dochkr1 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dochkr1		Structured version Visualization version GIF version

Theorem dochkr1 40983

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 38574. (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 2728	. . . 4 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
2		eqid 2728	. . . 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 40615	. . . 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 40961	. . . . 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 38499	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))𝑧 ≠ (0_g‘𝑈))
16		dochkr1.r	. . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈)
17		eqid 2728	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
18	5	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) ∧ 𝑧 ≠ (0_g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
19	12	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) ∧ 𝑧 ≠ (0_g‘𝑈)) → 𝐺 ∈ 𝐹)
20		eldifsn 4795	. . . . . . . 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 712	. . . . . 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 40982	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) ∧ 𝑧 ≠ (0_g‘𝑈)) → (𝐺‘𝑧) ≠ (0_g‘𝑅))
24	23	ex 411	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) → (𝑧 ≠ (0_g‘𝑈) → (𝐺‘𝑧) ≠ (0_g‘𝑅)))
25	24	reximdva 3165	. . 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 38600	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉)
28		eqid 2728	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
29	3, 4, 9, 28, 8	dochlss 40859	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSubSp‘𝑈))
30	5, 27, 29	syl2anc 582	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSubSp‘𝑈))
31	6, 30	jca 510	. . . . . . 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 40614	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LVec)
34	33	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → 𝑈 ∈ LVec)
35	16	lvecdrng 20997	. . . . . . . . 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 40860	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉)
40	5, 27, 39	syl2anc 582	. . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉)
41	40	sselda 3982	. . . . . . . . . 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 2728	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
44	16, 43, 9, 10	lflcl 38568	. . . . . . . . 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 2728	. . . . . . . . 9 ⊢ (inv_r‘𝑅) = (inv_r‘𝑅)
48	43, 17, 47	drnginvrcl 20653	. . . . . . . 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 510	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → (((inv_r‘𝑅)‘(𝐺‘𝑧)) ∈ (Base‘𝑅) ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))))
52		eqid 2728	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑈) = ( ·_𝑠 ‘𝑈)
53	16, 52, 43, 28	lssvscl 20846	. . . . . 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 582	. . . . 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 20654	. . . . . . 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 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) → 𝑈 ∈ LMod)
58	12	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) → 𝐺 ∈ 𝐹)
59		dochkr1.z	. . . . . . . . . . 11 ⊢ 0 = (0_g‘𝑈)
60	16, 17, 59, 10	lfl0 38569	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐺‘ 0 ) = (0_g‘𝑅))
61	57, 58, 60	syl2anc 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺))) → (𝐺‘ 0 ) = (0_g‘𝑅))
62		fveqeq2 6911	. . . . . . . . 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 2958	. . . . . . 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 21004	. . . . . 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 711	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → (((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧) ≠ 0 )
68		eldifsn 4795	. . . . 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 581	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → (((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧) ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }))
70		eqid 2728	. . . . . . 7 ⊢ (._r‘𝑅) = (._r‘𝑅)
71	16, 43, 70, 9, 52, 10	lflmul 38572	. . . . . 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 20656	. . . . . 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 2768	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → (𝐺‘(((inv_r‘𝑅)‘(𝐺‘𝑧))( ·_𝑠 ‘𝑈)𝑧)) = 1 )
77		fveqeq2 6911	. . . . 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 3611	. . . 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 582	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ (𝐺‘𝑧) ≠ (0_g‘𝑅)) → ∃𝑥 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })(𝐺‘𝑥) = 1 )
80	79	rexlimdv3a 3156	. 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∃wrex 3067 ∖ cdif 3946 ⊆ wss 3949 {csn 4632 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 ._rcmulr 17241 Scalarcsca 17243 ·_𝑠 cvsca 17244 0_gc0g 17428 1_rcur 20128 inv_rcinvr 20333 DivRingcdr 20631 LModclmod 20750 LSubSpclss 20822 LVecclvec 20994 LSAtomsclsa 38478 LFnlclfn 38561 LKerclk 38589 HLchlt 38854 LHypclh 39489 DVecHcdvh 40583 ocHcoch 40852

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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-riotaBAD 38457

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-undef 8285 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-0g 17430 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-p1 18425 df-lat 18431 df-clat 18498 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-cntz 19275 df-lsm 19598 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-dvr 20347 df-drng 20633 df-lmod 20752 df-lss 20823 df-lsp 20863 df-lvec 20995 df-lsatoms 38480 df-lshyp 38481 df-lfl 38562 df-lkr 38590 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-llines 39003 df-lplanes 39004 df-lvols 39005 df-lines 39006 df-psubsp 39008 df-pmap 39009 df-padd 39301 df-lhyp 39493 df-laut 39494 df-ldil 39609 df-ltrn 39610 df-trl 39664 df-tgrp 40248 df-tendo 40260 df-edring 40262 df-dveca 40508 df-disoa 40534 df-dvech 40584 df-dib 40644 df-dic 40678 df-dih 40734 df-doch 40853 df-djh 40900

This theorem is referenced by: lcfl6 41005

W3C validator