Theorem dihmeetlem20N 41025

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dihmeetlem14.b	⊢ 𝐵 = (Base‘𝐾)
dihmeetlem14.l	⊢ ≤ = (le‘𝐾)
dihmeetlem14.h	⊢ 𝐻 = (LHyp‘𝐾)
dihmeetlem14.j	⊢ ∨ = (join‘𝐾)
dihmeetlem14.m	⊢ ∧ = (meet‘𝐾)
dihmeetlem14.a	⊢ 𝐴 = (Atoms‘𝐾)
dihmeetlem14.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihmeetlem14.s	⊢ ⊕ = (LSSum‘𝑈)
dihmeetlem14.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)

Assertion

Ref	Expression
dihmeetlem20N	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))

Proof of Theorem dihmeetlem20N

Dummy variables 𝑟 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1133	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp2 1134	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))
3		simp3ll 1241	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → 𝑌 ∈ 𝐵)
4		simp3r 1199	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝑋 ∧ 𝑌) ≤ 𝑊)
5		dihmeetlem14.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
6		dihmeetlem14.l	. . . 4 ⊢ ≤ = (le‘𝐾)
7		dihmeetlem14.j	. . . 4 ⊢ ∨ = (join‘𝐾)
8		dihmeetlem14.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
9		dihmeetlem14.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
10		dihmeetlem14.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
11	5, 6, 7, 8, 9, 10	lhpmcvr6N 39727	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋))
12	1, 2, 3, 4, 11	syl112anc 1371	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋))
13		simp3l 1198	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊))
14		simp2l 1196	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → 𝑋 ∈ 𝐵)
15		simp1l 1194	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → 𝐾 ∈ HL)
16	15	hllatd 39062	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → 𝐾 ∈ Lat)
17	5, 8	latmcom 18488	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌 ∧ 𝑋) = (𝑋 ∧ 𝑌))
18	16, 3, 14, 17	syl3anc 1368	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝑌 ∧ 𝑋) = (𝑋 ∧ 𝑌))
19	18, 4	eqbrtrd 5175	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝑌 ∧ 𝑋) ≤ 𝑊)
20	5, 6, 7, 8, 9, 10	lhpmcvr6N 39727	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ (𝑌 ∧ 𝑋) ≤ 𝑊)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌))
21	1, 13, 14, 19, 20	syl112anc 1371	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌))
22		reeanv 3217	. . 3 ⊢ (∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌)) ↔ (∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌)))
23		simp11 1200	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
24		simp12 1201	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌))) → (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))
25	3	3ad2ant1 1130	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌))) → 𝑌 ∈ 𝐵)
26		simp2l 1196	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌))) → 𝑞 ∈ 𝐴)
27		simp3l1 1275	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌))) → ¬ 𝑞 ≤ 𝑊)
28	26, 27	jca 510	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌))) → (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊))
29		simp2r 1197	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌))) → 𝑟 ∈ 𝐴)
30		simp3r1 1278	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌))) → ¬ 𝑟 ≤ 𝑊)
31	29, 30	jca 510	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌))) → (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊))
32		simp3l3 1277	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌))) → 𝑞 ≤ 𝑋)
33		simp3r3 1280	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌))) → 𝑟 ≤ 𝑌)
34		simp13r 1286	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌))) → (𝑋 ∧ 𝑌) ≤ 𝑊)
35	32, 33, 34	3jca 1125	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌))) → (𝑞 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊))
36		dihmeetlem14.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
37		dihmeetlem14.s	. . . . . . 7 ⊢ ⊕ = (LSSum‘𝑈)
38		dihmeetlem14.i	. . . . . . 7 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
39	5, 6, 10, 7, 8, 9, 36, 37, 38	dihmeetlem19N 41024	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑞 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊))) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
40	23, 24, 25, 28, 31, 35, 39	syl33anc 1382	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌))) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
41	40	3exp 1116	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → ((𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → (((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))))
42	41	rexlimdvv 3201	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))))
43	22, 42	biimtrrid 242	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → ((∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))))
44	12, 21, 43	mp2and 697	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  ∃wrex 3060   ∩ cin 3946   class class class wbr 5153  ‘cfv 6554  (class class class)co 7424  Basecbs 17213  lecple 17273  joincjn 18336  meetcmee 18337  Latclat 18456  LSSumclsm 19632  Atomscatm 38961  HLchlt 39048  LHypclh 39683  DVecHcdvh 40777  DIsoHcdih 40927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-riotaBAD 38651

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-tpos 8241  df-undef 8288  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-n0 12525  df-z 12611  df-uz 12875  df-fz 13539  df-struct 17149  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ress 17243  df-plusg 17279  df-mulr 17280  df-sca 17282  df-vsca 17283  df-0g 17456  df-mre 17599  df-mrc 17600  df-acs 17602  df-proset 18320  df-poset 18338  df-plt 18355  df-lub 18371  df-glb 18372  df-join 18373  df-meet 18374  df-p0 18450  df-p1 18451  df-lat 18457  df-clat 18524  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-submnd 18774  df-grp 18931  df-minusg 18932  df-sbg 18933  df-subg 19117  df-cntz 19311  df-lsm 19634  df-cmn 19780  df-abl 19781  df-mgp 20118  df-rng 20136  df-ur 20165  df-ring 20218  df-oppr 20316  df-dvdsr 20339  df-unit 20340  df-invr 20370  df-dvr 20383  df-drng 20709  df-lmod 20838  df-lss 20909  df-lsp 20949  df-lvec 21081  df-oposet 38874  df-ol 38876  df-oml 38877  df-covers 38964  df-ats 38965  df-atl 38996  df-cvlat 39020  df-hlat 39049  df-llines 39197  df-lplanes 39198  df-lvols 39199  df-lines 39200  df-psubsp 39202  df-pmap 39203  df-padd 39495  df-lhyp 39687  df-laut 39688  df-ldil 39803  df-ltrn 39804  df-trl 39858  df-tendo 40454  df-edring 40456  df-disoa 40728  df-dvech 40778  df-dib 40838  df-dic 40872  df-dih 40928

This theorem is referenced by:  dihmeetALTN  41026