paddunN - Mathbox for Norm Megill

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Theorem paddunN 40563

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 6875.) (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 1152	. . 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 40450	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ 𝐴)
5	2, 3	paddunssN 40444	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ∪ 𝑇) ⊆ (𝑆 + 𝑇))
6		paddun.o	. . . 4 ⊢ ⊥ = (⊥_𝑃‘𝐾)
7	2, 6	polcon3N 40553	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑆 + 𝑇) ⊆ 𝐴 ∧ (𝑆 ∪ 𝑇) ⊆ (𝑆 + 𝑇)) → ( ⊥ ‘(𝑆 + 𝑇)) ⊆ ( ⊥ ‘(𝑆 ∪ 𝑇)))
8	1, 4, 5, 7	syl3anc 1394	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘(𝑆 + 𝑇)) ⊆ ( ⊥ ‘(𝑆 ∪ 𝑇)))
9		hlclat 39994	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
10	9	3ad2ant1 1149	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝐾 ∈ CLat)
11		unss 4145	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) ↔ (𝑆 ∪ 𝑇) ⊆ 𝐴)
12	11	biimpi 219	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ∪ 𝑇) ⊆ 𝐴)
13	12	3adant1 1146	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ∪ 𝑇) ⊆ 𝐴)
14		eqid 2765	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
15	14, 2	atssbase 39926	. . . . . . . . 9 ⊢ 𝐴 ⊆ (Base‘𝐾)
16	13, 15	sstrdi 3951	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ∪ 𝑇) ⊆ (Base‘𝐾))
17		eqid 2765	. . . . . . . . 9 ⊢ (lub‘𝐾) = (lub‘𝐾)
18	14, 17	clatlubcl 18549	. . . . . . . 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 2765	. . . . . . . 8 ⊢ (pmap‘𝐾) = (pmap‘𝐾)
21	14, 20	pmapssbaN 40396	. . . . . . 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 40544	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ( ⊥ ‘𝑆) ⊆ 𝐴)
24	23	3adant3 1148	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘𝑆) ⊆ 𝐴)
25	2, 6	polssatN 40544	. . . . . . . . . . . 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 40544	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘𝑇) ⊆ 𝐴)
28	27	3adant2 1147	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘𝑇) ⊆ 𝐴)
29	2, 6	polssatN 40544	. . . . . . . . . . . 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 1144	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝐾 ∈ HL ∧ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑇)) ⊆ 𝐴))
32	2, 6	2polssN 40551	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)))
33	32	3adant3 1148	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)))
34	2, 6	2polssN 40551	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ⊆ 𝐴) → 𝑇 ⊆ ( ⊥ ‘( ⊥ ‘𝑇)))
35	34	3adant2 1147	. . . . . . . . . . 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 40455	. . . . . . . . . 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 40550	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑆)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑆)))
40	39	3adant3 1148	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑆)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑆)))
41	17, 2, 20, 6	2polvalN 40550	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑇)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑇)))
42	41	3adant2 1147	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑇)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑇)))
43	40, 42	oveq12d 7418	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑆)) + ( ⊥ ‘( ⊥ ‘𝑇))) = (((pmap‘𝐾)‘((lub‘𝐾)‘𝑆)) + ((pmap‘𝐾)‘((lub‘𝐾)‘𝑇))))
44	38, 43	sseqtrd 3975	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ (((pmap‘𝐾)‘((lub‘𝐾)‘𝑆)) + ((pmap‘𝐾)‘((lub‘𝐾)‘𝑇))))
45		hllat 39999	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
46	45	3ad2ant1 1149	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝐾 ∈ Lat)
47		simp2 1153	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑆 ⊆ 𝐴)
48	47, 15	sstrdi 3951	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑆 ⊆ (Base‘𝐾))
49	14, 17	clatlubcl 18549	. . . . . . . . . 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 1154	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑇 ⊆ 𝐴)
52	51, 15	sstrdi 3951	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑇 ⊆ (Base‘𝐾))
53	14, 17	clatlubcl 18549	. . . . . . . . . 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 2765	. . . . . . . . . 10 ⊢ (join‘𝐾) = (join‘𝐾)
56	14, 55, 20, 3	pmapjoin 40488	. . . . . . . . 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 1394	. . . . . . . 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 3949	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ ((pmap‘𝐾)‘(((lub‘𝐾)‘𝑆)(join‘𝐾)((lub‘𝐾)‘𝑇))))
59	14, 55, 17	lubun 18561	. . . . . . . . 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 1394	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘(𝑆 ∪ 𝑇)) = (((lub‘𝐾)‘𝑆)(join‘𝐾)((lub‘𝐾)‘𝑇)))
61	60	fveq2d 6875	. . . . . . 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 3976	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))
63		eqid 2765	. . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾)
64	14, 63, 17	lubss 18559	. . . . . 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 1394	. . . . 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 3951	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ (Base‘𝐾))
67	14, 17	clatlubcl 18549	. . . . . . 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 18549	. . . . . . 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 40397	. . . . . 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 1394	. . . . 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 40550	. . . . 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 40550	. . . . . 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 40562	. . . . . 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 2803	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))) = ((pmap‘𝐾)‘((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))))
81	73, 75, 80	3sstr4d 3994	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 + 𝑇))) ⊆ ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))))
82	2, 6	2polcon4bN 40554	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑆 + 𝑇) ⊆ 𝐴 ∧ (𝑆 ∪ 𝑇) ⊆ 𝐴) → (( ⊥ ‘( ⊥ ‘(𝑆 + 𝑇))) ⊆ ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))) ↔ ( ⊥ ‘(𝑆 ∪ 𝑇)) ⊆ ( ⊥ ‘(𝑆 + 𝑇))))
83	1, 4, 13, 82	syl3anc 1394	. . 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 3956	1 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘(𝑆 + 𝑇)) = ( ⊥ ‘(𝑆 ∪ 𝑇)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ∪ cun 3905 ⊆ wss 3907 class class class wbr 5105 ‘cfv 6525 (class class class)co 7400 Basecbs 17259 lecple 17307 lubclub 18355 joincjn 18357 Latclat 18477 CLatccla 18544 Atomscatm 39899 HLchlt 39986 pmapcpmap 40133 +_𝑃cpadd 40431 ⊥_𝑃cpolN 40538

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-id 5547 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-1st 7974 df-2nd 7975 df-proset 18340 df-poset 18359 df-plt 18374 df-lub 18390 df-glb 18391 df-join 18392 df-meet 18393 df-p0 18469 df-p1 18470 df-lat 18478 df-clat 18545 df-oposet 39812 df-ol 39814 df-oml 39815 df-covers 39902 df-ats 39903 df-atl 39934 df-cvlat 39958 df-hlat 39987 df-psubsp 40139 df-pmap 40140 df-padd 40432 df-polarityN 40539

This theorem is referenced by: poldmj1N 40564

W3C validator