Theorem dihmeetlem2N 36588

Description: Isomorphism H of a conjunction. (Contributed by NM, 22-Mar-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dihmeetlem2.b	⊢ 𝐵 = (Base‘𝐾)
dihmeetlem2.m	⊢ ∧ = (meet‘𝐾)
dihmeetlem2.h	⊢ 𝐻 = (LHyp‘𝐾)
dihmeetlem2.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihmeetlem2.l	⊢ ≤ = (le‘𝐾)
dihmeetlem2.j	⊢ ∨ = (join‘𝐾)
dihmeetlem2.a	⊢ 𝐴 = (Atoms‘𝐾)
dihmeetlem2.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
dihmeetlem2.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihmeetlem2.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dihmeetlem2.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dihmeetlem2.g	⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑞)
dihmeetlem2.o	⊢ 0 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))

Assertion

Ref	Expression
dihmeetlem2N	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))

Proof of Theorem dihmeetlem2N

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . 6 ⊢ (glb‘𝐾) = (glb‘𝐾)
2		dihmeetlem2.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
3		simp1l 1085	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝐾 ∈ HL)
4		simp2l 1087	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑋 ∈ 𝐵)
5		simp3l 1089	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ∈ 𝐵)
6	1, 2, 3, 4, 5	meetval 17019	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌}))
7	6	fveq2d 6195	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (((DIsoB‘𝐾)‘𝑊)‘(𝑋 ∧ 𝑌)) = (((DIsoB‘𝐾)‘𝑊)‘((glb‘𝐾)‘{𝑋, 𝑌})))
8		simp1 1061	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9		dihmeetlem2.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
10		dihmeetlem2.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
11		dihmeetlem2.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
12		eqid 2622	. . . . . . . . 9 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
13	9, 10, 11, 12	dibeldmN 36447	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom ((DIsoB‘𝐾)‘𝑊) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)))
14	13	biimpar 502	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → 𝑋 ∈ dom ((DIsoB‘𝐾)‘𝑊))
15	14	3adant3 1081	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑋 ∈ dom ((DIsoB‘𝐾)‘𝑊))
16	9, 10, 11, 12	dibeldmN 36447	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑌 ∈ dom ((DIsoB‘𝐾)‘𝑊) ↔ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)))
17	16	biimpar 502	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ∈ dom ((DIsoB‘𝐾)‘𝑊))
18	17	3adant2 1080	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ∈ dom ((DIsoB‘𝐾)‘𝑊))
19		prssg 4350	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∈ dom ((DIsoB‘𝐾)‘𝑊) ∧ 𝑌 ∈ dom ((DIsoB‘𝐾)‘𝑊)) ↔ {𝑋, 𝑌} ⊆ dom ((DIsoB‘𝐾)‘𝑊)))
20	4, 5, 19	syl2anc 693	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝑋 ∈ dom ((DIsoB‘𝐾)‘𝑊) ∧ 𝑌 ∈ dom ((DIsoB‘𝐾)‘𝑊)) ↔ {𝑋, 𝑌} ⊆ dom ((DIsoB‘𝐾)‘𝑊)))
21	15, 18, 20	mpbi2and 956	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → {𝑋, 𝑌} ⊆ dom ((DIsoB‘𝐾)‘𝑊))
22		prnzg 4311	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → {𝑋, 𝑌} ≠ ∅)
23	4, 22	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → {𝑋, 𝑌} ≠ ∅)
24	1, 11, 12	dibglbN 36455	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ({𝑋, 𝑌} ⊆ dom ((DIsoB‘𝐾)‘𝑊) ∧ {𝑋, 𝑌} ≠ ∅)) → (((DIsoB‘𝐾)‘𝑊)‘((glb‘𝐾)‘{𝑋, 𝑌})) = ∩ 𝑥 ∈ {𝑋, 𝑌} (((DIsoB‘𝐾)‘𝑊)‘𝑥))
25	8, 21, 23, 24	syl12anc 1324	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (((DIsoB‘𝐾)‘𝑊)‘((glb‘𝐾)‘{𝑋, 𝑌})) = ∩ 𝑥 ∈ {𝑋, 𝑌} (((DIsoB‘𝐾)‘𝑊)‘𝑥))
26	7, 25	eqtrd 2656	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (((DIsoB‘𝐾)‘𝑊)‘(𝑋 ∧ 𝑌)) = ∩ 𝑥 ∈ {𝑋, 𝑌} (((DIsoB‘𝐾)‘𝑊)‘𝑥))
27		hllat 34650	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
28	3, 27	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝐾 ∈ Lat)
29	9, 2	latmcl 17052	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵)
30	28, 4, 5, 29	syl3anc 1326	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) ∈ 𝐵)
31		simp1r 1086	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
32	9, 11	lhpbase 35284	. . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
33	31, 32	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑊 ∈ 𝐵)
34	9, 10, 2	latmle1 17076	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑋)
35	28, 4, 5, 34	syl3anc 1326	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) ≤ 𝑋)
36		simp2r 1088	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑋 ≤ 𝑊)
37	9, 10, 28, 30, 4, 33, 35, 36	lattrd 17058	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) ≤ 𝑊)
38		dihmeetlem2.i	. . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
39	9, 10, 11, 38, 12	dihvalb 36526	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑌) ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = (((DIsoB‘𝐾)‘𝑊)‘(𝑋 ∧ 𝑌)))
40	8, 30, 37, 39	syl12anc 1324	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = (((DIsoB‘𝐾)‘𝑊)‘(𝑋 ∧ 𝑌)))
41		simpl1 1064	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 ∈ {𝑋, 𝑌}) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
42		vex 3203	. . . . . . 7 ⊢ 𝑥 ∈ V
43	42	elpr 4198	. . . . . 6 ⊢ (𝑥 ∈ {𝑋, 𝑌} ↔ (𝑥 = 𝑋 ∨ 𝑥 = 𝑌))
44		simpl2 1065	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 = 𝑋) → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
45		eleq1 2689	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵))
46		breq1 4656	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
47	45, 46	anbi12d 747	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)))
48	47	adantl 482	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 = 𝑋) → ((𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)))
49	44, 48	mpbird 247	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 = 𝑋) → (𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊))
50		simpl3 1066	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 = 𝑌) → (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊))
51		eleq1 2689	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → (𝑥 ∈ 𝐵 ↔ 𝑌 ∈ 𝐵))
52		breq1 4656	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → (𝑥 ≤ 𝑊 ↔ 𝑌 ≤ 𝑊))
53	51, 52	anbi12d 747	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → ((𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊) ↔ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)))
54	53	adantl 482	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 = 𝑌) → ((𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊) ↔ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)))
55	50, 54	mpbird 247	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 = 𝑌) → (𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊))
56	49, 55	jaodan 826	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑥 = 𝑋 ∨ 𝑥 = 𝑌)) → (𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊))
57	43, 56	sylan2b 492	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 ∈ {𝑋, 𝑌}) → (𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊))
58	9, 10, 11, 38, 12	dihvalb 36526	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊)) → (𝐼‘𝑥) = (((DIsoB‘𝐾)‘𝑊)‘𝑥))
59	41, 57, 58	syl2anc 693	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 ∈ {𝑋, 𝑌}) → (𝐼‘𝑥) = (((DIsoB‘𝐾)‘𝑊)‘𝑥))
60	59	iineq2dv 4543	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥) = ∩ 𝑥 ∈ {𝑋, 𝑌} (((DIsoB‘𝐾)‘𝑊)‘𝑥))
61	26, 40, 60	3eqtr4d 2666	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥))
62		fveq2 6191	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐼‘𝑥) = (𝐼‘𝑋))
63		fveq2 6191	. . . 4 ⊢ (𝑥 = 𝑌 → (𝐼‘𝑥) = (𝐼‘𝑌))
64	62, 63	iinxprg 4601	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
65	4, 5, 64	syl2anc 693	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
66	61, 65	eqtrd 2656	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {cpr 4179  ∩ ciin 4521   class class class wbr 4653   ↦ cmpt 4729   I cid 5023  dom cdm 5114   ↾ cres 5116  ‘cfv 5888  ℩crio 6610  (class class class)co 6650  Basecbs 15857  lecple 15948  occoc 15949  glbcglb 16943  joincjn 16944  meetcmee 16945  Latclat 17045  Atomscatm 34550  HLchlt 34637  LHypclh 35270  LTrncltrn 35387  trLctrl 35445  TEndoctendo 36040  DIsoBcdib 36427  DIsoHcdih 36517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-disoa 36318  df-dib 36428  df-dih 36518

This theorem is referenced by:  dihmeetbN  36592