Theorem dihmeetlem2N 38875

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 2758	. . . . . 6 ⊢ (glb‘𝐾) = (glb‘𝐾)
2		dihmeetlem2.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
3		simp1l 1194	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝐾 ∈ HL)
4		simp2l 1196	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑋 ∈ 𝐵)
5		simp3l 1198	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ∈ 𝐵)
6	1, 2, 3, 4, 5	meetval 17695	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌}))
7	6	fveq2d 6662	. . . 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 1133	. . . . 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 2758	. . . . . . . . 9 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
13	9, 10, 11, 12	dibeldmN 38734	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom ((DIsoB‘𝐾)‘𝑊) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)))
14	13	biimpar 481	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → 𝑋 ∈ dom ((DIsoB‘𝐾)‘𝑊))
15	14	3adant3 1129	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑋 ∈ dom ((DIsoB‘𝐾)‘𝑊))
16	9, 10, 11, 12	dibeldmN 38734	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑌 ∈ dom ((DIsoB‘𝐾)‘𝑊) ↔ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)))
17	16	biimpar 481	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ∈ dom ((DIsoB‘𝐾)‘𝑊))
18	17	3adant2 1128	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ∈ dom ((DIsoB‘𝐾)‘𝑊))
19		prssg 4709	. . . . . . 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 587	. . . . . 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 711	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → {𝑋, 𝑌} ⊆ dom ((DIsoB‘𝐾)‘𝑊))
22		prnzg 4671	. . . . . 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 38742	. . . . 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 835	. . . 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 2793	. . 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 36940	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝐾 ∈ Lat)
28	9, 2	latmcl 17728	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵)
29	27, 4, 5, 28	syl3anc 1368	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) ∈ 𝐵)
30		simp1r 1195	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
31	9, 11	lhpbase 37574	. . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
32	30, 31	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑊 ∈ 𝐵)
33	9, 10, 2	latmle1 17752	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑋)
34	27, 4, 5, 33	syl3anc 1368	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) ≤ 𝑋)
35		simp2r 1197	. . . . 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 17734	. . . 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 38813	. . . 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 835	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = (((DIsoB‘𝐾)‘𝑊)‘(𝑋 ∧ 𝑌)))
40		simpl1 1188	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 ∈ {𝑋, 𝑌}) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
41		vex 3413	. . . . . . 7 ⊢ 𝑥 ∈ V
42	41	elpr 4545	. . . . . 6 ⊢ (𝑥 ∈ {𝑋, 𝑌} ↔ (𝑥 = 𝑋 ∨ 𝑥 = 𝑌))
43		simpl2 1189	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 = 𝑋) → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
44		eleq1 2839	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵))
45		breq1 5035	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
46	44, 45	anbi12d 633	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)))
47	46	adantl 485	. . . . . . . 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 1190	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑥 = 𝑌) → (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊))
50		eleq1 2839	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → (𝑥 ∈ 𝐵 ↔ 𝑌 ∈ 𝐵))
51		breq1 5035	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → (𝑥 ≤ 𝑊 ↔ 𝑌 ≤ 𝑊))
52	50, 51	anbi12d 633	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → ((𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊) ↔ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)))
53	52	adantl 485	. . . . . . . 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 955	. . . . . 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 596	. . . . 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 38813	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≤ 𝑊)) → (𝐼‘𝑥) = (((DIsoB‘𝐾)‘𝑊)‘𝑥))
58	40, 56, 57	syl2anc 587	. . . 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 4908	. . 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 2803	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥))
61		fveq2 6658	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐼‘𝑥) = (𝐼‘𝑋))
62		fveq2 6658	. . . 4 ⊢ (𝑥 = 𝑌 → (𝐼‘𝑥) = (𝐼‘𝑌))
63	61, 62	iinxprg 4976	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
64	4, 5, 63	syl2anc 587	. 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 2793	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 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951   ∩ cin 3857   ⊆ wss 3858  ∅c0 4225  {cpr 4524  ∩ ciin 4884   class class class wbr 5032   ↦ cmpt 5112   I cid 5429  dom cdm 5524   ↾ cres 5526  ‘cfv 6335  ℩crio 7107  (class class class)co 7150  Basecbs 16541  lecple 16630  occoc 16631  glbcglb 17619  joincjn 17620  meetcmee 17621  Latclat 17721  Atomscatm 36839  HLchlt 36926  LHypclh 37560  LTrncltrn 37677  trLctrl 37734  TEndoctendo 38328  DIsoBcdib 38714  DIsoHcdih 38804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-iin 4886  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8418  df-proset 17604  df-poset 17622  df-plt 17634  df-lub 17650  df-glb 17651  df-join 17652  df-meet 17653  df-p0 17715  df-p1 17716  df-lat 17722  df-clat 17784  df-oposet 36752  df-ol 36754  df-oml 36755  df-covers 36842  df-ats 36843  df-atl 36874  df-cvlat 36898  df-hlat 36927  df-lhyp 37564  df-laut 37565  df-ldil 37680  df-ltrn 37681  df-trl 37735  df-disoa 38605  df-dib 38715  df-dih 38805

This theorem is referenced by:  dihmeetbN  38879