paddunN - Mathbox for Norm Megill

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Theorem paddunN 37062

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 6673.) (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 1132	. . 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 36949	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ 𝐴)
5	2, 3	paddunssN 36943	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ∪ 𝑇) ⊆ (𝑆 + 𝑇))
6		paddun.o	. . . 4 ⊢ ⊥ = (⊥_𝑃‘𝐾)
7	2, 6	polcon3N 37052	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑆 + 𝑇) ⊆ 𝐴 ∧ (𝑆 ∪ 𝑇) ⊆ (𝑆 + 𝑇)) → ( ⊥ ‘(𝑆 + 𝑇)) ⊆ ( ⊥ ‘(𝑆 ∪ 𝑇)))
8	1, 4, 5, 7	syl3anc 1367	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘(𝑆 + 𝑇)) ⊆ ( ⊥ ‘(𝑆 ∪ 𝑇)))
9		hlclat 36493	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
10	9	3ad2ant1 1129	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝐾 ∈ CLat)
11		unss 4159	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) ↔ (𝑆 ∪ 𝑇) ⊆ 𝐴)
12	11	biimpi 218	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ∪ 𝑇) ⊆ 𝐴)
13	12	3adant1 1126	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ∪ 𝑇) ⊆ 𝐴)
14		eqid 2821	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
15	14, 2	atssbase 36425	. . . . . . . . 9 ⊢ 𝐴 ⊆ (Base‘𝐾)
16	13, 15	sstrdi 3978	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ∪ 𝑇) ⊆ (Base‘𝐾))
17		eqid 2821	. . . . . . . . 9 ⊢ (lub‘𝐾) = (lub‘𝐾)
18	14, 17	clatlubcl 17721	. . . . . . . 8 ⊢ ((𝐾 ∈ CLat ∧ (𝑆 ∪ 𝑇) ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘(𝑆 ∪ 𝑇)) ∈ (Base‘𝐾))
19	10, 16, 18	syl2anc 586	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘(𝑆 ∪ 𝑇)) ∈ (Base‘𝐾))
20		eqid 2821	. . . . . . . 8 ⊢ (pmap‘𝐾) = (pmap‘𝐾)
21	14, 20	pmapssbaN 36895	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ ((lub‘𝐾)‘(𝑆 ∪ 𝑇)) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))) ⊆ (Base‘𝐾))
22	1, 19, 21	syl2anc 586	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))) ⊆ (Base‘𝐾))
23	2, 6	polssatN 37043	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ( ⊥ ‘𝑆) ⊆ 𝐴)
24	23	3adant3 1128	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘𝑆) ⊆ 𝐴)
25	2, 6	polssatN 37043	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑆) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝐴)
26	1, 24, 25	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝐴)
27	2, 6	polssatN 37043	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘𝑇) ⊆ 𝐴)
28	27	3adant2 1127	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘𝑇) ⊆ 𝐴)
29	2, 6	polssatN 37043	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑇) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑇)) ⊆ 𝐴)
30	1, 28, 29	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑇)) ⊆ 𝐴)
31	1, 26, 30	3jca 1124	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝐾 ∈ HL ∧ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑇)) ⊆ 𝐴))
32	2, 6	2polssN 37050	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)))
33	32	3adant3 1128	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)))
34	2, 6	2polssN 37050	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ⊆ 𝐴) → 𝑇 ⊆ ( ⊥ ‘( ⊥ ‘𝑇)))
35	34	3adant2 1127	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑇 ⊆ ( ⊥ ‘( ⊥ ‘𝑇)))
36	33, 35	jca 514	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)) ∧ 𝑇 ⊆ ( ⊥ ‘( ⊥ ‘𝑇))))
37	2, 3	paddss12 36954	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑇)) ⊆ 𝐴) → ((𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)) ∧ 𝑇 ⊆ ( ⊥ ‘( ⊥ ‘𝑇))) → (𝑆 + 𝑇) ⊆ (( ⊥ ‘( ⊥ ‘𝑆)) + ( ⊥ ‘( ⊥ ‘𝑇)))))
38	31, 36, 37	sylc 65	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ (( ⊥ ‘( ⊥ ‘𝑆)) + ( ⊥ ‘( ⊥ ‘𝑇))))
39	17, 2, 20, 6	2polvalN 37049	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑆)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑆)))
40	39	3adant3 1128	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑆)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑆)))
41	17, 2, 20, 6	2polvalN 37049	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑇)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑇)))
42	41	3adant2 1127	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑇)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑇)))
43	40, 42	oveq12d 7173	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑆)) + ( ⊥ ‘( ⊥ ‘𝑇))) = (((pmap‘𝐾)‘((lub‘𝐾)‘𝑆)) + ((pmap‘𝐾)‘((lub‘𝐾)‘𝑇))))
44	38, 43	sseqtrd 4006	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ (((pmap‘𝐾)‘((lub‘𝐾)‘𝑆)) + ((pmap‘𝐾)‘((lub‘𝐾)‘𝑇))))
45		hllat 36498	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
46	45	3ad2ant1 1129	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝐾 ∈ Lat)
47		simp2 1133	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑆 ⊆ 𝐴)
48	47, 15	sstrdi 3978	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑆 ⊆ (Base‘𝐾))
49	14, 17	clatlubcl 17721	. . . . . . . . . 10 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑆) ∈ (Base‘𝐾))
50	10, 48, 49	syl2anc 586	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘𝑆) ∈ (Base‘𝐾))
51		simp3 1134	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑇 ⊆ 𝐴)
52	51, 15	sstrdi 3978	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑇 ⊆ (Base‘𝐾))
53	14, 17	clatlubcl 17721	. . . . . . . . . 10 ⊢ ((𝐾 ∈ CLat ∧ 𝑇 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑇) ∈ (Base‘𝐾))
54	10, 52, 53	syl2anc 586	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘𝑇) ∈ (Base‘𝐾))
55		eqid 2821	. . . . . . . . . 10 ⊢ (join‘𝐾) = (join‘𝐾)
56	14, 55, 20, 3	pmapjoin 36987	. . . . . . . . 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 1367	. . . . . . . 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 3976	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ ((pmap‘𝐾)‘(((lub‘𝐾)‘𝑆)(join‘𝐾)((lub‘𝐾)‘𝑇))))
59	14, 55, 17	lubun 17732	. . . . . . . . 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 1367	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘(𝑆 ∪ 𝑇)) = (((lub‘𝐾)‘𝑆)(join‘𝐾)((lub‘𝐾)‘𝑇)))
61	60	fveq2d 6673	. . . . . . 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 4007	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))
63		eqid 2821	. . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾)
64	14, 63, 17	lubss 17730	. . . . . 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 1367	. . . . 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 3978	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ (Base‘𝐾))
67	14, 17	clatlubcl 17721	. . . . . . 7 ⊢ ((𝐾 ∈ CLat ∧ (𝑆 + 𝑇) ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘(𝑆 + 𝑇)) ∈ (Base‘𝐾))
68	10, 66, 67	syl2anc 586	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘(𝑆 + 𝑇)) ∈ (Base‘𝐾))
69	14, 17	clatlubcl 17721	. . . . . . 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 586	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇)))) ∈ (Base‘𝐾))
71	14, 63, 20	pmaple 36896	. . . . . 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 1367	. . . . 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 234	. . . 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 37049	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑆 + 𝑇) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 + 𝑇))) = ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 + 𝑇))))
75	1, 4, 74	syl2anc 586	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 + 𝑇))) = ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 + 𝑇))))
76	17, 2, 20, 6	2polvalN 37049	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∪ 𝑇) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))) = ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))
77	1, 13, 76	syl2anc 586	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))) = ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))
78	17, 2, 20	2pmaplubN 37061	. . . . . 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 586	. . . . 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 2859	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))) = ((pmap‘𝐾)‘((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))))
81	73, 75, 80	3sstr4d 4013	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 + 𝑇))) ⊆ ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))))
82	2, 6	2polcon4bN 37053	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑆 + 𝑇) ⊆ 𝐴 ∧ (𝑆 ∪ 𝑇) ⊆ 𝐴) → (( ⊥ ‘( ⊥ ‘(𝑆 + 𝑇))) ⊆ ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))) ↔ ( ⊥ ‘(𝑆 ∪ 𝑇)) ⊆ ( ⊥ ‘(𝑆 + 𝑇))))
83	1, 4, 13, 82	syl3anc 1367	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (( ⊥ ‘( ⊥ ‘(𝑆 + 𝑇))) ⊆ ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))) ↔ ( ⊥ ‘(𝑆 ∪ 𝑇)) ⊆ ( ⊥ ‘(𝑆 + 𝑇))))
84	81, 83	mpbid 234	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘(𝑆 ∪ 𝑇)) ⊆ ( ⊥ ‘(𝑆 + 𝑇)))
85	8, 84	eqssd 3983	1 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘(𝑆 + 𝑇)) = ( ⊥ ‘(𝑆 ∪ 𝑇)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∪ cun 3933 ⊆ wss 3935 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 lecple 16571 lubclub 17551 joincjn 17553 Latclat 17654 CLatccla 17716 Atomscatm 36398 HLchlt 36485 pmapcpmap 36632 +_𝑃cpadd 36930 ⊥_𝑃cpolN 37037

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-riotaBAD 36088

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-undef 7938 df-proset 17537 df-poset 17555 df-plt 17567 df-lub 17583 df-glb 17584 df-join 17585 df-meet 17586 df-p0 17648 df-p1 17649 df-lat 17655 df-clat 17717 df-oposet 36311 df-ol 36313 df-oml 36314 df-covers 36401 df-ats 36402 df-atl 36433 df-cvlat 36457 df-hlat 36486 df-psubsp 36638 df-pmap 36639 df-padd 36931 df-polarityN 37038

This theorem is referenced by: poldmj1N 37063

W3C validator