Theorem hlmod1i 41105

Description: A version of the modular law pmod1i 41097 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 40612	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
4	3	3ad2ant1 1151	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → 𝐾 ∈ Lat)
5		simp21 1225	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → 𝑋 ∈ 𝐵)
6		simp22 1226	. . . . 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 18642	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵)
9	4, 5, 6, 8	syl3anc 1399	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝑋 ∨ 𝑌) ∈ 𝐵)
10		simp23 1227	. . . 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 18643	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 ∨ 𝑌) ∧ 𝑍) ∈ 𝐵)
13	4, 9, 10, 12	syl3anc 1399	. . 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 18643	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 ∧ 𝑍) ∈ 𝐵)
15	4, 6, 10, 14	syl3anc 1399	. . . 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 18642	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑌 ∧ 𝑍) ∈ 𝐵) → (𝑋 ∨ (𝑌 ∧ 𝑍)) ∈ 𝐵)
17	4, 5, 15, 16	syl3anc 1399	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝑋 ∨ (𝑌 ∧ 𝑍)) ∈ 𝐵)
18		simp1 1154	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → 𝐾 ∈ HL)
19		eqid 2763	. . . . . . . . 9 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
20		hlmod.f	. . . . . . . . 9 ⊢ 𝐹 = (pmap‘𝐾)
21	1, 19, 20	pmapssat 41008	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ⊆ (Atoms‘𝐾))
22	18, 5, 21	syl2anc 596	. . . . . . 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 41008	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾))
24	18, 6, 23	syl2anc 596	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾))
25		eqid 2763	. . . . . . . . 9 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾)
26	1, 25, 20	pmapsub 41017	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑍 ∈ 𝐵) → (𝐹‘𝑍) ∈ (PSubSp‘𝐾))
27	4, 10, 26	syl2anc 596	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))) → (𝐹‘𝑍) ∈ (PSubSp‘𝐾))
28		simp3l 1220	. . . . . . . 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 41010	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 ≤ 𝑍 ↔ (𝐹‘𝑋) ⊆ (𝐹‘𝑍)))
30	18, 5, 10, 29	syl3anc 1399	. . . . . . . 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 235	. . . . . . 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 41097	. . . . . . . 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 1135	. . . . . . 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 1411	. . . . . 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 41022	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐹‘((𝑋 ∨ 𝑌) ∧ 𝑍)) = ((𝐹‘(𝑋 ∨ 𝑌)) ∩ (𝐹‘𝑍)))
37	18, 9, 10, 36	syl3anc 1399	. . . . . . 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 1221	. . . . . . . 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 4119	. . . . . . 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 2798	. . . . . 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 41022	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐹‘(𝑌 ∧ 𝑍)) = ((𝐹‘𝑌) ∩ (𝐹‘𝑍)))
42	18, 6, 10, 41	syl3anc 1399	. . . . . . 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 7459	. . . . . 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 2808	. . . . 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 41101	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑌 ∧ 𝑍) ∈ 𝐵) → ((𝐹‘𝑋) + (𝐹‘(𝑌 ∧ 𝑍))) ⊆ (𝐹‘(𝑋 ∨ (𝑌 ∧ 𝑍))))
46	4, 5, 15, 45	syl3anc 1399	. . . . 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 3947	. . . 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 41010	. . . . 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 1399	. . . 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 260	. . 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 18696	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑍 → (𝑋 ∨ (𝑌 ∧ 𝑍)) ≤ ((𝑋 ∨ 𝑌) ∧ 𝑍)))
52	51	3impia 1135	. . . 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 1411	. . 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 18648	. 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 1139	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌))) → ((𝑋 ∨ 𝑌) ∧ 𝑍) = (𝑋 ∨ (𝑌 ∧ 𝑍))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∩ cin 4092   class class class wbr 5106  ‘cfv 6558  (class class class)co 7443  Basecbs 17416  lecple 17464  joincjn 18514  meetcmee 18515  Latclat 18634  Atomscatm 40512  HLchlt 40599  PSubSpcpsubsp 40745  pmapcpmap 40746  +_𝑃cpadd 41044

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-proset 18497  df-poset 18516  df-plt 18531  df-lub 18547  df-glb 18548  df-join 18549  df-meet 18550  df-p0 18626  df-lat 18635  df-clat 18702  df-oposet 40425  df-ol 40427  df-oml 40428  df-covers 40515  df-ats 40516  df-atl 40547  df-cvlat 40571  df-hlat 40600  df-psubsp 40752  df-pmap 40753  df-padd 41045

This theorem is used by:  atmod1i1  41106  atmod1i2  41108  llnmod1i2  41109