dochkr1 - Mathbox for Norm Megill

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Theorem dochkr1 40862

Description: A nonzero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 38453. (Contributed by NM, 2-Jan-2015.)

**Hypotheses**
Ref	Expression
dochkr1.h	⊢ 𝐻 = (LHyp‘𝐾)
dochkr1.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
dochkr1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochkr1.v	⊢ 𝑉 = (Base‘𝑈)
dochkr1.r	⊢ 𝑅 = (Scalar‘𝑈)
dochkr1.z	⊢ 0 = (0_g‘𝑈)
dochkr1.i	⊢ 1 = (1_r‘𝑅)
dochkr1.f	⊢ 𝐹 = (LFnl‘𝑈)
dochkr1.l	⊢ 𝐿 = (LKer‘𝑈)
dochkr1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dochkr1.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
dochkr1.n	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)

**Assertion**
Ref	Expression
dochkr1	⊢ (𝜑 → ∃𝑥 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })(𝐺‘𝑥) = 1 )

Distinct variable groups: 𝑥, 0 𝑥,𝐺 𝑥,𝐿 𝑥,𝑅 𝑥,𝑈 𝑥, ⊥ 𝑥, 1 Allowed substitution hints: 𝜑(𝑥) 𝐹(𝑥) 𝐻(𝑥) 𝐾(𝑥) 𝑉(𝑥) 𝑊(𝑥)

Proof of Theorem dochkr1 Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2726	. . . 4 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
2		eqid 2726	. . . 4 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
3		dochkr1.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4		dochkr1.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		dochkr1.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	3, 4, 5	dvhlmod 40494	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		dochkr1.n	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)
8		dochkr1.o	. . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
9		dochkr1.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
10		dochkr1.f	. . . . . 6 ⊢ 𝐹 = (LFnl‘𝑈)
11		dochkr1.l	. . . . . 6 ⊢ 𝐿 = (LKer‘𝑈)
12		dochkr1.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
13	3, 8, 4, 9, 2, 10, 11, 5, 12	dochkrsat2 40840	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈)))
14	7, 13	mpbid 231	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈))
15	1, 2, 6, 14	lsateln0 38378	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))𝑧 ≠ (0_g‘𝑈))
16		dochkr1.r	. . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈)
17		eqid 2726	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
18	5	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) ∧ 𝑧 ≠ (0_g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
19	12	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) ∧ 𝑧 ≠ (0_g‘𝑈)) → 𝐺 ∈ 𝐹)
20		eldifsn 4785	. . . . . . . 8 ⊢ (𝑧 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ {(0_g‘𝑈)}) ↔ (𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ 𝑧 ≠ (0_g‘𝑈)))
21	20	biimpri 227	. . . . . . 7 ⊢ ((𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ 𝑧 ≠ (0_g‘𝑈)) → 𝑧 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ {(0_g‘𝑈)}))
22	21	adantll 711	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) ∧ 𝑧 ≠ (0_g‘𝑈)) → 𝑧 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ {(0_g‘𝑈)}))
23	3, 8, 4, 9, 16, 17, 1, 10, 11, 18, 19, 22	dochfln0 40861	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) ∧ 𝑧 ≠ (0_g‘𝑈)) → (𝐺‘𝑧) ≠ (0_g‘𝑅))
24	23	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) → (𝑧 ≠ (0_g‘𝑈) → (𝐺‘𝑧) ≠ (0_g‘𝑅)))
25	24	reximdva 3162	. . 3 ⊢ (𝜑 → (∃𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))𝑧 ≠ (0_g‘𝑈) → ∃𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))(𝐺‘𝑧) ≠ (0_g‘𝑅)))
26	15, 25	mpd 15	. 2 ⊢ (𝜑 → ∃𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))(𝐺‘𝑧) ≠ (0_g‘𝑅))
27	9, 10, 11, 6, 12	lkrssv 38479	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉)
28		eqid 2726	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
29	3, 4, 9, 28, 8	dochlss 40738	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSubSp‘𝑈))
30	5, 27, 29	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSubSp‘𝑈))
31	6, 30	jca 511	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∈ LMod ∧ ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSubSp‘𝑈)))
32	31	3ad2ant1 1130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → (𝑈 ∈ LMod ∧ ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSubSp‘𝑈)))
33	3, 4, 5	dvhlvec 40493	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LVec)
34	33	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → 𝑈 ∈ LVec)
35	16	lvecdrng 20953	. . . . . . . . 9 ⊢ (𝑈 ∈ LVec → 𝑅 ∈ DivRing)
36	34, 35	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → 𝑅 ∈ DivRing)
37	6	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → 𝑈 ∈ LMod)
38	12	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → 𝐺 ∈ 𝐹)
39	3, 4, 9, 8	dochssv 40739	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉)
40	5, 27, 39	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉)
41	40	sselda 3977	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) → 𝑧 ∈ 𝑉)
42	41	3adant3 1129	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → 𝑧 ∈ 𝑉)
43		eqid 2726	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
44	16, 43, 9, 10	lflcl 38447	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑧 ∈ 𝑉) → (𝐺‘𝑧) ∈ (Base‘𝑅))
45	37, 38, 42, 44	syl3anc 1368	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → (𝐺‘𝑧) ∈ (Base‘𝑅))
46		simp3 1135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → (𝐺‘𝑧) ≠ (0_g‘𝑅))
47		eqid 2726	. . . . . . . . 9 ⊢ (inv_r‘𝑅) = (inv_r‘𝑅)
48	43, 17, 47	drnginvrcl 20609	. . . . . . . 8 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑧) ∈ (Base‘𝑅) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → ((inv_r‘𝑅)‘(𝐺‘𝑧)) ∈ (Base‘𝑅))
49	36, 45, 46, 48	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → ((inv_r‘𝑅)‘(𝐺‘𝑧)) ∈ (Base‘𝑅))
50		simp2 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)))
51	49, 50	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → (((inv_r‘𝑅)‘(𝐺‘𝑧)) ∈ (Base‘𝑅) ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))))
52		eqid 2726	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑈) = ( ·_𝑠 ‘𝑈)
53	16, 52, 43, 28	lssvscl 20802	. . . . . 6 ⊢ (((𝑈 ∈ LMod ∧ ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSubSp‘𝑈)) ∧ (((inv_r‘𝑅)‘(𝐺‘𝑧)) ∈ (Base‘𝑅) ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)))) → (((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧) ∈ ( ⊥ ‘(𝐿‘𝐺)))
54	32, 51, 53	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → (((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧) ∈ ( ⊥ ‘(𝐿‘𝐺)))
55	43, 17, 47	drnginvrn0 20610	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑧) ∈ (Base‘𝑅) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → ((inv_r‘𝑅)‘(𝐺‘𝑧)) ≠ (0_g‘𝑅))
56	36, 45, 46, 55	syl3anc 1368	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → ((inv_r‘𝑅)‘(𝐺‘𝑧)) ≠ (0_g‘𝑅))
57	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) → 𝑈 ∈ LMod)
58	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) → 𝐺 ∈ 𝐹)
59		dochkr1.z	. . . . . . . . . . 11 ⊢ 0 = (0_g‘𝑈)
60	16, 17, 59, 10	lfl0 38448	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐺‘ 0 ) = (0_g‘𝑅))
61	57, 58, 60	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) → (𝐺‘ 0 ) = (0_g‘𝑅))
62		fveqeq2 6894	. . . . . . . . 9 ⊢ (𝑧 = 0 → ((𝐺‘𝑧) = (0_g‘𝑅) ↔ (𝐺‘ 0 ) = (0_g‘𝑅)))
63	61, 62	syl5ibrcom 246	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) → (𝑧 = 0 → (𝐺‘𝑧) = (0_g‘𝑅)))
64	63	necon3d 2955	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) → ((𝐺‘𝑧) ≠ (0_g‘𝑅) → 𝑧 ≠ 0 ))
65	64	3impia 1114	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → 𝑧 ≠ 0 )
66	9, 52, 16, 43, 17, 59, 34, 49, 42	lvecvsn0 20960	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → ((((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧) ≠ 0 ↔ (((inv_r‘𝑅)‘(𝐺‘𝑧)) ≠ (0_g‘𝑅) ∧ 𝑧 ≠ 0 )))
67	56, 65, 66	mpbir2and 710	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → (((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧) ≠ 0 )
68		eldifsn 4785	. . . . 5 ⊢ ((((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧) ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }) ↔ ((((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧) ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧) ≠ 0 ))
69	54, 67, 68	sylanbrc 582	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → (((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧) ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }))
70		eqid 2726	. . . . . . 7 ⊢ (._r‘𝑅) = (._r‘𝑅)
71	16, 43, 70, 9, 52, 10	lflmul 38451	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (((inv_r‘𝑅)‘(𝐺‘𝑧)) ∈ (Base‘𝑅) ∧ 𝑧 ∈ 𝑉)) → (𝐺‘(((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧)) = (((inv_r‘𝑅)‘(𝐺‘𝑧))(._r‘𝑅)(𝐺‘𝑧)))
72	37, 38, 49, 42, 71	syl112anc 1371	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → (𝐺‘(((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧)) = (((inv_r‘𝑅)‘(𝐺‘𝑧))(._r‘𝑅)(𝐺‘𝑧)))
73		dochkr1.i	. . . . . . 7 ⊢ 1 = (1_r‘𝑅)
74	43, 17, 70, 73, 47	drnginvrl 20612	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑧) ∈ (Base‘𝑅) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → (((inv_r‘𝑅)‘(𝐺‘𝑧))(._r‘𝑅)(𝐺‘𝑧)) = 1 )
75	36, 45, 46, 74	syl3anc 1368	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → (((inv_r‘𝑅)‘(𝐺‘𝑧))(._r‘𝑅)(𝐺‘𝑧)) = 1 )
76	72, 75	eqtrd 2766	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → (𝐺‘(((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧)) = 1 )
77		fveqeq2 6894	. . . . 5 ⊢ (𝑥 = (((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧) → ((𝐺‘𝑥) = 1 ↔ (𝐺‘(((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧)) = 1 ))
78	77	rspcev 3606	. . . 4 ⊢ (((((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧) ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }) ∧ (𝐺‘(((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧)) = 1 ) → ∃𝑥 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })(𝐺‘𝑥) = 1 )
79	69, 76, 78	syl2anc 583	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → ∃𝑥 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })(𝐺‘𝑥) = 1 )
80	79	rexlimdv3a 3153	. 2 ⊢ (𝜑 → (∃𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))(𝐺‘𝑧) ≠ (0_g‘𝑅) → ∃𝑥 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })(𝐺‘𝑥) = 1 ))
81	26, 80	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })(𝐺‘𝑥) = 1 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∃wrex 3064 ∖ cdif 3940 ⊆ wss 3943 {csn 4623 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 ._rcmulr 17207 Scalarcsca 17209 ·_𝑠 cvsca 17210 0_gc0g 17394 1_rcur 20086 inv_rcinvr 20289 DivRingcdr 20587 LModclmod 20706 LSubSpclss 20778 LVecclvec 20950 LSAtomsclsa 38357 LFnlclfn 38440 LKerclk 38468 HLchlt 38733 LHypclh 39368 DVecHcdvh 40462 ocHcoch 40731

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-riotaBAD 38336

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-undef 8259 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-0g 17396 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-cntz 19233 df-lsm 19556 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-drng 20589 df-lmod 20708 df-lss 20779 df-lsp 20819 df-lvec 20951 df-lsatoms 38359 df-lshyp 38360 df-lfl 38441 df-lkr 38469 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-llines 38882 df-lplanes 38883 df-lvols 38884 df-lines 38885 df-psubsp 38887 df-pmap 38888 df-padd 39180 df-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 df-trl 39543 df-tgrp 40127 df-tendo 40139 df-edring 40141 df-dveca 40387 df-disoa 40413 df-dvech 40463 df-dib 40523 df-dic 40557 df-dih 40613 df-doch 40732 df-djh 40779

This theorem is referenced by: lcfl6 40884

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