Theorem lfl1dim 38593

Description: Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)

Hypotheses

Ref	Expression
lfl1dim.v	⊢ 𝑉 = (Base‘𝑊)
lfl1dim.d	⊢ 𝐷 = (Scalar‘𝑊)
lfl1dim.f	⊢ 𝐹 = (LFnl‘𝑊)
lfl1dim.l	⊢ 𝐿 = (LKer‘𝑊)
lfl1dim.k	⊢ 𝐾 = (Base‘𝐷)
lfl1dim.t	⊢ · = (.r‘𝐷)
lfl1dim.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lfl1dim.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)

Assertion

Ref	Expression
lfl1dim	⊢ (𝜑 → {𝑔 ∈ 𝐹 ∣ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)} = {𝑔 ∣ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))})

Distinct variable groups:   𝐷,𝑘   𝑘,𝐹   𝑘,𝐺   𝑘,𝐾   𝑘,𝐿   𝑘,𝑉   𝑘,𝑊   𝑔,𝑘,𝜑   · ,𝑘

Allowed substitution hints:   𝐷(𝑔)   · (𝑔)   𝐹(𝑔)   𝐺(𝑔)   𝐾(𝑔)   𝐿(𝑔)   𝑉(𝑔)   𝑊(𝑔)

Proof of Theorem lfl1dim

Step	Hyp	Ref	Expression
1		df-rab 3430	. 2 ⊢ {𝑔 ∈ 𝐹 ∣ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)} = {𝑔 ∣ (𝑔 ∈ 𝐹 ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔))}
2		lfl1dim.w	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ LVec)
3		lveclmod 20991	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
4	2, 3	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ LMod)
5		lfl1dim.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (Scalar‘𝑊)
6		lfl1dim.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (Base‘𝐷)
7		eqid 2728	. . . . . . . . . . . 12 ⊢ (0g‘𝐷) = (0g‘𝐷)
8	5, 6, 7	lmod0cl 20771	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → (0g‘𝐷) ∈ 𝐾)
9	4, 8	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝐷) ∈ 𝐾)
10	9	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑔 = (𝑉 × {(0g‘𝐷)})) → (0g‘𝐷) ∈ 𝐾)
11		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑔 = (𝑉 × {(0g‘𝐷)})) → 𝑔 = (𝑉 × {(0g‘𝐷)}))
12		lfl1dim.v	. . . . . . . . . . 11 ⊢ 𝑉 = (Base‘𝑊)
13		lfl1dim.f	. . . . . . . . . . 11 ⊢ 𝐹 = (LFnl‘𝑊)
14		lfl1dim.t	. . . . . . . . . . 11 ⊢ · = (.r‘𝐷)
15	4	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑔 = (𝑉 × {(0g‘𝐷)})) → 𝑊 ∈ LMod)
16		lfl1dim.g	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
17	16	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑔 = (𝑉 × {(0g‘𝐷)})) → 𝐺 ∈ 𝐹)
18	12, 5, 13, 6, 14, 7, 15, 17	lfl0sc 38554	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑔 = (𝑉 × {(0g‘𝐷)})) → (𝐺 ∘f · (𝑉 × {(0g‘𝐷)})) = (𝑉 × {(0g‘𝐷)}))
19	11, 18	eqtr4d 2771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑔 = (𝑉 × {(0g‘𝐷)})) → 𝑔 = (𝐺 ∘f · (𝑉 × {(0g‘𝐷)})))
20		sneq 4639	. . . . . . . . . . . 12 ⊢ (𝑘 = (0g‘𝐷) → {𝑘} = {(0g‘𝐷)})
21	20	xpeq2d 5708	. . . . . . . . . . 11 ⊢ (𝑘 = (0g‘𝐷) → (𝑉 × {𝑘}) = (𝑉 × {(0g‘𝐷)}))
22	21	oveq2d 7436	. . . . . . . . . 10 ⊢ (𝑘 = (0g‘𝐷) → (𝐺 ∘f · (𝑉 × {𝑘})) = (𝐺 ∘f · (𝑉 × {(0g‘𝐷)})))
23	22	rspceeqv 3631	. . . . . . . . 9 ⊢ (((0g‘𝐷) ∈ 𝐾 ∧ 𝑔 = (𝐺 ∘f · (𝑉 × {(0g‘𝐷)}))) → ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})))
24	10, 19, 23	syl2anc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑔 = (𝑉 × {(0g‘𝐷)})) → ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})))
25	24	a1d 25	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑔 = (𝑉 × {(0g‘𝐷)})) → ((𝐿‘𝐺) ⊆ (𝐿‘𝑔) → ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))))
26	9	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → (0g‘𝐷) ∈ 𝐾)
27		lfl1dim.l	. . . . . . . . . . . . 13 ⊢ 𝐿 = (LKer‘𝑊)
28	4	ad3antrrr 729	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → 𝑊 ∈ LMod)
29		simpllr 775	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → 𝑔 ∈ 𝐹)
30	12, 13, 27, 28, 29	lkrssv 38568	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → (𝐿‘𝑔) ⊆ 𝑉)
31	4	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → 𝑊 ∈ LMod)
32	16	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → 𝐺 ∈ 𝐹)
33	5, 7, 12, 13, 27	lkr0f 38566	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐿‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × {(0g‘𝐷)})))
34	31, 32, 33	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → ((𝐿‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × {(0g‘𝐷)})))
35	34	biimpar 477	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) → (𝐿‘𝐺) = 𝑉)
36	35	sseq1d 4011	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) → ((𝐿‘𝐺) ⊆ (𝐿‘𝑔) ↔ 𝑉 ⊆ (𝐿‘𝑔)))
37	36	biimpa 476	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → 𝑉 ⊆ (𝐿‘𝑔))
38	30, 37	eqssd 3997	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → (𝐿‘𝑔) = 𝑉)
39	5, 7, 12, 13, 27	lkr0f 38566	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑔 ∈ 𝐹) → ((𝐿‘𝑔) = 𝑉 ↔ 𝑔 = (𝑉 × {(0g‘𝐷)})))
40	28, 29, 39	syl2anc 583	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → ((𝐿‘𝑔) = 𝑉 ↔ 𝑔 = (𝑉 × {(0g‘𝐷)})))
41	38, 40	mpbid 231	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → 𝑔 = (𝑉 × {(0g‘𝐷)}))
42	16	ad3antrrr 729	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → 𝐺 ∈ 𝐹)
43	12, 5, 13, 6, 14, 7, 28, 42	lfl0sc 38554	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → (𝐺 ∘f · (𝑉 × {(0g‘𝐷)})) = (𝑉 × {(0g‘𝐷)}))
44	41, 43	eqtr4d 2771	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → 𝑔 = (𝐺 ∘f · (𝑉 × {(0g‘𝐷)})))
45	26, 44, 23	syl2anc 583	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})))
46	45	ex 412	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) → ((𝐿‘𝐺) ⊆ (𝐿‘𝑔) → ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))))
47		eqid 2728	. . . . . . . . 9 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊)
48	2	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ (𝑔 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))) → 𝑊 ∈ LVec)
49	16	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ (𝑔 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))) → 𝐺 ∈ 𝐹)
50		simprr 772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ (𝑔 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))) → 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))
51	12, 5, 7, 47, 13, 27	lkrshp 38577	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)})) → (𝐿‘𝐺) ∈ (LSHyp‘𝑊))
52	48, 49, 50, 51	syl3anc 1369	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ (𝑔 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))) → (𝐿‘𝐺) ∈ (LSHyp‘𝑊))
53		simplr 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ (𝑔 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))) → 𝑔 ∈ 𝐹)
54		simprl 770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ (𝑔 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))) → 𝑔 ≠ (𝑉 × {(0g‘𝐷)}))
55	12, 5, 7, 47, 13, 27	lkrshp 38577	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LVec ∧ 𝑔 ∈ 𝐹 ∧ 𝑔 ≠ (𝑉 × {(0g‘𝐷)})) → (𝐿‘𝑔) ∈ (LSHyp‘𝑊))
56	48, 53, 54, 55	syl3anc 1369	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ (𝑔 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))) → (𝐿‘𝑔) ∈ (LSHyp‘𝑊))
57	47, 48, 52, 56	lshpcmp 38460	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ (𝑔 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))) → ((𝐿‘𝐺) ⊆ (𝐿‘𝑔) ↔ (𝐿‘𝐺) = (𝐿‘𝑔)))
58	2	ad3antrrr 729	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ (𝑔 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))) ∧ (𝐿‘𝐺) = (𝐿‘𝑔)) → 𝑊 ∈ LVec)
59	16	ad3antrrr 729	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ (𝑔 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))) ∧ (𝐿‘𝐺) = (𝐿‘𝑔)) → 𝐺 ∈ 𝐹)
60		simpllr 775	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ (𝑔 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))) ∧ (𝐿‘𝐺) = (𝐿‘𝑔)) → 𝑔 ∈ 𝐹)
61		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ (𝑔 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))) ∧ (𝐿‘𝐺) = (𝐿‘𝑔)) → (𝐿‘𝐺) = (𝐿‘𝑔))
62	5, 6, 14, 12, 13, 27	eqlkr2 38572	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝑔)) → ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})))
63	58, 59, 60, 61, 62	syl121anc 1373	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ (𝑔 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))) ∧ (𝐿‘𝐺) = (𝐿‘𝑔)) → ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})))
64	63	ex 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ (𝑔 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))) → ((𝐿‘𝐺) = (𝐿‘𝑔) → ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))))
65	57, 64	sylbid 239	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ (𝑔 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))) → ((𝐿‘𝐺) ⊆ (𝐿‘𝑔) → ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))))
66	25, 46, 65	pm2.61da2ne 3027	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → ((𝐿‘𝐺) ⊆ (𝐿‘𝑔) → ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))))
67	2	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑘 ∈ 𝐾) → 𝑊 ∈ LVec)
68	16	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑘 ∈ 𝐾) → 𝐺 ∈ 𝐹)
69		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑘 ∈ 𝐾) → 𝑘 ∈ 𝐾)
70	12, 5, 6, 14, 13, 27, 67, 68, 69	lkrscss 38570	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑘 ∈ 𝐾) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑘}))))
71	70	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → (𝑘 ∈ 𝐾 → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑘})))))
72		fveq2 6897	. . . . . . . . . 10 ⊢ (𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})) → (𝐿‘𝑔) = (𝐿‘(𝐺 ∘f · (𝑉 × {𝑘}))))
73	72	sseq2d 4012	. . . . . . . . 9 ⊢ (𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})) → ((𝐿‘𝐺) ⊆ (𝐿‘𝑔) ↔ (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑘})))))
74	73	biimprcd 249	. . . . . . . 8 ⊢ ((𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑘}))) → (𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})) → (𝐿‘𝐺) ⊆ (𝐿‘𝑔)))
75	71, 74	syl6 35	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → (𝑘 ∈ 𝐾 → (𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})) → (𝐿‘𝐺) ⊆ (𝐿‘𝑔))))
76	75	rexlimdv 3150	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → (∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})) → (𝐿‘𝐺) ⊆ (𝐿‘𝑔)))
77	66, 76	impbid 211	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → ((𝐿‘𝐺) ⊆ (𝐿‘𝑔) ↔ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))))
78	77	pm5.32da 578	. . . 4 ⊢ (𝜑 → ((𝑔 ∈ 𝐹 ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) ↔ (𝑔 ∈ 𝐹 ∧ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})))))
79	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝑊 ∈ LMod)
80	16	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐺 ∈ 𝐹)
81		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝑘 ∈ 𝐾)
82	12, 5, 6, 14, 13, 79, 80, 81	lflvscl 38549	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → (𝐺 ∘f · (𝑉 × {𝑘})) ∈ 𝐹)
83		eleq1a 2824	. . . . . . . 8 ⊢ ((𝐺 ∘f · (𝑉 × {𝑘})) ∈ 𝐹 → (𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})) → 𝑔 ∈ 𝐹))
84	82, 83	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → (𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})) → 𝑔 ∈ 𝐹))
85	84	pm4.71rd 562	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → (𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})) ↔ (𝑔 ∈ 𝐹 ∧ 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})))))
86	85	rexbidva 3173	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})) ↔ ∃𝑘 ∈ 𝐾 (𝑔 ∈ 𝐹 ∧ 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})))))
87		r19.42v 3187	. . . . 5 ⊢ (∃𝑘 ∈ 𝐾 (𝑔 ∈ 𝐹 ∧ 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))) ↔ (𝑔 ∈ 𝐹 ∧ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))))
88	86, 87	bitr2di 288	. . . 4 ⊢ (𝜑 → ((𝑔 ∈ 𝐹 ∧ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))) ↔ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))))
89	78, 88	bitrd 279	. . 3 ⊢ (𝜑 → ((𝑔 ∈ 𝐹 ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) ↔ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))))
90	89	abbidv 2797	. 2 ⊢ (𝜑 → {𝑔 ∣ (𝑔 ∈ 𝐹 ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔))} = {𝑔 ∣ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))})
91	1, 90	eqtrid 2780	1 ⊢ (𝜑 → {𝑔 ∈ 𝐹 ∣ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)} = {𝑔 ∣ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {cab 2705   ≠ wne 2937  ∃wrex 3067  {crab 3429   ⊆ wss 3947  {csn 4629   × cxp 5676  ‘cfv 6548  (class class class)co 7420   ∘_f cof 7683  Basecbs 17180  ._rcmulr 17234  Scalarcsca 17236  0_gc0g 17421  LModclmod 20743  LVecclvec 20987  LSHypclsh 38447  LFnlclfn 38529  LKerclk 38557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  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 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7685  df-om 7871  df-1st 7993  df-2nd 7994  df-tpos 8232  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-3 12307  df-sets 17133  df-slot 17151  df-ndx 17163  df-base 17181  df-ress 17210  df-plusg 17246  df-mulr 17247  df-0g 17423  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-submnd 18741  df-grp 18893  df-minusg 18894  df-sbg 18895  df-subg 19078  df-cntz 19268  df-lsm 19591  df-cmn 19737  df-abl 19738  df-mgp 20075  df-rng 20093  df-ur 20122  df-ring 20175  df-oppr 20273  df-dvdsr 20296  df-unit 20297  df-invr 20327  df-drng 20626  df-lmod 20745  df-lss 20816  df-lsp 20856  df-lvec 20988  df-lshyp 38449  df-lfl 38530  df-lkr 38558

This theorem is referenced by:  ldual1dim  38638

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