Theorem lfl1dim 36731

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 3079	. 2 ⊢ {𝑔 ∈ 𝐹 ∣ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)} = {𝑔 ∣ (𝑔 ∈ 𝐹 ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔))}
2		lfl1dim.w	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ LVec)
3		lveclmod 19959	. . . . . . . . . . . 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 2758	. . . . . . . . . . . 12 ⊢ (0g‘𝐷) = (0g‘𝐷)
8	5, 6, 7	lmod0cl 19741	. . . . . . . . . . 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 488	. . . . . . . . . 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 36692	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑔 = (𝑉 × {(0g‘𝐷)})) → (𝐺 ∘f · (𝑉 × {(0g‘𝐷)})) = (𝑉 × {(0g‘𝐷)}))
19	11, 18	eqtr4d 2796	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑔 = (𝑉 × {(0g‘𝐷)})) → 𝑔 = (𝐺 ∘f · (𝑉 × {(0g‘𝐷)})))
20		sneq 4535	. . . . . . . . . . . 12 ⊢ (𝑘 = (0g‘𝐷) → {𝑘} = {(0g‘𝐷)})
21	20	xpeq2d 5558	. . . . . . . . . . 11 ⊢ (𝑘 = (0g‘𝐷) → (𝑉 × {𝑘}) = (𝑉 × {(0g‘𝐷)}))
22	21	oveq2d 7172	. . . . . . . . . 10 ⊢ (𝑘 = (0g‘𝐷) → (𝐺 ∘f · (𝑉 × {𝑘})) = (𝐺 ∘f · (𝑉 × {(0g‘𝐷)})))
23	22	rspceeqv 3558	. . . . . . . . 9 ⊢ (((0g‘𝐷) ∈ 𝐾 ∧ 𝑔 = (𝐺 ∘f · (𝑉 × {(0g‘𝐷)}))) → ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})))
24	10, 19, 23	syl2anc 587	. . . . . . . 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 36706	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → (𝐿‘𝑔) ⊆ 𝑉)
31	4	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → 𝑊 ∈ LMod)
32	16	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → 𝐺 ∈ 𝐹)
33	5, 7, 12, 13, 27	lkr0f 36704	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐿‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × {(0g‘𝐷)})))
34	31, 32, 33	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → ((𝐿‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × {(0g‘𝐷)})))
35	34	biimpar 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) → (𝐿‘𝐺) = 𝑉)
36	35	sseq1d 3925	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) → ((𝐿‘𝐺) ⊆ (𝐿‘𝑔) ↔ 𝑉 ⊆ (𝐿‘𝑔)))
37	36	biimpa 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → 𝑉 ⊆ (𝐿‘𝑔))
38	30, 37	eqssd 3911	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → (𝐿‘𝑔) = 𝑉)
39	5, 7, 12, 13, 27	lkr0f 36704	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑔 ∈ 𝐹) → ((𝐿‘𝑔) = 𝑉 ↔ 𝑔 = (𝑉 × {(0g‘𝐷)})))
40	28, 29, 39	syl2anc 587	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → ((𝐿‘𝑔) = 𝑉 ↔ 𝑔 = (𝑉 × {(0g‘𝐷)})))
41	38, 40	mpbid 235	. . . . . . . . . 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 36692	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → (𝐺 ∘f · (𝑉 × {(0g‘𝐷)})) = (𝑉 × {(0g‘𝐷)}))
44	41, 43	eqtr4d 2796	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → 𝑔 = (𝐺 ∘f · (𝑉 × {(0g‘𝐷)})))
45	26, 44, 23	syl2anc 587	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) → ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})))
46	45	ex 416	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) → ((𝐿‘𝐺) ⊆ (𝐿‘𝑔) → ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))))
47		eqid 2758	. . . . . . . . 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 36715	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)})) → (𝐿‘𝐺) ∈ (LSHyp‘𝑊))
52	48, 49, 50, 51	syl3anc 1368	. . . . . . . . 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 36715	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LVec ∧ 𝑔 ∈ 𝐹 ∧ 𝑔 ≠ (𝑉 × {(0g‘𝐷)})) → (𝐿‘𝑔) ∈ (LSHyp‘𝑊))
56	48, 53, 54, 55	syl3anc 1368	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ (𝑔 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))) → (𝐿‘𝑔) ∈ (LSHyp‘𝑊))
57	47, 48, 52, 56	lshpcmp 36598	. . . . . . . 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 488	. . . . . . . . . 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 36710	. . . . . . . . . 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 1372	. . . . . . . . 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 416	. . . . . . . 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 243	. . . . . . 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 3039	. . . . . 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 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑘 ∈ 𝐾) → 𝑘 ∈ 𝐾)
70	12, 5, 6, 14, 13, 27, 67, 68, 69	lkrscss 36708	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑘 ∈ 𝐾) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑘}))))
71	70	ex 416	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → (𝑘 ∈ 𝐾 → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑘})))))
72		fveq2 6663	. . . . . . . . . 10 ⊢ (𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})) → (𝐿‘𝑔) = (𝐿‘(𝐺 ∘f · (𝑉 × {𝑘}))))
73	72	sseq2d 3926	. . . . . . . . 9 ⊢ (𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})) → ((𝐿‘𝐺) ⊆ (𝐿‘𝑔) ↔ (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑘})))))
74	73	biimprcd 253	. . . . . . . 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 3207	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → (∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})) → (𝐿‘𝐺) ⊆ (𝐿‘𝑔)))
77	66, 76	impbid 215	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → ((𝐿‘𝐺) ⊆ (𝐿‘𝑔) ↔ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))))
78	77	pm5.32da 582	. . . 4 ⊢ (𝜑 → ((𝑔 ∈ 𝐹 ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) ↔ (𝑔 ∈ 𝐹 ∧ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})))))
79	4	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝑊 ∈ LMod)
80	16	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐺 ∈ 𝐹)
81		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝑘 ∈ 𝐾)
82	12, 5, 6, 14, 13, 79, 80, 81	lflvscl 36687	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → (𝐺 ∘f · (𝑉 × {𝑘})) ∈ 𝐹)
83		eleq1a 2847	. . . . . . . 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 566	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → (𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})) ↔ (𝑔 ∈ 𝐹 ∧ 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})))))
86	85	rexbidva 3220	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})) ↔ ∃𝑘 ∈ 𝐾 (𝑔 ∈ 𝐹 ∧ 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘})))))
87		r19.42v 3268	. . . . 5 ⊢ (∃𝑘 ∈ 𝐾 (𝑔 ∈ 𝐹 ∧ 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))) ↔ (𝑔 ∈ 𝐹 ∧ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))))
88	86, 87	bitr2di 291	. . . 4 ⊢ (𝜑 → ((𝑔 ∈ 𝐹 ∧ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))) ↔ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))))
89	78, 88	bitrd 282	. . 3 ⊢ (𝜑 → ((𝑔 ∈ 𝐹 ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)) ↔ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))))
90	89	abbidv 2822	. 2 ⊢ (𝜑 → {𝑔 ∣ (𝑔 ∈ 𝐹 ∧ (𝐿‘𝐺) ⊆ (𝐿‘𝑔))} = {𝑔 ∣ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))})
91	1, 90	syl5eq 2805	1 ⊢ (𝜑 → {𝑔 ∈ 𝐹 ∣ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)} = {𝑔 ∣ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘f · (𝑉 × {𝑘}))})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735   ≠ wne 2951  ∃wrex 3071  {crab 3074   ⊆ wss 3860  {csn 4525   × cxp 5526  ‘cfv 6340  (class class class)co 7156   ∘_f cof 7409  Basecbs 16554  ._rcmulr 16637  Scalarcsca 16639  0_gc0g 16784  LModclmod 19715  LVecclvec 19955  LSHypclsh 36585  LFnlclfn 36667  LKerclk 36695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7411  df-om 7586  df-1st 7699  df-2nd 7700  df-tpos 7908  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-er 8305  df-map 8424  df-en 8541  df-dom 8542  df-sdom 8543  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688  df-2 11750  df-3 11751  df-ndx 16557  df-slot 16558  df-base 16560  df-sets 16561  df-ress 16562  df-plusg 16649  df-mulr 16650  df-0g 16786  df-mgm 17931  df-sgrp 17980  df-mnd 17991  df-submnd 18036  df-grp 18185  df-minusg 18186  df-sbg 18187  df-subg 18356  df-cntz 18527  df-lsm 18841  df-cmn 18988  df-abl 18989  df-mgp 19321  df-ur 19333  df-ring 19380  df-oppr 19457  df-dvdsr 19475  df-unit 19476  df-invr 19506  df-drng 19585  df-lmod 19717  df-lss 19785  df-lsp 19825  df-lvec 19956  df-lshyp 36587  df-lfl 36668  df-lkr 36696

This theorem is referenced by:  ldual1dim  36776