Theorem hlmod1i 39459

Description: A version of the modular law pmod1i 39451 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 38965	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
4	3	3ad2ant1 1130	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → 𝐾 ∈ Lat)
5		simp21 1203	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → 𝑋 ∈ 𝐵)
6		simp22 1204	. . . . 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 18434	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵)
9	4, 5, 6, 8	syl3anc 1368	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝑋 ∨ 𝑌) ∈ 𝐵)
10		simp23 1205	. . . 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 18435	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 ∨ 𝑌) ∧ 𝑍) ∈ 𝐵)
13	4, 9, 10, 12	syl3anc 1368	. . 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 18435	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 ∧ 𝑍) ∈ 𝐵)
15	4, 6, 10, 14	syl3anc 1368	. . . 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 18434	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑌 ∧ 𝑍) ∈ 𝐵) → (𝑋 ∨ (𝑌 ∧ 𝑍)) ∈ 𝐵)
17	4, 5, 15, 16	syl3anc 1368	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝑋 ∨ (𝑌 ∧ 𝑍)) ∈ 𝐵)
18		simp1 1133	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → 𝐾 ∈ HL)
19		eqid 2725	. . . . . . . . 9 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
20		hlmod.f	. . . . . . . . 9 ⊢ 𝐹 = (pmap‘𝐾)
21	1, 19, 20	pmapssat 39362	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ⊆ (Atoms‘𝐾))
22	18, 5, 21	syl2anc 582	. . . . . . 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 39362	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾))
24	18, 6, 23	syl2anc 582	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾))
25		eqid 2725	. . . . . . . . 9 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾)
26	1, 25, 20	pmapsub 39371	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑍 ∈ 𝐵) → (𝐹‘𝑍) ∈ (PSubSp‘𝐾))
27	4, 10, 26	syl2anc 582	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝐹‘𝑍) ∈ (PSubSp‘𝐾))
28		simp3l 1198	. . . . . . . 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 39364	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 ≤ 𝑍 ↔ (𝐹‘𝑋) ⊆ (𝐹‘𝑍)))
30	18, 5, 10, 29	syl3anc 1368	. . . . . . . 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 39451	. . . . . . . 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 1114	. . . . . . 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 1380	. . . . . 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 39376	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐹‘((𝑋 ∨ 𝑌) ∧ 𝑍)) = ((𝐹‘(𝑋 ∨ 𝑌)) ∩ (𝐹‘𝑍)))
37	18, 9, 10, 36	syl3anc 1368	. . . . . . 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 1199	. . . . . . . 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 4209	. . . . . . 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 2765	. . . . . 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 39376	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐹‘(𝑌 ∧ 𝑍)) = ((𝐹‘𝑌) ∩ (𝐹‘𝑍)))
42	18, 6, 10, 41	syl3anc 1368	. . . . . . 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 7435	. . . . . 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 2775	. . . . 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 39455	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑌 ∧ 𝑍) ∈ 𝐵) → ((𝐹‘𝑋) + (𝐹‘(𝑌 ∧ 𝑍))) ⊆ (𝐹‘(𝑋 ∨ (𝑌 ∧ 𝑍))))
46	4, 5, 15, 45	syl3anc 1368	. . . . 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 4015	. . . 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 39364	. . . . 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 1368	. . . 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 18488	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑍 → (𝑋 ∨ (𝑌 ∧ 𝑍)) ≤ ((𝑋 ∨ 𝑌) ∧ 𝑍)))
52	51	3impia 1114	. . . 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 1380	. . 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 18440	. 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 1118	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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ∩ cin 3943   ⊆ wss 3944   class class class wbr 5149  ‘cfv 6549  (class class class)co 7419  Basecbs 17183  lecple 17243  joincjn 18306  meetcmee 18307  Latclat 18426  Atomscatm 38865  HLchlt 38952  PSubSpcpsubsp 39099  pmapcpmap 39100  +_𝑃cpadd 39398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-proset 18290  df-poset 18308  df-plt 18325  df-lub 18341  df-glb 18342  df-join 18343  df-meet 18344  df-p0 18420  df-lat 18427  df-clat 18494  df-oposet 38778  df-ol 38780  df-oml 38781  df-covers 38868  df-ats 38869  df-atl 38900  df-cvlat 38924  df-hlat 38953  df-psubsp 39106  df-pmap 39107  df-padd 39399

This theorem is referenced by:  atmod1i1  39460  atmod1i2  39462  llnmod1i2  39463