Theorem tendoex 41073

Description: Generalization of Lemma K of [Crawley] p. 118, cdlemk 41072. TODO: can this be used to shorten uses of cdlemk 41072? (Contributed by NM, 15-Oct-2013.)

Hypotheses

Ref	Expression
tendoex.l	⊢ ≤ = (le‘𝐾)
tendoex.h	⊢ 𝐻 = (LHyp‘𝐾)
tendoex.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
tendoex.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
tendoex.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)

Assertion

Ref	Expression
tendoex	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)

Distinct variable groups:   𝑢,𝐸   𝑢,𝐹   𝑢,𝐾   𝑢,𝑁   𝑢,𝑅   𝑢,𝑇   𝑢,𝑊

Allowed substitution hints:   𝐻(𝑢)   ≤ (𝑢)

Proof of Theorem tendoex

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1l 1225	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
2		hlop 39460	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
3	1, 2	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP)
4		simpl1 1192	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
5		simpl2r 1228	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → 𝑁 ∈ 𝑇)
6		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		tendoex.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
8		tendoex.t	. . . . . . . 8 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
9		tendoex.r	. . . . . . . 8 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
10	6, 7, 8, 9	trlcl 40262	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑁 ∈ 𝑇) → (𝑅‘𝑁) ∈ (Base‘𝐾))
11	4, 5, 10	syl2anc 584	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → (𝑅‘𝑁) ∈ (Base‘𝐾))
12		simpr 484	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → (𝑅‘𝐹) ∈ (Atoms‘𝐾))
13		simpl3 1194	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → (𝑅‘𝑁) ≤ (𝑅‘𝐹))
14		tendoex.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
15		eqid 2731	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
16		eqid 2731	. . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
17	6, 14, 15, 16	leat 39391	. . . . . 6 ⊢ (((𝐾 ∈ OP ∧ (𝑅‘𝑁) ∈ (Base‘𝐾) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ((𝑅‘𝑁) = (𝑅‘𝐹) ∨ (𝑅‘𝑁) = (0.‘𝐾)))
18	3, 11, 12, 13, 17	syl31anc 1375	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → ((𝑅‘𝑁) = (𝑅‘𝐹) ∨ (𝑅‘𝑁) = (0.‘𝐾)))
19		simp3 1138	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → (𝑅‘𝑁) ≤ (𝑅‘𝐹))
20		breq2 5093	. . . . . . . . 9 ⊢ ((𝑅‘𝐹) = (0.‘𝐾) → ((𝑅‘𝑁) ≤ (𝑅‘𝐹) ↔ (𝑅‘𝑁) ≤ (0.‘𝐾)))
21	19, 20	syl5ibcom 245	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ((𝑅‘𝐹) = (0.‘𝐾) → (𝑅‘𝑁) ≤ (0.‘𝐾)))
22	21	imp 406	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝑅‘𝑁) ≤ (0.‘𝐾))
23		simpl1l 1225	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → 𝐾 ∈ HL)
24	23, 2	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → 𝐾 ∈ OP)
25		simpl1 1192	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
26		simpl2r 1228	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → 𝑁 ∈ 𝑇)
27	25, 26, 10	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝑅‘𝑁) ∈ (Base‘𝐾))
28	6, 14, 15	ople0 39285	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ (𝑅‘𝑁) ∈ (Base‘𝐾)) → ((𝑅‘𝑁) ≤ (0.‘𝐾) ↔ (𝑅‘𝑁) = (0.‘𝐾)))
29	24, 27, 28	syl2anc 584	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → ((𝑅‘𝑁) ≤ (0.‘𝐾) ↔ (𝑅‘𝑁) = (0.‘𝐾)))
30	22, 29	mpbid 232	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝑅‘𝑁) = (0.‘𝐾))
31	30	olcd 874	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → ((𝑅‘𝑁) = (𝑅‘𝐹) ∨ (𝑅‘𝑁) = (0.‘𝐾)))
32		simp1 1136	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
33		simp2l 1200	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → 𝐹 ∈ 𝑇)
34	15, 16, 7, 8, 9	trlator0 40269	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((𝑅‘𝐹) ∈ (Atoms‘𝐾) ∨ (𝑅‘𝐹) = (0.‘𝐾)))
35	32, 33, 34	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ((𝑅‘𝐹) ∈ (Atoms‘𝐾) ∨ (𝑅‘𝐹) = (0.‘𝐾)))
36	18, 31, 35	mpjaodan 960	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ((𝑅‘𝑁) = (𝑅‘𝐹) ∨ (𝑅‘𝑁) = (0.‘𝐾)))
37	36	3expa 1118	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ((𝑅‘𝑁) = (𝑅‘𝐹) ∨ (𝑅‘𝑁) = (0.‘𝐾)))
38		eqcom 2738	. . . . 5 ⊢ ((𝑅‘𝑁) = (𝑅‘𝐹) ↔ (𝑅‘𝐹) = (𝑅‘𝑁))
39		tendoex.e	. . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
40	7, 8, 9, 39	cdlemk 41072	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
41	40	3expa 1118	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
42	38, 41	sylan2b 594	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (𝑅‘𝐹)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
43		eqid 2731	. . . . . . 7 ⊢ (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) = (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))
44	6, 7, 8, 39, 43	tendo0cl 40888	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸)
45	44	ad2antrr 726	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸)
46		simplrl 776	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → 𝐹 ∈ 𝑇)
47	43, 6	tendo02 40885	. . . . . . 7 ⊢ (𝐹 ∈ 𝑇 → ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = ( I ↾ (Base‘𝐾)))
48	46, 47	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = ( I ↾ (Base‘𝐾)))
49	6, 15, 7, 8, 9	trlid0b 40276	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑁 ∈ 𝑇) → (𝑁 = ( I ↾ (Base‘𝐾)) ↔ (𝑅‘𝑁) = (0.‘𝐾)))
50	49	adantrl 716	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) → (𝑁 = ( I ↾ (Base‘𝐾)) ↔ (𝑅‘𝑁) = (0.‘𝐾)))
51	50	biimpar 477	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → 𝑁 = ( I ↾ (Base‘𝐾)))
52	48, 51	eqtr4d 2769	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁)
53		fveq1 6821	. . . . . . 7 ⊢ (𝑢 = (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) → (𝑢‘𝐹) = ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹))
54	53	eqeq1d 2733	. . . . . 6 ⊢ (𝑢 = (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) → ((𝑢‘𝐹) = 𝑁 ↔ ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁))
55	54	rspcev 3572	. . . . 5 ⊢ (((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸 ∧ ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
56	45, 52, 55	syl2anc 584	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
57	42, 56	jaodan 959	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ ((𝑅‘𝑁) = (𝑅‘𝐹) ∨ (𝑅‘𝑁) = (0.‘𝐾))) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
58	37, 57	syldan 591	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
59	58	3impa 1109	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∃wrex 3056   class class class wbr 5089   ↦ cmpt 5170   I cid 5508   ↾ cres 5616  ‘cfv 6481  Basecbs 17120  lecple 17168  0.cp0 18327  OPcops 39270  Atomscatm 39361  HLchlt 39448  LHypclh 40082  LTrncltrn 40199  trLctrl 40256  TEndoctendo 40850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-riotaBAD 39051

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-undef 8203  df-map 8752  df-proset 18200  df-poset 18219  df-plt 18234  df-lub 18250  df-glb 18251  df-join 18252  df-meet 18253  df-p0 18329  df-p1 18330  df-lat 18338  df-clat 18405  df-oposet 39274  df-ol 39276  df-oml 39277  df-covers 39364  df-ats 39365  df-atl 39396  df-cvlat 39420  df-hlat 39449  df-llines 39596  df-lplanes 39597  df-lvols 39598  df-lines 39599  df-psubsp 39601  df-pmap 39602  df-padd 39894  df-lhyp 40086  df-laut 40087  df-ldil 40202  df-ltrn 40203  df-trl 40257  df-tendo 40853

This theorem is referenced by:  dva1dim  41083  dihjatcclem4  41519