Theorem hlmod1i 40302

Description: A version of the modular law pmod1i 40294 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 39809	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
4	3	3ad2ant1 1134	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → 𝐾 ∈ Lat)
5		simp21 1208	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → 𝑋 ∈ 𝐵)
6		simp22 1209	. . . . 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 18405	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵)
9	4, 5, 6, 8	syl3anc 1374	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝑋 ∨ 𝑌) ∈ 𝐵)
10		simp23 1210	. . . 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 18406	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 ∨ 𝑌) ∧ 𝑍) ∈ 𝐵)
13	4, 9, 10, 12	syl3anc 1374	. . 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 18406	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 ∧ 𝑍) ∈ 𝐵)
15	4, 6, 10, 14	syl3anc 1374	. . . 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 18405	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑌 ∧ 𝑍) ∈ 𝐵) → (𝑋 ∨ (𝑌 ∧ 𝑍)) ∈ 𝐵)
17	4, 5, 15, 16	syl3anc 1374	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝑋 ∨ (𝑌 ∧ 𝑍)) ∈ 𝐵)
18		simp1 1137	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → 𝐾 ∈ HL)
19		eqid 2736	. . . . . . . . 9 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
20		hlmod.f	. . . . . . . . 9 ⊢ 𝐹 = (pmap‘𝐾)
21	1, 19, 20	pmapssat 40205	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ⊆ (Atoms‘𝐾))
22	18, 5, 21	syl2anc 585	. . . . . . 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 40205	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾))
24	18, 6, 23	syl2anc 585	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾))
25		eqid 2736	. . . . . . . . 9 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾)
26	1, 25, 20	pmapsub 40214	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑍 ∈ 𝐵) → (𝐹‘𝑍) ∈ (PSubSp‘𝐾))
27	4, 10, 26	syl2anc 585	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝐹‘𝑍) ∈ (PSubSp‘𝐾))
28		simp3l 1203	. . . . . . . 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 40207	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 ≤ 𝑍 ↔ (𝐹‘𝑋) ⊆ (𝐹‘𝑍)))
30	18, 5, 10, 29	syl3anc 1374	. . . . . . . 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 232	. . . . . . 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 40294	. . . . . . . 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 1118	. . . . . . 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 1386	. . . . . 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 40219	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐹‘((𝑋 ∨ 𝑌) ∧ 𝑍)) = ((𝐹‘(𝑋 ∨ 𝑌)) ∩ (𝐹‘𝑍)))
37	18, 9, 10, 36	syl3anc 1374	. . . . . . 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 1204	. . . . . . . 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 4159	. . . . . . 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 2771	. . . . . 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 40219	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐹‘(𝑌 ∧ 𝑍)) = ((𝐹‘𝑌) ∩ (𝐹‘𝑍)))
42	18, 6, 10, 41	syl3anc 1374	. . . . . . 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 7383	. . . . . 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 2781	. . . . 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 40298	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑌 ∧ 𝑍) ∈ 𝐵) → ((𝐹‘𝑋) + (𝐹‘(𝑌 ∧ 𝑍))) ⊆ (𝐹‘(𝑋 ∨ (𝑌 ∧ 𝑍))))
46	4, 5, 15, 45	syl3anc 1374	. . . . 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 3956	. . . 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 40207	. . . . 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 1374	. . . 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 257	. . 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 18459	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑍 → (𝑋 ∨ (𝑌 ∧ 𝑍)) ≤ ((𝑋 ∨ 𝑌) ∧ 𝑍)))
52	51	3impia 1118	. . . 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 1386	. . 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 18411	. 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 1122	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∩ cin 3888   ⊆ wss 3889   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  lecple 17227  joincjn 18277  meetcmee 18278  Latclat 18397  Atomscatm 39709  HLchlt 39796  PSubSpcpsubsp 39942  pmapcpmap 39943  +_𝑃cpadd 40241

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-proset 18260  df-poset 18279  df-plt 18294  df-lub 18310  df-glb 18311  df-join 18312  df-meet 18313  df-p0 18389  df-lat 18398  df-clat 18465  df-oposet 39622  df-ol 39624  df-oml 39625  df-covers 39712  df-ats 39713  df-atl 39744  df-cvlat 39768  df-hlat 39797  df-psubsp 39949  df-pmap 39950  df-padd 40242

This theorem is referenced by:  atmod1i1  40303  atmod1i2  40305  llnmod1i2  40306