 ⊢ (((𝑉 ∈ LMod ∧ (◡𝐹 “ { 0 }) ∈ (LSubSp‘𝑉) ∧ 𝑤 ⊆ (◡𝐹 “ { 0 })) ∧ 𝑤 ∈ (LIndS‘𝐾)) → 𝑤 ∈ (LIndS‘𝑉))
265	99, 102, 110, 262, 264	syl31anc 1371	. . . 4 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → 𝑤 ∈ (LIndS‘𝑉))
266	23	islinds4 21609	. . . . 5 ⊢ (𝑉 ∈ LVec → (𝑤 ∈ (LIndS‘𝑉) ↔ ∃𝑏 ∈ (LBasis‘𝑉)𝑤 ⊆ 𝑏))
267	266	ad2antrr 722	. . . 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 3160	. 2 ⊢ (((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) ∧ 𝑤 ∈ (LBasis‘𝐾)) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))
270	8, 269	exlimddv 1936	1 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539  ∃wex 1779   ∈ wcel 2104   ≠ wne 2938  ∃wrex 3068  Vcvv 3472   ∖ cdif 3944   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321  {csn 4627   class class class wbr 5147  ^◡ccnv 5674  ran crn 5676   ↾ cres 5677   “ cima 5678   Fn wfn 6537  ⟶wf 6538  –1-1→wf1 6539  –1-1-onto→wf1o 6541  ‘cfv 6542  (class class class)co 7411   ≈ cen 8938   +_𝑒 cxad 13094  ♯chash 14294  Basecbs 17148   ↾_s cress 17177  +_gcplusg 17201  0_gc0g 17389  Mndcmnd 18659  Grpcgrp 18855  SubGrpcsubg 19036   GrpHom cghm 19127  LSSumclsm 19543  LModclmod 20614  LSubSpclss 20686  LSpanclspn 20726   LMHom clmhm 20774   LMIso clmim 20775  LBasisclbs 20829  LVecclvec 20857  LIndSclinds 21579  dimcldim 32971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-reg 9589  ax-inf2 9638  ax-ac2 10460  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-rpss 7715  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-tpos 8213  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-sup 9439  df-oi 9507  df-r1 9761  df-rank 9762  df-dju 9898  df-card 9936  df-acn 9939  df-ac 10113  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-xnn0 12549  df-z 12563  df-dec 12682  df-uz 12827  df-xadd 13097  df-fz 13489  df-fzo 13632  df-seq 13971  df-hash 14295  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-sca 17217  df-vsca 17218  df-ip 17219  df-tset 17220  df-ple 17221  df-ocomp 17222  df-ds 17223  df-hom 17225  df-cco 17226  df-0g 17391  df-gsum 17392  df-prds 17397  df-pws 17399  df-mre 17534  df-mrc 17535  df-mri 17536  df-acs 17537  df-proset 18252  df-drs 18253  df-poset 18270  df-ipo 18485  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18705  df-submnd 18706  df-grp 18858  df-minusg 18859  df-sbg 18860  df-mulg 18987  df-subg 19039  df-ghm 19128  df-cntz 19222  df-lsm 19545  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-oppr 20225  df-dvdsr 20248  df-unit 20249  df-invr 20279  df-nzr 20404  df-subrg 20459  df-drng 20502  df-lmod 20616  df-lss 20687  df-lsp 20727  df-lmhm 20777  df-lmim 20778  df-lbs 20830  df-lvec 20858  df-sra 20930  df-rgmod 20931  df-dsmm 21506  df-frlm 21521  df-uvc 21557  df-lindf 21580  df-linds 21581  df-dim 32972

This theorem is referenced by:  qusdimsum  33001