Theorem tendoex 42298

Description: Generalization of Lemma K of [Crawley] p. 118, cdlemk 42297. TODO: can this be used to shorten uses of cdlemk 42297? (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 1243	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
2		hlop 40685	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
3	1, 2	syl 18	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP)
4		simpl1 1210	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
5		simpl2r 1246	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → 𝑁 ∈ 𝑇)
6		eqid 2763	. . . . . . . 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 41487	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑁 ∈ 𝑇) → (𝑅‘𝑁) ∈ (Base‘𝐾))
11	4, 5, 10	syl2anc 596	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → (𝑅‘𝑁) ∈ (Base‘𝐾))
12		simpr 490	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → (𝑅‘𝐹) ∈ (Atoms‘𝐾))
13		simpl3 1212	. . . . . 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 2763	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
16		eqid 2763	. . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
17	6, 14, 15, 16	leat 40616	. . . . . 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 1401	. . . . 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 1156	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → (𝑅‘𝑁) ≤ (𝑅‘𝐹))
20		breq2 5112	. . . . . . . . 9 ⊢ ((𝑅‘𝐹) = (0.‘𝐾) → ((𝑅‘𝑁) ≤ (𝑅‘𝐹) ↔ (𝑅‘𝑁) ≤ (0.‘𝐾)))
21	19, 20	syl5ibcom 248	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ((𝑅‘𝐹) = (0.‘𝐾) → (𝑅‘𝑁) ≤ (0.‘𝐾)))
22	21	imp 412	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝑅‘𝑁) ≤ (0.‘𝐾))
23		simpl1l 1243	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → 𝐾 ∈ HL)
24	23, 2	syl 18	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → 𝐾 ∈ OP)
25		simpl1 1210	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
26		simpl2r 1246	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → 𝑁 ∈ 𝑇)
27	25, 26, 10	syl2anc 596	. . . . . . . 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 40510	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ (𝑅‘𝑁) ∈ (Base‘𝐾)) → ((𝑅‘𝑁) ≤ (0.‘𝐾) ↔ (𝑅‘𝑁) = (0.‘𝐾)))
29	24, 27, 28	syl2anc 596	. . . . . . 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 235	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝑅‘𝑁) = (0.‘𝐾))
31	30	olcd 888	. . . . 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 1154	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
33		simp2l 1218	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → 𝐹 ∈ 𝑇)
34	15, 16, 7, 8, 9	trlator0 41494	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((𝑅‘𝐹) ∈ (Atoms‘𝐾) ∨ (𝑅‘𝐹) = (0.‘𝐾)))
35	32, 33, 34	syl2anc 596	. . . . 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 973	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ((𝑅‘𝑁) = (𝑅‘𝐹) ∨ (𝑅‘𝑁) = (0.‘𝐾)))
37	36	3expa 1136	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ((𝑅‘𝑁) = (𝑅‘𝐹) ∨ (𝑅‘𝑁) = (0.‘𝐾)))
38		eqcom 2770	. . . . 5 ⊢ ((𝑅‘𝑁) = (𝑅‘𝐹) ↔ (𝑅‘𝐹) = (𝑅‘𝑁))
39		tendoex.e	. . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
40	7, 8, 9, 39	cdlemk 42297	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
41	40	3expa 1136	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
42	38, 41	sylan2b 606	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (𝑅‘𝐹)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
43		eqid 2763	. . . . . . 7 ⊢ (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) = (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))
44	6, 7, 8, 39, 43	tendo0cl 42113	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸)
45	44	ad2antrr 739	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸)
46		simplrl 789	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → 𝐹 ∈ 𝑇)
47	43, 6	tendo02 42110	. . . . . . 7 ⊢ (𝐹 ∈ 𝑇 → ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = ( I ↾ (Base‘𝐾)))
48	46, 47	syl 18	. . . . . 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 41501	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑁 ∈ 𝑇) → (𝑁 = ( I ↾ (Base‘𝐾)) ↔ (𝑅‘𝑁) = (0.‘𝐾)))
50	49	adantrl 729	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) → (𝑁 = ( I ↾ (Base‘𝐾)) ↔ (𝑅‘𝑁) = (0.‘𝐾)))
51	50	biimpar 483	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → 𝑁 = ( I ↾ (Base‘𝐾)))
52	48, 51	eqtr4d 2801	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁)
53		fveq1 6915	. . . . . . 7 ⊢ (𝑢 = (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) → (𝑢‘𝐹) = ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹))
54	53	eqeq1d 2765	. . . . . 6 ⊢ (𝑢 = (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) → ((𝑢‘𝐹) = 𝑁 ↔ ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁))
55	54	rspcev 3580	. . . . 5 ⊢ (((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸 ∧ ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
56	45, 52, 55	syl2anc 596	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
57	42, 56	jaodan 972	. . 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 603	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
59	58	3impa 1127	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   class class class wbr 5108   ↦ cmpt 5192   I cid 5552   ↾ cres 5660  ‘cfv 6566  Basecbs 17431  lecple 17479  0.cp0 18645  OPcops 40495  Atomscatm 40586  HLchlt 40673  LHypclh 41307  LTrncltrn 41424  trLctrl 41481  TEndoctendo 42075

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-riotaBAD 40276

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-undef 8328  df-map 8887  df-proset 18518  df-poset 18537  df-plt 18552  df-lub 18568  df-glb 18569  df-join 18570  df-meet 18571  df-p0 18647  df-p1 18648  df-lat 18656  df-clat 18723  df-oposet 40499  df-ol 40501  df-oml 40502  df-covers 40589  df-ats 40590  df-atl 40621  df-cvlat 40645  df-hlat 40674  df-llines 40821  df-lplanes 40822  df-lvols 40823  df-lines 40824  df-psubsp 40826  df-pmap 40827  df-padd 41119  df-lhyp 41311  df-laut 41312  df-ldil 41427  df-ltrn 41428  df-trl 41482  df-tendo 42078

This theorem is used by:  dva1dim  42308  dihjatcclem4  42744