lss1d - Metamath Proof Explorer

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Theorem lss1d 21289

Description: One-dimensional subspace (or zero-dimensional if 𝑋 is the zero vector). (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)

**Hypotheses**
Ref	Expression
lss1d.v	⊢ 𝑉 = (Base‘𝑊)
lss1d.f	⊢ 𝐹 = (Scalar‘𝑊)
lss1d.t	⊢ · = ( ·_𝑠 ‘𝑊)
lss1d.k	⊢ 𝐾 = (Base‘𝐹)
lss1d.s	⊢ 𝑆 = (LSubSp‘𝑊)

**Assertion**
Ref	Expression
lss1d	⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ∈ 𝑆)

Distinct variable groups: 𝑣,𝑘,𝐾 · ,𝑘,𝑣 𝑘,𝑉,𝑣 𝑘,𝐹 𝑘,𝑊,𝑣 𝑘,𝑋,𝑣 Allowed substitution hints: 𝑆(𝑣, 𝑘) 𝐹(𝑣)

Proof of Theorem lss1d Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lss1d.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
2	1	a1i 11	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝐹 = (Scalar‘𝑊))
3		lss1d.k	. . 3 ⊢ 𝐾 = (Base‘𝐹)
4	3	a1i 11	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝐾 = (Base‘𝐹))
5		lss1d.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
6	5	a1i 11	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑉 = (Base‘𝑊))
7		eqidd 2764	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (+_g‘𝑊) = (+_g‘𝑊))
8		lss1d.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝑊)
9	8	a1i 11	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → · = ( ·_𝑠 ‘𝑊))
10		lss1d.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑊)
11	10	a1i 11	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑆 = (LSubSp‘𝑊))
12	5, 1, 8, 3	lmodvscl 21204	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑘 · 𝑋) ∈ 𝑉)
13	12	3expa 1136	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑘 ∈ 𝐾) ∧ 𝑋 ∈ 𝑉) → (𝑘 · 𝑋) ∈ 𝑉)
14	13	an32s 665	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ 𝑘 ∈ 𝐾) → (𝑘 · 𝑋) ∈ 𝑉)
15		eleq1a 2859	. . . . 5 ⊢ ((𝑘 · 𝑋) ∈ 𝑉 → (𝑣 = (𝑘 · 𝑋) → 𝑣 ∈ 𝑉))
16	14, 15	syl 18	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ 𝑘 ∈ 𝐾) → (𝑣 = (𝑘 · 𝑋) → 𝑣 ∈ 𝑉))
17	16	rexlimdva 3167	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) → 𝑣 ∈ 𝑉))
18	17	abssdv 3995	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ⊆ 𝑉)
19		eqid 2763	. . . . 5 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
20	1, 3, 19	lmod0cl 21214	. . . 4 ⊢ (𝑊 ∈ LMod → (0_g‘𝐹) ∈ 𝐾)
21	20	adantr 486	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (0_g‘𝐹) ∈ 𝐾)
22		nfcv 2926	. . . 4 ⊢ Ⅎ𝑘(0_g‘𝐹)
23		nfre1 3291	. . . . . 6 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)
24	23	nfab 2932	. . . . 5 ⊢ Ⅎ𝑘{𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)}
25		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑘∅
26	24, 25	nfne 3062	. . . 4 ⊢ Ⅎ𝑘{𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅
27		biidd 265	. . . 4 ⊢ (𝑘 = (0_g‘𝐹) → ({𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅ ↔ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅))
28		ovex 7484	. . . . . 6 ⊢ (𝑘 · 𝑋) ∈ V
29	28	elabrex 7277	. . . . 5 ⊢ (𝑘 ∈ 𝐾 → (𝑘 · 𝑋) ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)})
30	29	ne0d 4291	. . . 4 ⊢ (𝑘 ∈ 𝐾 → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅)
31	22, 26, 27, 30	vtoclgaf 3539	. . 3 ⊢ ((0_g‘𝐹) ∈ 𝐾 → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅)
32	21, 31	syl 18	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅)
33		vex 3458	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
34		eqeq1 2767	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑎 → (𝑣 = (𝑘 · 𝑋) ↔ 𝑎 = (𝑘 · 𝑋)))
35	34	rexbidv 3190	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑎 → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) ↔ ∃𝑘 ∈ 𝐾 𝑎 = (𝑘 · 𝑋)))
36	33, 35	elab 3636	. . . . . . . . . 10 ⊢ (𝑎 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑘 ∈ 𝐾 𝑎 = (𝑘 · 𝑋))
37		oveq1 7458	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑦 → (𝑘 · 𝑋) = (𝑦 · 𝑋))
38	37	eqeq2d 2774	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑦 → (𝑎 = (𝑘 · 𝑋) ↔ 𝑎 = (𝑦 · 𝑋)))
39	38	cbvrexvw 3245	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ 𝐾 𝑎 = (𝑘 · 𝑋) ↔ ∃𝑦 ∈ 𝐾 𝑎 = (𝑦 · 𝑋))
40	36, 39	bitri 278	. . . . . . . . 9 ⊢ (𝑎 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑦 ∈ 𝐾 𝑎 = (𝑦 · 𝑋))
41		vex 3458	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
42		eqeq1 2767	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑏 → (𝑣 = (𝑘 · 𝑋) ↔ 𝑏 = (𝑘 · 𝑋)))
43	42	rexbidv 3190	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑏 → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) ↔ ∃𝑘 ∈ 𝐾 𝑏 = (𝑘 · 𝑋)))
44	41, 43	elab 3636	. . . . . . . . . 10 ⊢ (𝑏 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑘 ∈ 𝐾 𝑏 = (𝑘 · 𝑋))
45		oveq1 7458	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → (𝑘 · 𝑋) = (𝑧 · 𝑋))
46	45	eqeq2d 2774	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → (𝑏 = (𝑘 · 𝑋) ↔ 𝑏 = (𝑧 · 𝑋)))
47	46	cbvrexvw 3245	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ 𝐾 𝑏 = (𝑘 · 𝑋) ↔ ∃𝑧 ∈ 𝐾 𝑏 = (𝑧 · 𝑋))
48	44, 47	bitri 278	. . . . . . . . 9 ⊢ (𝑏 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑧 ∈ 𝐾 𝑏 = (𝑧 · 𝑋))
49	40, 48	anbi12i 640	. . . . . . . 8 ⊢ ((𝑎 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ∧ 𝑏 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)}) ↔ (∃𝑦 ∈ 𝐾 𝑎 = (𝑦 · 𝑋) ∧ ∃𝑧 ∈ 𝐾 𝑏 = (𝑧 · 𝑋)))
50		reeanv 3238	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐾 ∃𝑧 ∈ 𝐾 (𝑎 = (𝑦 · 𝑋) ∧ 𝑏 = (𝑧 · 𝑋)) ↔ (∃𝑦 ∈ 𝐾 𝑎 = (𝑦 · 𝑋) ∧ ∃𝑧 ∈ 𝐾 𝑏 = (𝑧 · 𝑋)))
51	49, 50	bitr4i 281	. . . . . . 7 ⊢ ((𝑎 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ∧ 𝑏 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)}) ↔ ∃𝑦 ∈ 𝐾 ∃𝑧 ∈ 𝐾 (𝑎 = (𝑦 · 𝑋) ∧ 𝑏 = (𝑧 · 𝑋)))
52		simpll 779	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑊 ∈ LMod)
53		simprr 785	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑥 ∈ 𝐾)
54		simprll 791	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑦 ∈ 𝐾)
55		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝐹) = (._r‘𝐹)
56	1, 3, 55	lmodmcl 21199	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → (𝑥(._r‘𝐹)𝑦) ∈ 𝐾)
57	52, 53, 54, 56	syl3anc 1399	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → (𝑥(._r‘𝐹)𝑦) ∈ 𝐾)
58		simprlr 792	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑧 ∈ 𝐾)
59		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐹) = (+_g‘𝐹)
60	1, 3, 59	lmodacl 21198	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ (𝑥(._r‘𝐹)𝑦) ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) → ((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) ∈ 𝐾)
61	52, 57, 58, 60	syl3anc 1399	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) ∈ 𝐾)
62		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑋 ∈ 𝑉)
63		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
64	5, 63, 1, 8, 3, 59	lmodvsdir 21212	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ ((𝑥(._r‘𝐹)𝑦) ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) · 𝑋) = (((𝑥(._r‘𝐹)𝑦) · 𝑋)(+_g‘𝑊)(𝑧 · 𝑋)))
65	52, 57, 58, 62, 64	syl13anc 1400	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → (((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) · 𝑋) = (((𝑥(._r‘𝐹)𝑦) · 𝑋)(+_g‘𝑊)(𝑧 · 𝑋)))
66	5, 1, 8, 3, 55	lmodvsass 21213	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑥(._r‘𝐹)𝑦) · 𝑋) = (𝑥 · (𝑦 · 𝑋)))
67	52, 53, 54, 62, 66	syl13anc 1400	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ((𝑥(._r‘𝐹)𝑦) · 𝑋) = (𝑥 · (𝑦 · 𝑋)))
68	67	oveq1d 7466	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → (((𝑥(._r‘𝐹)𝑦) · 𝑋)(+_g‘𝑊)(𝑧 · 𝑋)) = ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)))
69	65, 68	eqtr2d 2799	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)) = (((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) · 𝑋))
70		oveq1 7458	. . . . . . . . . . . . 13 ⊢ (𝑘 = ((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) → (𝑘 · 𝑋) = (((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) · 𝑋))
71	70	rspceeqv 3602	. . . . . . . . . . . 12 ⊢ ((((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) ∈ 𝐾 ∧ ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)) = (((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) · 𝑋)) → ∃𝑘 ∈ 𝐾 ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)) = (𝑘 · 𝑋))
72	61, 69, 71	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ∃𝑘 ∈ 𝐾 ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)) = (𝑘 · 𝑋))
73		oveq2 7459	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑦 · 𝑋) → (𝑥 · 𝑎) = (𝑥 · (𝑦 · 𝑋)))
74		oveq12 7460	. . . . . . . . . . . . . 14 ⊢ (((𝑥 · 𝑎) = (𝑥 · (𝑦 · 𝑋)) ∧ 𝑏 = (𝑧 · 𝑋)) → ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)))
75	73, 74	sylan 592	. . . . . . . . . . . . 13 ⊢ ((𝑎 = (𝑦 · 𝑋) ∧ 𝑏 = (𝑧 · 𝑋)) → ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)))
76	75	eqeq1d 2765	. . . . . . . . . . . 12 ⊢ ((𝑎 = (𝑦 · 𝑋) ∧ 𝑏 = (𝑧 · 𝑋)) → (((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋) ↔ ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)) = (𝑘 · 𝑋)))
77	76	rexbidv 3190	. . . . . . . . . . 11 ⊢ ((𝑎 = (𝑦 · 𝑋) ∧ 𝑏 = (𝑧 · 𝑋)) → (∃𝑘 ∈ 𝐾 ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋) ↔ ∃𝑘 ∈ 𝐾 ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)) = (𝑘 · 𝑋)))
78	72, 77	syl5ibrcom 250	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ((𝑎 = (𝑦 · 𝑋) ∧ 𝑏 = (𝑧 · 𝑋)) → ∃𝑘 ∈ 𝐾 ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋)))
79	78	expr 462	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (𝑥 ∈ 𝐾 → ((𝑎 = (𝑦 · 𝑋) ∧ 𝑏 = (𝑧 · 𝑋)) → ∃𝑘 ∈ 𝐾 ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋))))
80	79	com23 87	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ (𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑎 = (𝑦 · 𝑋) ∧ 𝑏 = (𝑧 · 𝑋)) → (𝑥 ∈ 𝐾 → ∃𝑘 ∈ 𝐾 ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋))))
81	80	rexlimdvva 3223	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (∃𝑦 ∈ 𝐾 ∃𝑧 ∈ 𝐾 (𝑎 = (𝑦 · 𝑋) ∧ 𝑏 = (𝑧 · 𝑋)) → (𝑥 ∈ 𝐾 → ∃𝑘 ∈ 𝐾 ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋))))
82	51, 81	biimtrid 245	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑎 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ∧ 𝑏 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)}) → (𝑥 ∈ 𝐾 → ∃𝑘 ∈ 𝐾 ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋))))
83	82	expcomd 422	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑏 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} → (𝑎 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} → (𝑥 ∈ 𝐾 → ∃𝑘 ∈ 𝐾 ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋)))))
84	83	com24 96	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ 𝐾 → (𝑎 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} → (𝑏 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} → ∃𝑘 ∈ 𝐾 ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋)))))
85	84	3imp2 1368	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ (𝑥 ∈ 𝐾 ∧ 𝑎 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ∧ 𝑏 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)})) → ∃𝑘 ∈ 𝐾 ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋))
86		ovex 7484	. . . 4 ⊢ ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) ∈ V
87		eqeq1 2767	. . . . 5 ⊢ (𝑣 = ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) → (𝑣 = (𝑘 · 𝑋) ↔ ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋)))
88	87	rexbidv 3190	. . . 4 ⊢ (𝑣 = ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) ↔ ∃𝑘 ∈ 𝐾 ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋)))
89	86, 88	elab 3636	. . 3 ⊢ (((𝑥 · 𝑎)(+_g‘𝑊)𝑏) ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑘 ∈ 𝐾 ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋))
90	85, 89	sylibr 237	. 2 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ (𝑥 ∈ 𝐾 ∧ 𝑎 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ∧ 𝑏 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)})) → ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)})
91	2, 4, 6, 7, 9, 11, 18, 32, 90	islssd 21261	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ∈ 𝑆)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 {cab 2741 ≠ wne 2959 ∃wrex 3090 ∅c0 4281 ‘cfv 6566 (class class class)co 7451 Basecbs 17431 +_gcplusg 17472 ._rcmulr 17473 Scalarcsca 17475 ·_𝑠 cvsca 17476 0_gc0g 17660 LModclmod 21186 LSubSpclss 21257

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-plusg 17486 df-0g 17662 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-grp 19197 df-mgp 20412 df-ring 20512 df-lmod 21188 df-lss 21258

This theorem is used by: lspsn 21328

W3C validator