lss1d - Metamath Proof Explorer

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Theorem lss1d 20949

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 2738	. 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 20864	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑘 · 𝑋) ∈ 𝑉)
13	12	3expa 1119	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑘 ∈ 𝐾) ∧ 𝑋 ∈ 𝑉) → (𝑘 · 𝑋) ∈ 𝑉)
14	13	an32s 653	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ 𝑘 ∈ 𝐾) → (𝑘 · 𝑋) ∈ 𝑉)
15		eleq1a 2832	. . . . 5 ⊢ ((𝑘 · 𝑋) ∈ 𝑉 → (𝑣 = (𝑘 · 𝑋) → 𝑣 ∈ 𝑉))
16	14, 15	syl 17	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ 𝑘 ∈ 𝐾) → (𝑣 = (𝑘 · 𝑋) → 𝑣 ∈ 𝑉))
17	16	rexlimdva 3139	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) → 𝑣 ∈ 𝑉))
18	17	abssdv 4008	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ⊆ 𝑉)
19		eqid 2737	. . . . 5 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
20	1, 3, 19	lmod0cl 20874	. . . 4 ⊢ (𝑊 ∈ LMod → (0_g‘𝐹) ∈ 𝐾)
21	20	adantr 480	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (0_g‘𝐹) ∈ 𝐾)
22		nfcv 2899	. . . 4 ⊢ Ⅎ𝑘(0_g‘𝐹)
23		nfre1 3263	. . . . . 6 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)
24	23	nfab 2905	. . . . 5 ⊢ Ⅎ𝑘{𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)}
25		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑘∅
26	24, 25	nfne 3034	. . . 4 ⊢ Ⅎ𝑘{𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅
27		biidd 262	. . . 4 ⊢ (𝑘 = (0_g‘𝐹) → ({𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅ ↔ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅))
28		ovex 7393	. . . . . 6 ⊢ (𝑘 · 𝑋) ∈ V
29	28	elabrex 7190	. . . . 5 ⊢ (𝑘 ∈ 𝐾 → (𝑘 · 𝑋) ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)})
30	29	ne0d 4283	. . . 4 ⊢ (𝑘 ∈ 𝐾 → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅)
31	22, 26, 27, 30	vtoclgaf 3520	. . 3 ⊢ ((0_g‘𝐹) ∈ 𝐾 → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅)
32	21, 31	syl 17	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅)
33		vex 3434	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
34		eqeq1 2741	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑎 → (𝑣 = (𝑘 · 𝑋) ↔ 𝑎 = (𝑘 · 𝑋)))
35	34	rexbidv 3162	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑎 → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) ↔ ∃𝑘 ∈ 𝐾 𝑎 = (𝑘 · 𝑋)))
36	33, 35	elab 3623	. . . . . . . . . 10 ⊢ (𝑎 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑘 ∈ 𝐾 𝑎 = (𝑘 · 𝑋))
37		oveq1 7367	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑦 → (𝑘 · 𝑋) = (𝑦 · 𝑋))
38	37	eqeq2d 2748	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑦 → (𝑎 = (𝑘 · 𝑋) ↔ 𝑎 = (𝑦 · 𝑋)))
39	38	cbvrexvw 3217	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ 𝐾 𝑎 = (𝑘 · 𝑋) ↔ ∃𝑦 ∈ 𝐾 𝑎 = (𝑦 · 𝑋))
40	36, 39	bitri 275	. . . . . . . . 9 ⊢ (𝑎 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑦 ∈ 𝐾 𝑎 = (𝑦 · 𝑋))
41		vex 3434	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
42		eqeq1 2741	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑏 → (𝑣 = (𝑘 · 𝑋) ↔ 𝑏 = (𝑘 · 𝑋)))
43	42	rexbidv 3162	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑏 → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) ↔ ∃𝑘 ∈ 𝐾 𝑏 = (𝑘 · 𝑋)))
44	41, 43	elab 3623	. . . . . . . . . 10 ⊢ (𝑏 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑘 ∈ 𝐾 𝑏 = (𝑘 · 𝑋))
45		oveq1 7367	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → (𝑘 · 𝑋) = (𝑧 · 𝑋))
46	45	eqeq2d 2748	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → (𝑏 = (𝑘 · 𝑋) ↔ 𝑏 = (𝑧 · 𝑋)))
47	46	cbvrexvw 3217	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ 𝐾 𝑏 = (𝑘 · 𝑋) ↔ ∃𝑧 ∈ 𝐾 𝑏 = (𝑧 · 𝑋))
48	44, 47	bitri 275	. . . . . . . . 9 ⊢ (𝑏 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑧 ∈ 𝐾 𝑏 = (𝑧 · 𝑋))
49	40, 48	anbi12i 629	. . . . . . . 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 3210	. . . . . . . 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 278	. . . . . . 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 767	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑊 ∈ LMod)
53		simprr 773	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑥 ∈ 𝐾)
54		simprll 779	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑦 ∈ 𝐾)
55		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝐹) = (._r‘𝐹)
56	1, 3, 55	lmodmcl 20859	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → (𝑥(._r‘𝐹)𝑦) ∈ 𝐾)
57	52, 53, 54, 56	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → (𝑥(._r‘𝐹)𝑦) ∈ 𝐾)
58		simprlr 780	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑧 ∈ 𝐾)
59		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐹) = (+_g‘𝐹)
60	1, 3, 59	lmodacl 20858	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ (𝑥(._r‘𝐹)𝑦) ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) → ((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) ∈ 𝐾)
61	52, 57, 58, 60	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) ∈ 𝐾)
62		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑋 ∈ 𝑉)
63		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
64	5, 63, 1, 8, 3, 59	lmodvsdir 20872	. . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . 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 20873	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑥(._r‘𝐹)𝑦) · 𝑋) = (𝑥 · (𝑦 · 𝑋)))
67	52, 53, 54, 62, 66	syl13anc 1375	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ((𝑥(._r‘𝐹)𝑦) · 𝑋) = (𝑥 · (𝑦 · 𝑋)))
68	67	oveq1d 7375	. . . . . . . . . . . . 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 2773	. . . . . . . . . . . 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 7367	. . . . . . . . . . . . 13 ⊢ (𝑘 = ((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) → (𝑘 · 𝑋) = (((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) · 𝑋))
71	70	rspceeqv 3588	. . . . . . . . . . . 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 585	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ∃𝑘 ∈ 𝐾 ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)) = (𝑘 · 𝑋))
73		oveq2 7368	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑦 · 𝑋) → (𝑥 · 𝑎) = (𝑥 · (𝑦 · 𝑋)))
74		oveq12 7369	. . . . . . . . . . . . . 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 581	. . . . . . . . . . . . 13 ⊢ ((𝑎 = (𝑦 · 𝑋) ∧ 𝑏 = (𝑧 · 𝑋)) → ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)))
76	75	eqeq1d 2739	. . . . . . . . . . . 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 3162	. . . . . . . . . . 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 247	. . . . . . . . . 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 456	. . . . . . . . 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 86	. . . . . . . 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 3195	. . . . . . 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 242	. . . . . 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 416	. . . . 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 95	. . . 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 1351	. . 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 7393	. . . 4 ⊢ ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) ∈ V
87		eqeq1 2741	. . . . 5 ⊢ (𝑣 = ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) → (𝑣 = (𝑘 · 𝑋) ↔ ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋)))
88	87	rexbidv 3162	. . . 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 3623	. . 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 234	. 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 20921	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {cab 2715 ≠ wne 2933 ∃wrex 3062 ∅c0 4274 ‘cfv 6492 (class class class)co 7360 Basecbs 17170 +_gcplusg 17211 ._rcmulr 17212 Scalarcsca 17214 ·_𝑠 cvsca 17215 0_gc0g 17393 LModclmod 20846 LSubSpclss 20917

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-plusg 17224 df-0g 17395 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18903 df-mgp 20113 df-ring 20207 df-lmod 20848 df-lss 20918

This theorem is referenced by: lspsn 20988

W3C validator