Theorem hlmod1i 37849

Description: A version of the modular law pmod1i 37841 that holds in a Hilbert lattice. (Contributed by NM, 13-May-2012.)

Hypotheses

Ref	Expression
hlmod.b	⊢ 𝐵 = (Base‘𝐾)
hlmod.l	⊢ ≤ = (le‘𝐾)
hlmod.j	⊢ ∨ = (join‘𝐾)
hlmod.m	⊢ ∧ = (meet‘𝐾)
hlmod.f	⊢ 𝐹 = (pmap‘𝐾)
hlmod.p	⊢ + = (+𝑃‘𝐾)

Assertion

Ref	Expression
hlmod1i	⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌))) → ((𝑋 ∨ 𝑌) ∧ 𝑍) = (𝑋 ∨ (𝑌 ∧ 𝑍))))

Proof of Theorem hlmod1i

Step	Hyp	Ref	Expression
1		hlmod.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		hlmod.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		hllat 37356	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
4	3	3ad2ant1 1131	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → 𝐾 ∈ Lat)
5		simp21 1204	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → 𝑋 ∈ 𝐵)
6		simp22 1205	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → 𝑌 ∈ 𝐵)
7		hlmod.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
8	1, 7	latjcl 18138	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵)
9	4, 5, 6, 8	syl3anc 1369	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝑋 ∨ 𝑌) ∈ 𝐵)
10		simp23 1206	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → 𝑍 ∈ 𝐵)
11		hlmod.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
12	1, 11	latmcl 18139	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 ∨ 𝑌) ∧ 𝑍) ∈ 𝐵)
13	4, 9, 10, 12	syl3anc 1369	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → ((𝑋 ∨ 𝑌) ∧ 𝑍) ∈ 𝐵)
14	1, 11	latmcl 18139	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 ∧ 𝑍) ∈ 𝐵)
15	4, 6, 10, 14	syl3anc 1369	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝑌 ∧ 𝑍) ∈ 𝐵)
16	1, 7	latjcl 18138	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑌 ∧ 𝑍) ∈ 𝐵) → (𝑋 ∨ (𝑌 ∧ 𝑍)) ∈ 𝐵)
17	4, 5, 15, 16	syl3anc 1369	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝑋 ∨ (𝑌 ∧ 𝑍)) ∈ 𝐵)
18		simp1 1134	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → 𝐾 ∈ HL)
19		eqid 2739	. . . . . . . . 9 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
20		hlmod.f	. . . . . . . . 9 ⊢ 𝐹 = (pmap‘𝐾)
21	1, 19, 20	pmapssat 37752	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ⊆ (Atoms‘𝐾))
22	18, 5, 21	syl2anc 583	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝐹‘𝑋) ⊆ (Atoms‘𝐾))
23	1, 19, 20	pmapssat 37752	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾))
24	18, 6, 23	syl2anc 583	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾))
25		eqid 2739	. . . . . . . . 9 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾)
26	1, 25, 20	pmapsub 37761	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑍 ∈ 𝐵) → (𝐹‘𝑍) ∈ (PSubSp‘𝐾))
27	4, 10, 26	syl2anc 583	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝐹‘𝑍) ∈ (PSubSp‘𝐾))
28		simp3l 1199	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → 𝑋 ≤ 𝑍)
29	1, 2, 20	pmaple 37754	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 ≤ 𝑍 ↔ (𝐹‘𝑋) ⊆ (𝐹‘𝑍)))
30	18, 5, 10, 29	syl3anc 1369	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝑋 ≤ 𝑍 ↔ (𝐹‘𝑋) ⊆ (𝐹‘𝑍)))
31	28, 30	mpbid 231	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝐹‘𝑋) ⊆ (𝐹‘𝑍))
32		hlmod.p	. . . . . . . . 9 ⊢ + = (+𝑃‘𝐾)
33	19, 25, 32	pmod1i 37841	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ ((𝐹‘𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹‘𝑌) ⊆ (Atoms‘𝐾) ∧ (𝐹‘𝑍) ∈ (PSubSp‘𝐾))) → ((𝐹‘𝑋) ⊆ (𝐹‘𝑍) → (((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)) = ((𝐹‘𝑋) + ((𝐹‘𝑌) ∩ (𝐹‘𝑍)))))
34	33	3impia 1115	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ ((𝐹‘𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹‘𝑌) ⊆ (Atoms‘𝐾) ∧ (𝐹‘𝑍) ∈ (PSubSp‘𝐾)) ∧ (𝐹‘𝑋) ⊆ (𝐹‘𝑍)) → (((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)) = ((𝐹‘𝑋) + ((𝐹‘𝑌) ∩ (𝐹‘𝑍))))
35	18, 22, 24, 27, 31, 34	syl131anc 1381	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)) = ((𝐹‘𝑋) + ((𝐹‘𝑌) ∩ (𝐹‘𝑍))))
36	1, 11, 19, 20	pmapmeet 37766	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐹‘((𝑋 ∨ 𝑌) ∧ 𝑍)) = ((𝐹‘(𝑋 ∨ 𝑌)) ∩ (𝐹‘𝑍)))
37	18, 9, 10, 36	syl3anc 1369	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝐹‘((𝑋 ∨ 𝑌) ∧ 𝑍)) = ((𝐹‘(𝑋 ∨ 𝑌)) ∩ (𝐹‘𝑍)))
38		simp3r 1200	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))
39	38	ineq1d 4150	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → ((𝐹‘(𝑋 ∨ 𝑌)) ∩ (𝐹‘𝑍)) = (((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)))
40	37, 39	eqtrd 2779	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝐹‘((𝑋 ∨ 𝑌) ∧ 𝑍)) = (((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)))
41	1, 11, 19, 20	pmapmeet 37766	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐹‘(𝑌 ∧ 𝑍)) = ((𝐹‘𝑌) ∩ (𝐹‘𝑍)))
42	18, 6, 10, 41	syl3anc 1369	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝐹‘(𝑌 ∧ 𝑍)) = ((𝐹‘𝑌) ∩ (𝐹‘𝑍)))
43	42	oveq2d 7284	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → ((𝐹‘𝑋) + (𝐹‘(𝑌 ∧ 𝑍))) = ((𝐹‘𝑋) + ((𝐹‘𝑌) ∩ (𝐹‘𝑍))))
44	35, 40, 43	3eqtr4d 2789	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝐹‘((𝑋 ∨ 𝑌) ∧ 𝑍)) = ((𝐹‘𝑋) + (𝐹‘(𝑌 ∧ 𝑍))))
45	1, 7, 20, 32	pmapjoin 37845	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑌 ∧ 𝑍) ∈ 𝐵) → ((𝐹‘𝑋) + (𝐹‘(𝑌 ∧ 𝑍))) ⊆ (𝐹‘(𝑋 ∨ (𝑌 ∧ 𝑍))))
46	4, 5, 15, 45	syl3anc 1369	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → ((𝐹‘𝑋) + (𝐹‘(𝑌 ∧ 𝑍))) ⊆ (𝐹‘(𝑋 ∨ (𝑌 ∧ 𝑍))))
47	44, 46	eqsstrd 3963	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝐹‘((𝑋 ∨ 𝑌) ∧ 𝑍)) ⊆ (𝐹‘(𝑋 ∨ (𝑌 ∧ 𝑍))))
48	1, 2, 20	pmaple 37754	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ ((𝑋 ∨ 𝑌) ∧ 𝑍) ∈ 𝐵 ∧ (𝑋 ∨ (𝑌 ∧ 𝑍)) ∈ 𝐵) → (((𝑋 ∨ 𝑌) ∧ 𝑍) ≤ (𝑋 ∨ (𝑌 ∧ 𝑍)) ↔ (𝐹‘((𝑋 ∨ 𝑌) ∧ 𝑍)) ⊆ (𝐹‘(𝑋 ∨ (𝑌 ∧ 𝑍)))))
49	18, 13, 17, 48	syl3anc 1369	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (((𝑋 ∨ 𝑌) ∧ 𝑍) ≤ (𝑋 ∨ (𝑌 ∧ 𝑍)) ↔ (𝐹‘((𝑋 ∨ 𝑌) ∧ 𝑍)) ⊆ (𝐹‘(𝑋 ∨ (𝑌 ∧ 𝑍)))))
50	47, 49	mpbird 256	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → ((𝑋 ∨ 𝑌) ∧ 𝑍) ≤ (𝑋 ∨ (𝑌 ∧ 𝑍)))
51	1, 2, 7, 11	mod1ile 18192	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑍 → (𝑋 ∨ (𝑌 ∧ 𝑍)) ≤ ((𝑋 ∨ 𝑌) ∧ 𝑍)))
52	51	3impia 1115	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑍) → (𝑋 ∨ (𝑌 ∧ 𝑍)) ≤ ((𝑋 ∨ 𝑌) ∧ 𝑍))
53	4, 5, 6, 10, 28, 52	syl131anc 1381	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝑋 ∨ (𝑌 ∧ 𝑍)) ≤ ((𝑋 ∨ 𝑌) ∧ 𝑍))
54	1, 2, 4, 13, 17, 50, 53	latasymd 18144	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → ((𝑋 ∨ 𝑌) ∧ 𝑍) = (𝑋 ∨ (𝑌 ∧ 𝑍)))
55	54	3expia 1119	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌))) → ((𝑋 ∨ 𝑌) ∧ 𝑍) = (𝑋 ∨ (𝑌 ∧ 𝑍))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109   ∩ cin 3890   ⊆ wss 3891   class class class wbr 5078  ‘cfv 6430  (class class class)co 7268  Basecbs 16893  lecple 16950  joincjn 18010  meetcmee 18011  Latclat 18130  Atomscatm 37256  HLchlt 37343  PSubSpcpsubsp 37489  pmapcpmap 37490  +_𝑃cpadd 37788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-iin 4932  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-proset 17994  df-poset 18012  df-plt 18029  df-lub 18045  df-glb 18046  df-join 18047  df-meet 18048  df-p0 18124  df-lat 18131  df-clat 18198  df-oposet 37169  df-ol 37171  df-oml 37172  df-covers 37259  df-ats 37260  df-atl 37291  df-cvlat 37315  df-hlat 37344  df-psubsp 37496  df-pmap 37497  df-padd 37789

This theorem is referenced by:  atmod1i1  37850  atmod1i2  37852  llnmod1i2  37853