Theorem cdlemg7fvbwN 41488

Description: Properties of a translation of an element not under 𝑊. TODO: Fix comment. Can this be simplified? Perhaps derived from cdleme48bw 41383? Done with a *ltrn* theorem? (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemg4.l	⊢ ≤ = (le‘𝐾)
cdlemg4.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg4.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg4.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg4.b	⊢ 𝐵 = (Base‘𝐾)

Assertion

Ref	Expression
cdlemg7fvbwN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑋) ∈ 𝐵 ∧ ¬ (𝐹‘𝑋) ≤ 𝑊))

Proof of Theorem cdlemg7fvbwN

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cdlemg4.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemg4.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		eqid 2762	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2762	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		cdlemg4.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemg4.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
7	1, 2, 3, 4, 5, 6	lhpmcvr2 40905	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋))
8	7	3adant3 1150	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋))
9		simp11 1222	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10		simp2 1155	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑟 ∈ 𝐴)
11		simp3l 1220	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ¬ 𝑟 ≤ 𝑊)
12	10, 11	jca 521	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊))
13		simp12 1223	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))
14		simp13 1224	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝐹 ∈ 𝑇)
15		simp3r 1221	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)
16		cdlemg4.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
17	6, 16, 2, 3, 5, 4, 1	cdlemg2fv 41480	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹‘𝑋) = ((𝐹‘𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)))
18	9, 12, 13, 14, 15, 17	syl122anc 1406	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹‘𝑋) = ((𝐹‘𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)))
19		simp11l 1303	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝐾 ∈ HL)
20	19	hllatd 40245	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝐾 ∈ Lat)
21	2, 5, 6, 16	ltrnel 41020	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊)) → ((𝐹‘𝑟) ∈ 𝐴 ∧ ¬ (𝐹‘𝑟) ≤ 𝑊))
22	21	simpld 500	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊)) → (𝐹‘𝑟) ∈ 𝐴)
23	9, 14, 12, 22	syl3anc 1398	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹‘𝑟) ∈ 𝐴)
24	1, 5	atbase 40170	. . . . . . 7 ⊢ ((𝐹‘𝑟) ∈ 𝐴 → (𝐹‘𝑟) ∈ 𝐵)
25	23, 24	syl 18	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹‘𝑟) ∈ 𝐵)
26		simp12l 1305	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑋 ∈ 𝐵)
27		simp11r 1304	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑊 ∈ 𝐻)
28	1, 6	lhpbase 40879	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
29	27, 28	syl 18	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑊 ∈ 𝐵)
30	1, 4	latmcl 18534	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑊) ∈ 𝐵)
31	20, 26, 29, 30	syl3anc 1398	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑋(meet‘𝐾)𝑊) ∈ 𝐵)
32	1, 3	latjcl 18533	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝐹‘𝑟) ∈ 𝐵 ∧ (𝑋(meet‘𝐾)𝑊) ∈ 𝐵) → ((𝐹‘𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) ∈ 𝐵)
33	20, 25, 31, 32	syl3anc 1398	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ((𝐹‘𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) ∈ 𝐵)
34	18, 33	eqeltrd 2862	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹‘𝑋) ∈ 𝐵)
35	21	simprd 501	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊)) → ¬ (𝐹‘𝑟) ≤ 𝑊)
36	9, 14, 12, 35	syl3anc 1398	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ¬ (𝐹‘𝑟) ≤ 𝑊)
37	1, 2, 3	latlej1 18542	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝐹‘𝑟) ∈ 𝐵 ∧ (𝑋(meet‘𝐾)𝑊) ∈ 𝐵) → (𝐹‘𝑟) ≤ ((𝐹‘𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)))
38	20, 25, 31, 37	syl3anc 1398	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹‘𝑟) ≤ ((𝐹‘𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)))
39	1, 2	lattr 18538	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ ((𝐹‘𝑟) ∈ 𝐵 ∧ ((𝐹‘𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (((𝐹‘𝑟) ≤ ((𝐹‘𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) ∧ ((𝐹‘𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) ≤ 𝑊) → (𝐹‘𝑟) ≤ 𝑊))
40	20, 25, 33, 29, 39	syl13anc 1399	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (((𝐹‘𝑟) ≤ ((𝐹‘𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) ∧ ((𝐹‘𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) ≤ 𝑊) → (𝐹‘𝑟) ≤ 𝑊))
41	38, 40	mpand 708	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (((𝐹‘𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) ≤ 𝑊 → (𝐹‘𝑟) ≤ 𝑊))
42	36, 41	mtod 201	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ¬ ((𝐹‘𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) ≤ 𝑊)
43	18	breq1d 5117	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ((𝐹‘𝑋) ≤ 𝑊 ↔ ((𝐹‘𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) ≤ 𝑊))
44	42, 43	mtbird 328	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ¬ (𝐹‘𝑋) ≤ 𝑊)
45	34, 44	jca 521	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ((𝐹‘𝑋) ∈ 𝐵 ∧ ¬ (𝐹‘𝑋) ≤ 𝑊))
46	45	rexlimdv3a 3169	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → (∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) → ((𝐹‘𝑋) ∈ 𝐵 ∧ ¬ (𝐹‘𝑋) ≤ 𝑊)))
47	8, 46	mpd 16	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑋) ∈ 𝐵 ∧ ¬ (𝐹‘𝑋) ≤ 𝑊))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3088   class class class wbr 5107  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  lecple 17355  joincjn 18405  meetcmee 18406  Latclat 18525  Atomscatm 40144  HLchlt 40231  LHypclh 40865  LTrncltrn 40982

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-riotaBAD 39834

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-undef 8275  df-map 8832  df-proset 18388  df-poset 18407  df-plt 18422  df-lub 18438  df-glb 18439  df-join 18440  df-meet 18441  df-p0 18517  df-p1 18518  df-lat 18526  df-clat 18593  df-oposet 40057  df-ol 40059  df-oml 40060  df-covers 40147  df-ats 40148  df-atl 40179  df-cvlat 40203  df-hlat 40232  df-llines 40379  df-lplanes 40380  df-lvols 40381  df-lines 40382  df-psubsp 40384  df-pmap 40385  df-padd 40677  df-lhyp 40869  df-laut 40870  df-ldil 40985  df-ltrn 40986  df-trl 41040

This theorem is used by:  cdlemg7fvN  41505