Theorem dihmeetlem2N 42357

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 2760	. . . . . 6 ⊢ (glb‘𝐾) = (glb‘𝐾)
2		dihmeetlem2.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
3		simp1l 1216	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝐾 ∈ HL)
4		simp2l 1218	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑋 ∈ 𝐵)
5		simp3l 1220	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ∈ 𝐵)
6	1, 2, 3, 4, 5	meetval 18563	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌}))
7	6	fveq2d 6889	. . . 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 1154	. . . . 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 2760	. . . . . . . . 9 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
13	9, 10, 11, 12	dibeldmN 42216	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom ((DIsoB‘𝐾)‘𝑊) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)))
14	13	biimpar 483	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → 𝑋 ∈ dom ((DIsoB‘𝐾)‘𝑊))
15	14	3adant3 1150	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑋 ∈ dom ((DIsoB‘𝐾)‘𝑊))
16	9, 10, 11, 12	dibeldmN 42216	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑌 ∈ dom ((DIsoB‘𝐾)‘𝑊) ↔ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)))
17	16	biimpar 483	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ∈ dom ((DIsoB‘𝐾)‘𝑊))
18	17	3adant2 1149	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ∈ dom ((DIsoB‘𝐾)‘𝑊))
19		prssg 4779	. . . . . . 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 596	. . . . . 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 725	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → {𝑋, 𝑌} ⊆ dom ((DIsoB‘𝐾)‘𝑊))
22		prnzg 4738	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → {𝑋, 𝑌} ≠ ∅)
23	4, 22	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → {𝑋, 𝑌} ≠ ∅)
24	1, 11, 12	dibglbN 42224	. . . . 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 850	. . . 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 2795	. . 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	3	hllatd 40422	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝐾 ∈ Lat)
28	9, 2	latmcl 18614	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵)
29	27, 4, 5, 28	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) ∈ 𝐵)
30		simp1r 1217	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
31	9, 11	lhpbase 41056	. . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
32	30, 31	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑊 ∈ 𝐵)
33	9, 10, 2	latmle1 18638	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑋)
34	27, 4, 5, 33	syl3anc 1398	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) ≤ 𝑋)
35		simp2r 1219	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑋 ≤ 𝑊)
36	9, 10, 27, 29, 4, 32, 34, 35	lattrd 18620	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) ≤ 𝑊)
37		dihmeetlem2.i	. . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
38	9, 10, 11, 37, 12	dihvalb 42295	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑌) ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = (((DIsoB‘𝐾)‘𝑊)‘(𝑋 ∧ 𝑌)))
39	8, 29, 36, 38	syl12anc 850	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = (((DIsoB‘𝐾)‘𝑊)‘(𝑋 ∧ 𝑌)))
40		simpl1 1210	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 ∈ {𝑋, 𝑌}) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
41		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
42	41	elpr 4608	. . . . . 6 ⊢ (𝑥 ∈ {𝑋, 𝑌} ↔ (𝑥 = 𝑋 ∨ 𝑥 = 𝑌))
43		simpl2 1211	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 = 𝑋) → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
44		eleq1 2848	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵))
45		breq1 5105	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
46	44, 45	anbi12d 644	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)))
47	46	adantl 487	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 = 𝑋) → ((𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)))
48	43, 47	mpbird 260	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 = 𝑋) → (𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊))
49		simpl3 1212	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 = 𝑌) → (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊))
50		eleq1 2848	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → (𝑥 ∈ 𝐵 ↔ 𝑌 ∈ 𝐵))
51		breq1 5105	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → (𝑥 ≤ 𝑊 ↔ 𝑌 ≤ 𝑊))
52	50, 51	anbi12d 644	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → ((𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊) ↔ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)))
53	52	adantl 487	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 = 𝑌) → ((𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊) ↔ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)))
54	49, 53	mpbird 260	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 = 𝑌) → (𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊))
55	48, 54	jaodan 972	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑥 = 𝑋 ∨ 𝑥 = 𝑌)) → (𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊))
56	42, 55	sylan2b 606	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 ∈ {𝑋, 𝑌}) → (𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊))
57	9, 10, 11, 37, 12	dihvalb 42295	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊)) → (𝐼‘𝑥) = (((DIsoB‘𝐾)‘𝑊)‘𝑥))
58	40, 56, 57	syl2anc 596	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 ∈ {𝑋, 𝑌}) → (𝐼‘𝑥) = (((DIsoB‘𝐾)‘𝑊)‘𝑥))
59	58	iineq2dv 4976	. . 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))
60	26, 39, 59	3eqtr4d 2805	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥))
61		fveq2 6885	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐼‘𝑥) = (𝐼‘𝑋))
62		fveq2 6885	. . . 4 ⊢ (𝑥 = 𝑌 → (𝐼‘𝑥) = (𝐼‘𝑌))
63	61, 62	iinxprg 5048	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
64	4, 5, 63	syl2anc 596	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
65	60, 64	eqtrd 2795	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278  {cpr 4585  ∩ ciin 4951   class class class wbr 5102   ↦ cmpt 5185   I cid 5544  dom cdm 5650   ↾ cres 5652  ‘cfv 6538  ℩crio 7376  (class class class)co 7420  Basecbs 17387  lecple 17435  occoc 17436  glbcglb 18484  joincjn 18485  meetcmee 18486  Latclat 18605  Atomscatm 40321  HLchlt 40408  LHypclh 41042  LTrncltrn 41159  trLctrl 41216  TEndoctendo 41810  DIsoBcdib 42196  DIsoHcdih 42286

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8849  df-proset 18468  df-poset 18487  df-plt 18502  df-lub 18518  df-glb 18519  df-join 18520  df-meet 18521  df-p0 18597  df-p1 18598  df-lat 18606  df-clat 18673  df-oposet 40234  df-ol 40236  df-oml 40237  df-covers 40324  df-ats 40325  df-atl 40356  df-cvlat 40380  df-hlat 40409  df-lhyp 41046  df-laut 41047  df-ldil 41162  df-ltrn 41163  df-trl 41217  df-disoa 42087  df-dib 42197  df-dih 42287

This theorem is used by:  dihmeetbN  42361