lss1d - Metamath Proof Explorer

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Theorem lss1d 20841

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 2729	. 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 20755	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑘 · 𝑋) ∈ 𝑉)
13	12	3expa 1116	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑘 ∈ 𝐾) ∧ 𝑋 ∈ 𝑉) → (𝑘 · 𝑋) ∈ 𝑉)
14	13	an32s 651	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ 𝑘 ∈ 𝐾) → (𝑘 · 𝑋) ∈ 𝑉)
15		eleq1a 2824	. . . . 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 3151	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) → 𝑣 ∈ 𝑉))
18	17	abssdv 4062	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ⊆ 𝑉)
19		eqid 2728	. . . . 5 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
20	1, 3, 19	lmod0cl 20765	. . . 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 3278	. . . . . 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 3039	. . . 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 7448	. . . . . 6 ⊢ (𝑘 · 𝑋) ∈ V
29	28	elabrex 7247	. . . . 5 ⊢ (𝑘 ∈ 𝐾 → (𝑘 · 𝑋) ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)})
30	29	ne0d 4332	. . . 4 ⊢ (𝑘 ∈ 𝐾 → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ≠ ∅)
31	22, 26, 27, 30	vtoclgaf 3561	. . 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 3474	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
34		eqeq1 2732	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑎 → (𝑣 = (𝑘 · 𝑋) ↔ 𝑎 = (𝑘 · 𝑋)))
35	34	rexbidv 3174	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑎 → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) ↔ ∃𝑘 ∈ 𝐾 𝑎 = (𝑘 · 𝑋)))
36	33, 35	elab 3666	. . . . . . . . . 10 ⊢ (𝑎 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑘 ∈ 𝐾 𝑎 = (𝑘 · 𝑋))
37		oveq1 7422	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑦 → (𝑘 · 𝑋) = (𝑦 · 𝑋))
38	37	eqeq2d 2739	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑦 → (𝑎 = (𝑘 · 𝑋) ↔ 𝑎 = (𝑦 · 𝑋)))
39	38	cbvrexvw 3231	. . . . . . . . . 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 3474	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
42		eqeq1 2732	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑏 → (𝑣 = (𝑘 · 𝑋) ↔ 𝑏 = (𝑘 · 𝑋)))
43	42	rexbidv 3174	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑏 → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) ↔ ∃𝑘 ∈ 𝐾 𝑏 = (𝑘 · 𝑋)))
44	41, 43	elab 3666	. . . . . . . . . 10 ⊢ (𝑏 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑘 ∈ 𝐾 𝑏 = (𝑘 · 𝑋))
45		oveq1 7422	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → (𝑘 · 𝑋) = (𝑧 · 𝑋))
46	45	eqeq2d 2739	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → (𝑏 = (𝑘 · 𝑋) ↔ 𝑏 = (𝑧 · 𝑋)))
47	46	cbvrexvw 3231	. . . . . . . . . 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 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 3222	. . . . . . . 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 766	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑊 ∈ LMod)
53		simprr 772	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑥 ∈ 𝐾)
54		simprll 778	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑦 ∈ 𝐾)
55		eqid 2728	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝐹) = (._r‘𝐹)
56	1, 3, 55	lmodmcl 20750	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → (𝑥(._r‘𝐹)𝑦) ∈ 𝐾)
57	52, 53, 54, 56	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → (𝑥(._r‘𝐹)𝑦) ∈ 𝐾)
58		simprlr 779	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑧 ∈ 𝐾)
59		eqid 2728	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐹) = (+_g‘𝐹)
60	1, 3, 59	lmodacl 20749	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ (𝑥(._r‘𝐹)𝑦) ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) → ((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) ∈ 𝐾)
61	52, 57, 58, 60	syl3anc 1369	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) ∈ 𝐾)
62		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → 𝑋 ∈ 𝑉)
63		eqid 2728	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
64	5, 63, 1, 8, 3, 59	lmodvsdir 20763	. . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . 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 20764	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑥(._r‘𝐹)𝑦) · 𝑋) = (𝑥 · (𝑦 · 𝑋)))
67	52, 53, 54, 62, 66	syl13anc 1370	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ((𝑥(._r‘𝐹)𝑦) · 𝑋) = (𝑥 · (𝑦 · 𝑋)))
68	67	oveq1d 7430	. . . . . . . . . . . . 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 2769	. . . . . . . . . . . 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 7422	. . . . . . . . . . . . 13 ⊢ (𝑘 = ((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) → (𝑘 · 𝑋) = (((𝑥(._r‘𝐹)𝑦)(+_g‘𝐹)𝑧) · 𝑋))
71	70	rspceeqv 3630	. . . . . . . . . . . 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 583	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) ∧ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) ∧ 𝑥 ∈ 𝐾)) → ∃𝑘 ∈ 𝐾 ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)) = (𝑘 · 𝑋))
73		oveq2 7423	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑦 · 𝑋) → (𝑥 · 𝑎) = (𝑥 · (𝑦 · 𝑋)))
74		oveq12 7424	. . . . . . . . . . . . . 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 579	. . . . . . . . . . . . 13 ⊢ ((𝑎 = (𝑦 · 𝑋) ∧ 𝑏 = (𝑧 · 𝑋)) → ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = ((𝑥 · (𝑦 · 𝑋))(+_g‘𝑊)(𝑧 · 𝑋)))
76	75	eqeq1d 2730	. . . . . . . . . . . 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 3174	. . . . . . . . . . 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 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 3207	. . . . . . 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 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 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 1347	. . 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 7448	. . . 4 ⊢ ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) ∈ V
87		eqeq1 2732	. . . . 5 ⊢ (𝑣 = ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) → (𝑣 = (𝑘 · 𝑋) ↔ ((𝑥 · 𝑎)(+_g‘𝑊)𝑏) = (𝑘 · 𝑋)))
88	87	rexbidv 3174	. . . 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 3666	. . 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 20813	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 1085 = wceq 1534 ∈ wcel 2099 {cab 2705 ≠ wne 2936 ∃wrex 3066 ∅c0 4319 ‘cfv 6543 (class class class)co 7415 Basecbs 17174 +_gcplusg 17227 ._rcmulr 17228 Scalarcsca 17230 ·_𝑠 cvsca 17231 0_gc0g 17415 LModclmod 20737 LSubSpclss 20809

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18887 df-mgp 20069 df-ring 20169 df-lmod 20739 df-lss 20810

This theorem is referenced by: lspsn 20880

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