lss1d - Metamath Proof Explorer

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Theorem lss1d 20225

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 2739	. 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 20140	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑘 · 𝑋) ∈ 𝑉)
13	12	3expa 1117	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑘 ∈ 𝐾) ∧ 𝑋 ∈ 𝑉) → (𝑘 · 𝑋) ∈ 𝑉)
14	13	an32s 649	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ 𝑘 ∈ 𝐾) → (𝑘 · 𝑋) ∈ 𝑉)
15		eleq1a 2834	. . . . 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 3213	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) → 𝑣 ∈ 𝑉))
18	17	abssdv 4002	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ⊆ 𝑉)
19		eqid 2738	. . . . 5 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
20	1, 3, 19	lmod0cl 20149	. . . 4 ⊢ (𝑊 ∈ LMod → (0_g‘𝐹) ∈ 𝐾)
21	20	adantr 481	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (0_g‘𝐹) ∈ 𝐾)
22		nfcv 2907	. . . 4 ⊢ Ⅎ𝑘(0_g‘𝐹)
23		nfre1 3239	. . . . . 6 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)
24	23	nfab 2913	. . . . 5 ⊢ Ⅎ𝑘{𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)}
25		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑘∅
26	24, 25	nfne 3045	. . . 4 ⊢ Ⅎ𝑘{𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅
27		biidd 261	. . . 4 ⊢ (𝑘 = (0_g‘𝐹) → ({𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅ ↔ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅))
28		ovex 7308	. . . . . 6 ⊢ (𝑘 · 𝑋) ∈ V
29	28	elabrex 7116	. . . . 5 ⊢ (𝑘 ∈ 𝐾 → (𝑘 · 𝑋) ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)})
30	29	ne0d 4269	. . . 4 ⊢ (𝑘 ∈ 𝐾 → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅)
31	22, 26, 27, 30	vtoclgaf 3512	. . 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 3436	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
34		eqeq1 2742	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑎 → (𝑣 = (𝑘 · 𝑋) ↔ 𝑎 = (𝑘 · 𝑋)))
35	34	rexbidv 3226	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑎 → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) ↔ ∃𝑘 ∈ 𝐾 𝑎 = (𝑘 · 𝑋)))
36	33, 35	elab 3609	. . . . . . . . . 10 ⊢ (𝑎 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑘 ∈ 𝐾 𝑎 = (𝑘 · 𝑋))
37		oveq1 7282	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑦 → (𝑘 · 𝑋) = (𝑦 · 𝑋))
38	37	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑦 → (𝑎 = (𝑘 · 𝑋) ↔ 𝑎 = (𝑦 · 𝑋)))
39	38	cbvrexvw 3384	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ 𝐾 𝑎 = (𝑘 · 𝑋) ↔ ∃𝑦 ∈ 𝐾 𝑎 = (𝑦 · 𝑋))
40	36, 39	bitri 274	. . . . . . . . 9 ⊢ (𝑎 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑦 ∈ 𝐾 𝑎 = (𝑦 · 𝑋))
41		vex 3436	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
42		eqeq1 2742	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑏 → (𝑣 = (𝑘 · 𝑋) ↔ 𝑏 = (𝑘 · 𝑋)))
43	42	rexbidv 3226	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑏 → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) ↔ ∃𝑘 ∈ 𝐾 𝑏 = (𝑘 · 𝑋)))
44	41, 43	elab 3609	. . . . . . . . . 10 ⊢ (𝑏 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑘 ∈ 𝐾 𝑏 = (𝑘 · 𝑋))
45		oveq1 7282	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → (𝑘 · 𝑋) = (𝑧 · 𝑋))
46	45	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → (𝑏 = (𝑘 · 𝑋) ↔ 𝑏 = (𝑧 · 𝑋)))
47	46	cbvrexvw 3384	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ 𝐾 𝑏 = (𝑘 · 𝑋) ↔ ∃𝑧 ∈ 𝐾 𝑏 = (𝑧 · 𝑋))
48	44, 47	bitri 274	. . . . . . . . 9 ⊢ (𝑏 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑧 ∈ 𝐾 𝑏 = (𝑧 · 𝑋))
49	40, 48	anbi12i 627	. . . . . . . 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 3294	. . . . . . . 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 277	. . . . . . 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 764	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑊 ∈ LMod)
53		simprr 770	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑥 ∈ 𝐾)
54		simprll 776	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑦 ∈ 𝐾)
55		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝐹) = (._r‘𝐹)
56	1, 3, 55	lmodmcl 20135	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → (𝑥(._r‘𝐹)𝑦) ∈ 𝐾)
57	52, 53, 54, 56	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → (𝑥(._r‘𝐹)𝑦) ∈ 𝐾)
58		simprlr 777	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑧 ∈ 𝐾)
59		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐹) = (+_g‘𝐹)
60	1, 3, 59	lmodacl 20134	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ (𝑥(._r‘𝐹)𝑦) ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) → ((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) ∈ 𝐾)
61	52, 57, 58, 60	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) ∈ 𝐾)
62		simplr 766	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑋 ∈ 𝑉)
63		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
64	5, 63, 1, 8, 3, 59	lmodvsdir 20147	. . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . 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 20148	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑥(._r‘𝐹)𝑦) · 𝑋) = (𝑥 · (𝑦 · 𝑋)))
67	52, 53, 54, 62, 66	syl13anc 1371	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ((𝑥(._r‘𝐹)𝑦) · 𝑋) = (𝑥 · (𝑦 · 𝑋)))
68	67	oveq1d 7290	. . . . . . . . . . . . 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 2779	. . . . . . . . . . . 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 7282	. . . . . . . . . . . . 13 ⊢ (𝑘 = ((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) → (𝑘 · 𝑋) = (((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) · 𝑋))
71	70	rspceeqv 3575	. . . . . . . . . . . 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 584	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ∃𝑘 ∈ 𝐾 ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)) = (𝑘 · 𝑋))
73		oveq2 7283	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑦 · 𝑋) → (𝑥 · 𝑎) = (𝑥 · (𝑦 · 𝑋)))
74		oveq12 7284	. . . . . . . . . . . . . 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 580	. . . . . . . . . . . . 13 ⊢ ((𝑎 = (𝑦 · 𝑋) ∧ 𝑏 = (𝑧 · 𝑋)) → ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)))
76	75	eqeq1d 2740	. . . . . . . . . . . 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 3226	. . . . . . . . . . 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 246	. . . . . . . . . 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 457	. . . . . . . . 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 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	syl5bi 241	. . . . . 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 417	. . . . 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 1348	. . 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 7308	. . . 4 ⊢ ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) ∈ V
87		eqeq1 2742	. . . . 5 ⊢ (𝑣 = ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) → (𝑣 = (𝑘 · 𝑋) ↔ ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋)))
88	87	rexbidv 3226	. . . 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 3609	. . 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 233	. 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 20197	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 {cab 2715 ≠ wne 2943 ∃wrex 3065 ∅c0 4256 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 +_gcplusg 16962 ._rcmulr 16963 Scalarcsca 16965 ·_𝑠 cvsca 16966 0_gc0g 17150 LModclmod 20123 LSubSpclss 20193

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-mgp 19721 df-ring 19785 df-lmod 20125 df-lss 20194

This theorem is referenced by: lspsn 20264

W3C validator