Theorem tendoex 41848

Description: Generalization of Lemma K of [Crawley] p. 118, cdlemk 41847. TODO: can this be used to shorten uses of cdlemk 41847? (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 40235	. . . . . . 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 2760	. . . . . . . 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 41037	. . . . . . 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 2760	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
16		eqid 2760	. . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
17	6, 14, 15, 16	leat 40166	. . . . . 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 1400	. . . . 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 5107	. . . . . . . . 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 40060	. . . . . . . 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 41044	. . . . . 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 2767	. . . . 5 ⊢ ((𝑅‘𝑁) = (𝑅‘𝐹) ↔ (𝑅‘𝐹) = (𝑅‘𝑁))
39		tendoex.e	. . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
40	7, 8, 9, 39	cdlemk 41847	. . . . . 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 2760	. . . . . . 7 ⊢ (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) = (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))
44	6, 7, 8, 39, 43	tendo0cl 41663	. . . . . 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 41660	. . . . . . 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 41051	. . . . . . . 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 2798	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁)
53		fveq1 6877	. . . . . . 7 ⊢ (𝑢 = (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) → (𝑢‘𝐹) = ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹))
54	53	eqeq1d 2762	. . . . . 6 ⊢ (𝑢 = (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) → ((𝑢‘𝐹) = 𝑁 ↔ ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁))
55	54	rspcev 3576	. . . . 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 2145  ∃wrex 3086   class class class wbr 5103   ↦ cmpt 5186   I cid 5549   ↾ cres 5657  ‘cfv 6533  Basecbs 17301  lecple 17349  0.cp0 18509  OPcops 40045  Atomscatm 40136  HLchlt 40223  LHypclh 40857  LTrncltrn 40974  trLctrl 41031  TEndoctendo 41625

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-riotaBAD 39826

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-undef 8271  df-map 8828  df-proset 18382  df-poset 18401  df-plt 18416  df-lub 18432  df-glb 18433  df-join 18434  df-meet 18435  df-p0 18511  df-p1 18512  df-lat 18520  df-clat 18587  df-oposet 40049  df-ol 40051  df-oml 40052  df-covers 40139  df-ats 40140  df-atl 40171  df-cvlat 40195  df-hlat 40224  df-llines 40371  df-lplanes 40372  df-lvols 40373  df-lines 40374  df-psubsp 40376  df-pmap 40377  df-padd 40669  df-lhyp 40861  df-laut 40862  df-ldil 40977  df-ltrn 40978  df-trl 41032  df-tendo 41628

This theorem is used by:  dva1dim  41858  dihjatcclem4  42294