Theorem tendoex 38916

Description: Generalization of Lemma K of [Crawley] p. 118, cdlemk 38915. TODO: can this be used to shorten uses of cdlemk 38915? (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 1222	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
2		hlop 37303	. . . . . . 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 1189	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
5		simpl2r 1225	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → 𝑁 ∈ 𝑇)
6		eqid 2738	. . . . . . . 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 38105	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑁 ∈ 𝑇) → (𝑅‘𝑁) ∈ (Base‘𝐾))
11	4, 5, 10	syl2anc 583	. . . . . 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 1191	. . . . . 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 2738	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
16		eqid 2738	. . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
17	6, 14, 15, 16	leat 37234	. . . . . 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 1371	. . . . 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 1136	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → (𝑅‘𝑁) ≤ (𝑅‘𝐹))
20		breq2 5074	. . . . . . . . 9 ⊢ ((𝑅‘𝐹) = (0.‘𝐾) → ((𝑅‘𝑁) ≤ (𝑅‘𝐹) ↔ (𝑅‘𝑁) ≤ (0.‘𝐾)))
21	19, 20	syl5ibcom 244	. . . . . . . 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 1222	. . . . . . . . 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 1189	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
26		simpl2r 1225	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → 𝑁 ∈ 𝑇)
27	25, 26, 10	syl2anc 583	. . . . . . . 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 37128	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ (𝑅‘𝑁) ∈ (Base‘𝐾)) → ((𝑅‘𝑁) ≤ (0.‘𝐾) ↔ (𝑅‘𝑁) = (0.‘𝐾)))
29	24, 27, 28	syl2anc 583	. . . . . . 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 231	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝑅‘𝑁) = (0.‘𝐾))
31	30	olcd 870	. . . . 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 1134	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
33		simp2l 1197	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → 𝐹 ∈ 𝑇)
34	15, 16, 7, 8, 9	trlator0 38112	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((𝑅‘𝐹) ∈ (Atoms‘𝐾) ∨ (𝑅‘𝐹) = (0.‘𝐾)))
35	32, 33, 34	syl2anc 583	. . . . 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 955	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ((𝑅‘𝑁) = (𝑅‘𝐹) ∨ (𝑅‘𝑁) = (0.‘𝐾)))
37	36	3expa 1116	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ((𝑅‘𝑁) = (𝑅‘𝐹) ∨ (𝑅‘𝑁) = (0.‘𝐾)))
38		eqcom 2745	. . . . 5 ⊢ ((𝑅‘𝑁) = (𝑅‘𝐹) ↔ (𝑅‘𝐹) = (𝑅‘𝑁))
39		tendoex.e	. . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
40	7, 8, 9, 39	cdlemk 38915	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
41	40	3expa 1116	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
42	38, 41	sylan2b 593	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (𝑅‘𝐹)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
43		eqid 2738	. . . . . . 7 ⊢ (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) = (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))
44	6, 7, 8, 39, 43	tendo0cl 38731	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸)
45	44	ad2antrr 722	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸)
46		simplrl 773	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → 𝐹 ∈ 𝑇)
47	43, 6	tendo02 38728	. . . . . . 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 38119	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑁 ∈ 𝑇) → (𝑁 = ( I ↾ (Base‘𝐾)) ↔ (𝑅‘𝑁) = (0.‘𝐾)))
50	49	adantrl 712	. . . . . . 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 2781	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁)
53		fveq1 6755	. . . . . . 7 ⊢ (𝑢 = (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) → (𝑢‘𝐹) = ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹))
54	53	eqeq1d 2740	. . . . . 6 ⊢ (𝑢 = (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) → ((𝑢‘𝐹) = 𝑁 ↔ ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁))
55	54	rspcev 3552	. . . . 5 ⊢ (((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸 ∧ ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
56	45, 52, 55	syl2anc 583	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
57	42, 56	jaodan 954	. . 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 590	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
59	58	3impa 1108	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∃wrex 3064   class class class wbr 5070   ↦ cmpt 5153   I cid 5479   ↾ cres 5582  ‘cfv 6418  Basecbs 16840  lecple 16895  0.cp0 18056  OPcops 37113  Atomscatm 37204  HLchlt 37291  LHypclh 37925  LTrncltrn 38042  trLctrl 38099  TEndoctendo 38693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-riotaBAD 36894

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-undef 8060  df-map 8575  df-proset 17928  df-poset 17946  df-plt 17963  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p0 18058  df-p1 18059  df-lat 18065  df-clat 18132  df-oposet 37117  df-ol 37119  df-oml 37120  df-covers 37207  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292  df-llines 37439  df-lplanes 37440  df-lvols 37441  df-lines 37442  df-psubsp 37444  df-pmap 37445  df-padd 37737  df-lhyp 37929  df-laut 37930  df-ldil 38045  df-ltrn 38046  df-trl 38100  df-tendo 38696

This theorem is referenced by:  dva1dim  38926  dihjatcclem4  39362