paddunN - Mathbox for Norm Megill

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Theorem paddunN 40729

Description: The closure of the projective sum of two sets of atoms is the same as the closure of their union. (Closure is actually double polarity, which can be trivially inferred from this theorem using fveq2d 6885.) (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
paddun.a	⊢ 𝐴 = (Atoms‘𝐾)
paddun.p	⊢ + = (+_𝑃‘𝐾)
paddun.o	⊢ ⊥ = (⊥_𝑃‘𝐾)

**Assertion**
Ref	Expression
paddunN	⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘(𝑆 + 𝑇)) = ( ⊥ ‘(𝑆 ∪ 𝑇)))

**Proof of Theorem paddunN**
Step	Hyp	Ref	Expression
1		simp1 1154	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝐾 ∈ HL)
2		paddun.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
3		paddun.p	. . . 4 ⊢ + = (+_𝑃‘𝐾)
4	2, 3	paddssat 40616	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ 𝐴)
5	2, 3	paddunssN 40610	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ∪ 𝑇) ⊆ (𝑆 + 𝑇))
6		paddun.o	. . . 4 ⊢ ⊥ = (⊥_𝑃‘𝐾)
7	2, 6	polcon3N 40719	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑆 + 𝑇) ⊆ 𝐴 ∧ (𝑆 ∪ 𝑇) ⊆ (𝑆 + 𝑇)) → ( ⊥ ‘(𝑆 + 𝑇)) ⊆ ( ⊥ ‘(𝑆 ∪ 𝑇)))
8	1, 4, 5, 7	syl3anc 1398	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘(𝑆 + 𝑇)) ⊆ ( ⊥ ‘(𝑆 ∪ 𝑇)))
9		hlclat 40160	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
10	9	3ad2ant1 1151	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝐾 ∈ CLat)
11		unss 4143	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) ↔ (𝑆 ∪ 𝑇) ⊆ 𝐴)
12	11	biimpi 219	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ∪ 𝑇) ⊆ 𝐴)
13	12	3adant1 1148	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ∪ 𝑇) ⊆ 𝐴)
14		eqid 2763	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
15	14, 2	atssbase 40092	. . . . . . . . 9 ⊢ 𝐴 ⊆ (Base‘𝐾)
16	13, 15	sstrdi 3949	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ∪ 𝑇) ⊆ (Base‘𝐾))
17		eqid 2763	. . . . . . . . 9 ⊢ (lub‘𝐾) = (lub‘𝐾)
18	14, 17	clatlubcl 18563	. . . . . . . 8 ⊢ ((𝐾 ∈ CLat ∧ (𝑆 ∪ 𝑇) ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘(𝑆 ∪ 𝑇)) ∈ (Base‘𝐾))
19	10, 16, 18	syl2anc 595	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘(𝑆 ∪ 𝑇)) ∈ (Base‘𝐾))
20		eqid 2763	. . . . . . . 8 ⊢ (pmap‘𝐾) = (pmap‘𝐾)
21	14, 20	pmapssbaN 40562	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ ((lub‘𝐾)‘(𝑆 ∪ 𝑇)) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))) ⊆ (Base‘𝐾))
22	1, 19, 21	syl2anc 595	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))) ⊆ (Base‘𝐾))
23	2, 6	polssatN 40710	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ( ⊥ ‘𝑆) ⊆ 𝐴)
24	23	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘𝑆) ⊆ 𝐴)
25	2, 6	polssatN 40710	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑆) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝐴)
26	1, 24, 25	syl2anc 595	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝐴)
27	2, 6	polssatN 40710	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘𝑇) ⊆ 𝐴)
28	27	3adant2 1149	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘𝑇) ⊆ 𝐴)
29	2, 6	polssatN 40710	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑇) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑇)) ⊆ 𝐴)
30	1, 28, 29	syl2anc 595	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑇)) ⊆ 𝐴)
31	1, 26, 30	3jca 1146	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝐾 ∈ HL ∧ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑇)) ⊆ 𝐴))
32	2, 6	2polssN 40717	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)))
33	32	3adant3 1150	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)))
34	2, 6	2polssN 40717	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ⊆ 𝐴) → 𝑇 ⊆ ( ⊥ ‘( ⊥ ‘𝑇)))
35	34	3adant2 1149	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑇 ⊆ ( ⊥ ‘( ⊥ ‘𝑇)))
36	33, 35	jca 520	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)) ∧ 𝑇 ⊆ ( ⊥ ‘( ⊥ ‘𝑇))))
37	2, 3	paddss12 40621	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑇)) ⊆ 𝐴) → ((𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)) ∧ 𝑇 ⊆ ( ⊥ ‘( ⊥ ‘𝑇))) → (𝑆 + 𝑇) ⊆ (( ⊥ ‘( ⊥ ‘𝑆)) + ( ⊥ ‘( ⊥ ‘𝑇)))))
38	31, 36, 37	sylc 66	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ (( ⊥ ‘( ⊥ ‘𝑆)) + ( ⊥ ‘( ⊥ ‘𝑇))))
39	17, 2, 20, 6	2polvalN 40716	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑆)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑆)))
40	39	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑆)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑆)))
41	17, 2, 20, 6	2polvalN 40716	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑇)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑇)))
42	41	3adant2 1149	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑇)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑇)))
43	40, 42	oveq12d 7428	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑆)) + ( ⊥ ‘( ⊥ ‘𝑇))) = (((pmap‘𝐾)‘((lub‘𝐾)‘𝑆)) + ((pmap‘𝐾)‘((lub‘𝐾)‘𝑇))))
44	38, 43	sseqtrd 3973	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ (((pmap‘𝐾)‘((lub‘𝐾)‘𝑆)) + ((pmap‘𝐾)‘((lub‘𝐾)‘𝑇))))
45		hllat 40165	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
46	45	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝐾 ∈ Lat)
47		simp2 1155	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑆 ⊆ 𝐴)
48	47, 15	sstrdi 3949	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑆 ⊆ (Base‘𝐾))
49	14, 17	clatlubcl 18563	. . . . . . . . . 10 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑆) ∈ (Base‘𝐾))
50	10, 48, 49	syl2anc 595	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘𝑆) ∈ (Base‘𝐾))
51		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑇 ⊆ 𝐴)
52	51, 15	sstrdi 3949	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑇 ⊆ (Base‘𝐾))
53	14, 17	clatlubcl 18563	. . . . . . . . . 10 ⊢ ((𝐾 ∈ CLat ∧ 𝑇 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑇) ∈ (Base‘𝐾))
54	10, 52, 53	syl2anc 595	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘𝑇) ∈ (Base‘𝐾))
55		eqid 2763	. . . . . . . . . 10 ⊢ (join‘𝐾) = (join‘𝐾)
56	14, 55, 20, 3	pmapjoin 40654	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ ((lub‘𝐾)‘𝑆) ∈ (Base‘𝐾) ∧ ((lub‘𝐾)‘𝑇) ∈ (Base‘𝐾)) → (((pmap‘𝐾)‘((lub‘𝐾)‘𝑆)) + ((pmap‘𝐾)‘((lub‘𝐾)‘𝑇))) ⊆ ((pmap‘𝐾)‘(((lub‘𝐾)‘𝑆)(join‘𝐾)((lub‘𝐾)‘𝑇))))
57	46, 50, 54, 56	syl3anc 1398	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (((pmap‘𝐾)‘((lub‘𝐾)‘𝑆)) + ((pmap‘𝐾)‘((lub‘𝐾)‘𝑇))) ⊆ ((pmap‘𝐾)‘(((lub‘𝐾)‘𝑆)(join‘𝐾)((lub‘𝐾)‘𝑇))))
58	44, 57	sstrd 3947	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ ((pmap‘𝐾)‘(((lub‘𝐾)‘𝑆)(join‘𝐾)((lub‘𝐾)‘𝑇))))
59	14, 55, 17	lubun 18575	. . . . . . . . 9 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾) ∧ 𝑇 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘(𝑆 ∪ 𝑇)) = (((lub‘𝐾)‘𝑆)(join‘𝐾)((lub‘𝐾)‘𝑇)))
60	10, 48, 52, 59	syl3anc 1398	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘(𝑆 ∪ 𝑇)) = (((lub‘𝐾)‘𝑆)(join‘𝐾)((lub‘𝐾)‘𝑇)))
61	60	fveq2d 6885	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))) = ((pmap‘𝐾)‘(((lub‘𝐾)‘𝑆)(join‘𝐾)((lub‘𝐾)‘𝑇))))
62	58, 61	sseqtrrd 3974	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))
63		eqid 2763	. . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾)
64	14, 63, 17	lubss 18573	. . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))) ⊆ (Base‘𝐾) ∧ (𝑆 + 𝑇) ⊆ ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇)))) → ((lub‘𝐾)‘(𝑆 + 𝑇))(le‘𝐾)((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇)))))
65	10, 22, 62, 64	syl3anc 1398	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘(𝑆 + 𝑇))(le‘𝐾)((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇)))))
66	4, 15	sstrdi 3949	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ (Base‘𝐾))
67	14, 17	clatlubcl 18563	. . . . . . 7 ⊢ ((𝐾 ∈ CLat ∧ (𝑆 + 𝑇) ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘(𝑆 + 𝑇)) ∈ (Base‘𝐾))
68	10, 66, 67	syl2anc 595	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘(𝑆 + 𝑇)) ∈ (Base‘𝐾))
69	14, 17	clatlubcl 18563	. . . . . . 7 ⊢ ((𝐾 ∈ CLat ∧ ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))) ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇)))) ∈ (Base‘𝐾))
70	10, 22, 69	syl2anc 595	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇)))) ∈ (Base‘𝐾))
71	14, 63, 20	pmaple 40563	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ ((lub‘𝐾)‘(𝑆 + 𝑇)) ∈ (Base‘𝐾) ∧ ((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇)))) ∈ (Base‘𝐾)) → (((lub‘𝐾)‘(𝑆 + 𝑇))(le‘𝐾)((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇)))) ↔ ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 + 𝑇))) ⊆ ((pmap‘𝐾)‘((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇)))))))
72	1, 68, 70, 71	syl3anc 1398	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (((lub‘𝐾)‘(𝑆 + 𝑇))(le‘𝐾)((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇)))) ↔ ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 + 𝑇))) ⊆ ((pmap‘𝐾)‘((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇)))))))
73	65, 72	mpbid 235	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 + 𝑇))) ⊆ ((pmap‘𝐾)‘((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))))
74	17, 2, 20, 6	2polvalN 40716	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑆 + 𝑇) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 + 𝑇))) = ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 + 𝑇))))
75	1, 4, 74	syl2anc 595	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 + 𝑇))) = ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 + 𝑇))))
76	17, 2, 20, 6	2polvalN 40716	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∪ 𝑇) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))) = ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))
77	1, 13, 76	syl2anc 595	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))) = ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))
78	17, 2, 20	2pmaplubN 40728	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∪ 𝑇) ⊆ 𝐴) → ((pmap‘𝐾)‘((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))) = ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))
79	1, 13, 78	syl2anc 595	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((pmap‘𝐾)‘((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))) = ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))
80	77, 79	eqtr4d 2801	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))) = ((pmap‘𝐾)‘((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))))
81	73, 75, 80	3sstr4d 3992	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 + 𝑇))) ⊆ ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))))
82	2, 6	2polcon4bN 40720	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑆 + 𝑇) ⊆ 𝐴 ∧ (𝑆 ∪ 𝑇) ⊆ 𝐴) → (( ⊥ ‘( ⊥ ‘(𝑆 + 𝑇))) ⊆ ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))) ↔ ( ⊥ ‘(𝑆 ∪ 𝑇)) ⊆ ( ⊥ ‘(𝑆 + 𝑇))))
83	1, 4, 13, 82	syl3anc 1398	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (( ⊥ ‘( ⊥ ‘(𝑆 + 𝑇))) ⊆ ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))) ↔ ( ⊥ ‘(𝑆 ∪ 𝑇)) ⊆ ( ⊥ ‘(𝑆 + 𝑇))))
84	81, 83	mpbid 235	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘(𝑆 ∪ 𝑇)) ⊆ ( ⊥ ‘(𝑆 + 𝑇)))
85	8, 84	eqssd 3954	1 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘(𝑆 + 𝑇)) = ( ⊥ ‘(𝑆 ∪ 𝑇)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ∪ cun 3903 ⊆ wss 3905 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 Basecbs 17273 lecple 17321 lubclub 18369 joincjn 18371 Latclat 18491 CLatccla 18558 Atomscatm 40065 HLchlt 40152 pmapcpmap 40299 +_𝑃cpadd 40597 ⊥_𝑃cpolN 40704

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7982 df-2nd 7983 df-proset 18354 df-poset 18373 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18492 df-clat 18559 df-oposet 39978 df-ol 39980 df-oml 39981 df-covers 40068 df-ats 40069 df-atl 40100 df-cvlat 40124 df-hlat 40153 df-psubsp 40305 df-pmap 40306 df-padd 40598 df-polarityN 40705

This theorem is used by: poldmj1N 40730

W3C validator