Theorem tendoex 38115

Description: Generalization of Lemma K of [Crawley] p. 118, cdlemk 38114. TODO: can this be used to shorten uses of cdlemk 38114? (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 1220	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
2		hlop 36502	. . . . . . 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 1187	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
5		simpl2r 1223	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → 𝑁 ∈ 𝑇)
6		eqid 2824	. . . . . . . 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 37304	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑁 ∈ 𝑇) → (𝑅‘𝑁) ∈ (Base‘𝐾))
11	4, 5, 10	syl2anc 586	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → (𝑅‘𝑁) ∈ (Base‘𝐾))
12		simpr 487	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → (𝑅‘𝐹) ∈ (Atoms‘𝐾))
13		simpl3 1189	. . . . . 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 2824	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
16		eqid 2824	. . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
17	6, 14, 15, 16	leat 36433	. . . . . 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 1369	. . . . 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 1134	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → (𝑅‘𝑁) ≤ (𝑅‘𝐹))
20		breq2 5073	. . . . . . . . 9 ⊢ ((𝑅‘𝐹) = (0.‘𝐾) → ((𝑅‘𝑁) ≤ (𝑅‘𝐹) ↔ (𝑅‘𝑁) ≤ (0.‘𝐾)))
21	19, 20	syl5ibcom 247	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ((𝑅‘𝐹) = (0.‘𝐾) → (𝑅‘𝑁) ≤ (0.‘𝐾)))
22	21	imp 409	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝑅‘𝑁) ≤ (0.‘𝐾))
23		simpl1l 1220	. . . . . . . . 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 1187	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
26		simpl2r 1223	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → 𝑁 ∈ 𝑇)
27	25, 26, 10	syl2anc 586	. . . . . . . 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 36327	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ (𝑅‘𝑁) ∈ (Base‘𝐾)) → ((𝑅‘𝑁) ≤ (0.‘𝐾) ↔ (𝑅‘𝑁) = (0.‘𝐾)))
29	24, 27, 28	syl2anc 586	. . . . . . 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 234	. . . . . 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 1132	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
33		simp2l 1195	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → 𝐹 ∈ 𝑇)
34	15, 16, 7, 8, 9	trlator0 37311	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((𝑅‘𝐹) ∈ (Atoms‘𝐾) ∨ (𝑅‘𝐹) = (0.‘𝐾)))
35	32, 33, 34	syl2anc 586	. . . . 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 1114	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ((𝑅‘𝑁) = (𝑅‘𝐹) ∨ (𝑅‘𝑁) = (0.‘𝐾)))
38		eqcom 2831	. . . . 5 ⊢ ((𝑅‘𝑁) = (𝑅‘𝐹) ↔ (𝑅‘𝐹) = (𝑅‘𝑁))
39		tendoex.e	. . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
40	7, 8, 9, 39	cdlemk 38114	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
41	40	3expa 1114	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
42	38, 41	sylan2b 595	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (𝑅‘𝐹)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
43		eqid 2824	. . . . . . 7 ⊢ (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) = (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))
44	6, 7, 8, 39, 43	tendo0cl 37930	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸)
45	44	ad2antrr 724	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸)
46		simplrl 775	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → 𝐹 ∈ 𝑇)
47	43, 6	tendo02 37927	. . . . . . 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 37318	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑁 ∈ 𝑇) → (𝑁 = ( I ↾ (Base‘𝐾)) ↔ (𝑅‘𝑁) = (0.‘𝐾)))
50	49	adantrl 714	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) → (𝑁 = ( I ↾ (Base‘𝐾)) ↔ (𝑅‘𝑁) = (0.‘𝐾)))
51	50	biimpar 480	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → 𝑁 = ( I ↾ (Base‘𝐾)))
52	48, 51	eqtr4d 2862	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) = (0.‘𝐾)) → ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁)
53		fveq1 6672	. . . . . . 7 ⊢ (𝑢 = (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) → (𝑢‘𝐹) = ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹))
54	53	eqeq1d 2826	. . . . . 6 ⊢ (𝑢 = (ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) → ((𝑢‘𝐹) = 𝑁 ↔ ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁))
55	54	rspcev 3626	. . . . 5 ⊢ (((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) ∈ 𝐸 ∧ ((ℎ ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))‘𝐹) = 𝑁) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
56	45, 52, 55	syl2anc 586	. . . 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 593	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)
59	58	3impa 1106	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝑁) ≤ (𝑅‘𝐹)) → ∃𝑢 ∈ 𝐸 (𝑢‘𝐹) = 𝑁)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ∃wrex 3142   class class class wbr 5069   ↦ cmpt 5149   I cid 5462   ↾ cres 5560  ‘cfv 6358  Basecbs 16486  lecple 16575  0.cp0 17650  OPcops 36312  Atomscatm 36403  HLchlt 36490  LHypclh 37124  LTrncltrn 37241  trLctrl 37298  TEndoctendo 37892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-riotaBAD 36093

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-undef 7942  df-map 8411  df-proset 17541  df-poset 17559  df-plt 17571  df-lub 17587  df-glb 17588  df-join 17589  df-meet 17590  df-p0 17652  df-p1 17653  df-lat 17659  df-clat 17721  df-oposet 36316  df-ol 36318  df-oml 36319  df-covers 36406  df-ats 36407  df-atl 36438  df-cvlat 36462  df-hlat 36491  df-llines 36638  df-lplanes 36639  df-lvols 36640  df-lines 36641  df-psubsp 36643  df-pmap 36644  df-padd 36936  df-lhyp 37128  df-laut 37129  df-ldil 37244  df-ltrn 37245  df-trl 37299  df-tendo 37895

This theorem is referenced by:  dva1dim  38125  dihjatcclem4  38561