lss1d - Metamath Proof Explorer

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Theorem lss1d 20960

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 2741	. 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 20875	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑘 · 𝑋) ∈ 𝑉)
13	12	3expa 1124	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑘 ∈ 𝐾) ∧ 𝑋 ∈ 𝑉) → (𝑘 · 𝑋) ∈ 𝑉)
14	13	an32s 658	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ 𝑘 ∈ 𝐾) → (𝑘 · 𝑋) ∈ 𝑉)
15		eleq1a 2835	. . . . 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 3141	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) → 𝑣 ∈ 𝑉))
18	17	abssdv 4005	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ⊆ 𝑉)
19		eqid 2740	. . . . 5 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
20	1, 3, 19	lmod0cl 20885	. . . 4 ⊢ (𝑊 ∈ LMod → (0_g‘𝐹) ∈ 𝐾)
21	20	adantr 481	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (0_g‘𝐹) ∈ 𝐾)
22		nfcv 2902	. . . 4 ⊢ Ⅎ𝑘(0_g‘𝐹)
23		nfre1 3265	. . . . . 6 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)
24	23	nfab 2908	. . . . 5 ⊢ Ⅎ𝑘{𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)}
25		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑘∅
26	24, 25	nfne 3036	. . . 4 ⊢ Ⅎ𝑘{𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅
27		biidd 263	. . . 4 ⊢ (𝑘 = (0_g‘𝐹) → ({𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅ ↔ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅))
28		ovex 7396	. . . . . 6 ⊢ (𝑘 · 𝑋) ∈ V
29	28	elabrex 7193	. . . . 5 ⊢ (𝑘 ∈ 𝐾 → (𝑘 · 𝑋) ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)})
30	29	ne0d 4277	. . . 4 ⊢ (𝑘 ∈ 𝐾 → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅)
31	22, 26, 27, 30	vtoclgaf 3522	. . 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 2744	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑎 → (𝑣 = (𝑘 · 𝑋) ↔ 𝑎 = (𝑘 · 𝑋)))
35	34	rexbidv 3164	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑎 → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) ↔ ∃𝑘 ∈ 𝐾 𝑎 = (𝑘 · 𝑋)))
36	33, 35	elab 3624	. . . . . . . . . 10 ⊢ (𝑎 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑘 ∈ 𝐾 𝑎 = (𝑘 · 𝑋))
37		oveq1 7370	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑦 → (𝑘 · 𝑋) = (𝑦 · 𝑋))
38	37	eqeq2d 2751	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑦 → (𝑎 = (𝑘 · 𝑋) ↔ 𝑎 = (𝑦 · 𝑋)))
39	38	cbvrexvw 3219	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ 𝐾 𝑎 = (𝑘 · 𝑋) ↔ ∃𝑦 ∈ 𝐾 𝑎 = (𝑦 · 𝑋))
40	36, 39	bitri 276	. . . . . . . . 9 ⊢ (𝑎 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑦 ∈ 𝐾 𝑎 = (𝑦 · 𝑋))
41		vex 3436	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
42		eqeq1 2744	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑏 → (𝑣 = (𝑘 · 𝑋) ↔ 𝑏 = (𝑘 · 𝑋)))
43	42	rexbidv 3164	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑏 → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) ↔ ∃𝑘 ∈ 𝐾 𝑏 = (𝑘 · 𝑋)))
44	41, 43	elab 3624	. . . . . . . . . 10 ⊢ (𝑏 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑘 ∈ 𝐾 𝑏 = (𝑘 · 𝑋))
45		oveq1 7370	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → (𝑘 · 𝑋) = (𝑧 · 𝑋))
46	45	eqeq2d 2751	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → (𝑏 = (𝑘 · 𝑋) ↔ 𝑏 = (𝑧 · 𝑋)))
47	46	cbvrexvw 3219	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ 𝐾 𝑏 = (𝑘 · 𝑋) ↔ ∃𝑧 ∈ 𝐾 𝑏 = (𝑧 · 𝑋))
48	44, 47	bitri 276	. . . . . . . . 9 ⊢ (𝑏 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑧 ∈ 𝐾 𝑏 = (𝑧 · 𝑋))
49	40, 48	anbi12i 634	. . . . . . . 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 3212	. . . . . . . 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 279	. . . . . . 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 772	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑊 ∈ LMod)
53		simprr 778	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑥 ∈ 𝐾)
54		simprll 784	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑦 ∈ 𝐾)
55		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝐹) = (._r‘𝐹)
56	1, 3, 55	lmodmcl 20870	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → (𝑥(._r‘𝐹)𝑦) ∈ 𝐾)
57	52, 53, 54, 56	syl3anc 1379	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → (𝑥(._r‘𝐹)𝑦) ∈ 𝐾)
58		simprlr 785	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑧 ∈ 𝐾)
59		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐹) = (+_g‘𝐹)
60	1, 3, 59	lmodacl 20869	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ (𝑥(._r‘𝐹)𝑦) ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) → ((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) ∈ 𝐾)
61	52, 57, 58, 60	syl3anc 1379	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) ∈ 𝐾)
62		simplr 774	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑋 ∈ 𝑉)
63		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
64	5, 63, 1, 8, 3, 59	lmodvsdir 20883	. . . . . . . . . . . . . 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 1380	. . . . . . . . . . . . 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 20884	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑥(._r‘𝐹)𝑦) · 𝑋) = (𝑥 · (𝑦 · 𝑋)))
67	52, 53, 54, 62, 66	syl13anc 1380	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ((𝑥(._r‘𝐹)𝑦) · 𝑋) = (𝑥 · (𝑦 · 𝑋)))
68	67	oveq1d 7378	. . . . . . . . . . . . 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 2776	. . . . . . . . . . . 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 7370	. . . . . . . . . . . . 13 ⊢ (𝑘 = ((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) → (𝑘 · 𝑋) = (((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) · 𝑋))
71	70	rspceeqv 3590	. . . . . . . . . . . 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 590	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ∃𝑘 ∈ 𝐾 ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)) = (𝑘 · 𝑋))
73		oveq2 7371	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑦 · 𝑋) → (𝑥 · 𝑎) = (𝑥 · (𝑦 · 𝑋)))
74		oveq12 7372	. . . . . . . . . . . . . 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 586	. . . . . . . . . . . . 13 ⊢ ((𝑎 = (𝑦 · 𝑋) ∧ 𝑏 = (𝑧 · 𝑋)) → ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)))
76	75	eqeq1d 2742	. . . . . . . . . . . 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 3164	. . . . . . . . . . 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 248	. . . . . . . . . 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 3197	. . . . . . 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 243	. . . . . 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 1356	. . 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 7396	. . . 4 ⊢ ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) ∈ V
87		eqeq1 2744	. . . . 5 ⊢ (𝑣 = ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) → (𝑣 = (𝑘 · 𝑋) ↔ ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋)))
88	87	rexbidv 3164	. . . 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 3624	. . 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 235	. 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 20932	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 1092 = wceq 1547 ∈ wcel 2119 {cab 2718 ≠ wne 2935 ∃wrex 3064 ∅c0 4268 ‘cfv 6492 (class class class)co 7363 Basecbs 17177 +_gcplusg 17218 ._rcmulr 17219 Scalarcsca 17221 ·_𝑠 cvsca 17222 0_gc0g 17400 LModclmod 20857 LSubSpclss 20928

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 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 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-plusg 17231 df-0g 17402 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-grp 18910 df-mgp 20120 df-ring 20214 df-lmod 20859 df-lss 20929

This theorem is referenced by: lspsn 20999

W3C validator