paddunN - Mathbox for Norm Megill

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Theorem paddunN 41060

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 6889.) (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 40947	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ 𝐴)
5	2, 3	paddunssN 40941	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ∪ 𝑇) ⊆ (𝑆 + 𝑇))
6		paddun.o	. . . 4 ⊢ ⊥ = (⊥_𝑃‘𝐾)
7	2, 6	polcon3N 41050	. . 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 40491	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
10	9	3ad2ant1 1151	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝐾 ∈ CLat)
11		unss 4188	. . . . . . . . . . 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 2760	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
15	14, 2	atssbase 40423	. . . . . . . . 9 ⊢ 𝐴 ⊆ (Base‘𝐾)
16	13, 15	sstrdi 3920	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ∪ 𝑇) ⊆ (Base‘𝐾))
17		eqid 2760	. . . . . . . . 9 ⊢ (lub‘𝐾) = (lub‘𝐾)
18	14, 17	clatlubcl 18678	. . . . . . . 8 ⊢ ((𝐾 ∈ CLat ∧ (𝑆 ∪ 𝑇) ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘(𝑆 ∪ 𝑇)) ∈ (Base‘𝐾))
19	10, 16, 18	syl2anc 596	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘(𝑆 ∪ 𝑇)) ∈ (Base‘𝐾))
20		eqid 2760	. . . . . . . 8 ⊢ (pmap‘𝐾) = (pmap‘𝐾)
21	14, 20	pmapssbaN 40893	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ ((lub‘𝐾)‘(𝑆 ∪ 𝑇)) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))) ⊆ (Base‘𝐾))
22	1, 19, 21	syl2anc 596	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))) ⊆ (Base‘𝐾))
23	2, 6	polssatN 41041	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ( ⊥ ‘𝑆) ⊆ 𝐴)
24	23	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘𝑆) ⊆ 𝐴)
25	2, 6	polssatN 41041	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑆) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝐴)
26	1, 24, 25	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝐴)
27	2, 6	polssatN 41041	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘𝑇) ⊆ 𝐴)
28	27	3adant2 1149	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘𝑇) ⊆ 𝐴)
29	2, 6	polssatN 41041	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑇) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑇)) ⊆ 𝐴)
30	1, 28, 29	syl2anc 596	. . . . . . . . . . 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 41048	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)))
33	32	3adant3 1150	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)))
34	2, 6	2polssN 41048	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ⊆ 𝐴) → 𝑇 ⊆ ( ⊥ ‘( ⊥ ‘𝑇)))
35	34	3adant2 1149	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑇 ⊆ ( ⊥ ‘( ⊥ ‘𝑇)))
36	33, 35	jca 521	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)) ∧ 𝑇 ⊆ ( ⊥ ‘( ⊥ ‘𝑇))))
37	2, 3	paddss12 40952	. . . . . . . . . 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 41047	. . . . . . . . . . 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 41047	. . . . . . . . . . 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 7438	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑆)) + ( ⊥ ‘( ⊥ ‘𝑇))) = (((pmap‘𝐾)‘((lub‘𝐾)‘𝑆)) + ((pmap‘𝐾)‘((lub‘𝐾)‘𝑇))))
44	38, 43	sseqtrd 3944	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ (((pmap‘𝐾)‘((lub‘𝐾)‘𝑆)) + ((pmap‘𝐾)‘((lub‘𝐾)‘𝑇))))
45		hllat 40496	. . . . . . . . . 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 3920	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑆 ⊆ (Base‘𝐾))
49	14, 17	clatlubcl 18678	. . . . . . . . . 10 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑆) ∈ (Base‘𝐾))
50	10, 48, 49	syl2anc 596	. . . . . . . . 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 3920	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → 𝑇 ⊆ (Base‘𝐾))
53	14, 17	clatlubcl 18678	. . . . . . . . . 10 ⊢ ((𝐾 ∈ CLat ∧ 𝑇 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑇) ∈ (Base‘𝐾))
54	10, 52, 53	syl2anc 596	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘𝑇) ∈ (Base‘𝐾))
55		eqid 2760	. . . . . . . . . 10 ⊢ (join‘𝐾) = (join‘𝐾)
56	14, 55, 20, 3	pmapjoin 40985	. . . . . . . . 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 3918	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ ((pmap‘𝐾)‘(((lub‘𝐾)‘𝑆)(join‘𝐾)((lub‘𝐾)‘𝑇))))
59	14, 55, 17	lubun 18690	. . . . . . . . 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 6889	. . . . . . 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 3945	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))
63		eqid 2760	. . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾)
64	14, 63, 17	lubss 18688	. . . . . 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 3920	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → (𝑆 + 𝑇) ⊆ (Base‘𝐾))
67	14, 17	clatlubcl 18678	. . . . . . 7 ⊢ ((𝐾 ∈ CLat ∧ (𝑆 + 𝑇) ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘(𝑆 + 𝑇)) ∈ (Base‘𝐾))
68	10, 66, 67	syl2anc 596	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘(𝑆 + 𝑇)) ∈ (Base‘𝐾))
69	14, 17	clatlubcl 18678	. . . . . . 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 596	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇)))) ∈ (Base‘𝐾))
71	14, 63, 20	pmaple 40894	. . . . . 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 41047	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑆 + 𝑇) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 + 𝑇))) = ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 + 𝑇))))
75	1, 4, 74	syl2anc 596	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 + 𝑇))) = ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 + 𝑇))))
76	17, 2, 20, 6	2polvalN 41047	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∪ 𝑇) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))) = ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))
77	1, 13, 76	syl2anc 596	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))) = ((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))
78	17, 2, 20	2pmaplubN 41059	. . . . . 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 596	. . . . 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 2798	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))) = ((pmap‘𝐾)‘((lub‘𝐾)‘((pmap‘𝐾)‘((lub‘𝐾)‘(𝑆 ∪ 𝑇))))))
81	73, 75, 80	3sstr4d 3963	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑆 + 𝑇))) ⊆ ( ⊥ ‘( ⊥ ‘(𝑆 ∪ 𝑇))))
82	2, 6	2polcon4bN 41051	. . . 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 3925	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 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ⊆ wss 3895 ∪ cun 4149 class class class wbr 5102 ‘cfv 6538 (class class class)co 7420 Basecbs 17388 lecple 17436 lubclub 18484 joincjn 18486 Latclat 18606 CLatccla 18673 Atomscatm 40396 HLchlt 40483 pmapcpmap 40630 +_𝑃cpadd 40928 ⊥_𝑃cpolN 41035

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-proset 18469 df-poset 18488 df-plt 18503 df-lub 18519 df-glb 18520 df-join 18521 df-meet 18522 df-p0 18598 df-p1 18599 df-lat 18607 df-clat 18674 df-oposet 40309 df-ol 40311 df-oml 40312 df-covers 40399 df-ats 40400 df-atl 40431 df-cvlat 40455 df-hlat 40484 df-psubsp 40636 df-pmap 40637 df-padd 40929 df-polarityN 41036

This theorem is used by: poldmj1N 41061

W3C validator