poml4N - Mathbox for Norm Megill

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Theorem poml4N 38416

Description: Orthomodular law for projective lattices. Lemma 3.3(1) in [Holland95] p. 215. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
poml4.a	⊢ 𝐴 = (Atoms‘𝐾)
poml4.p	⊢ ⊥ = (⊥_𝑃‘𝐾)

**Assertion**
Ref	Expression
poml4N	⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝑋 ⊆ 𝑌 ∧ ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))))

**Proof of Theorem poml4N**
Step	Hyp	Ref	Expression
1		eqcom 2743	. . 3 ⊢ (( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌 ↔ 𝑌 = ( ⊥ ‘( ⊥ ‘𝑌)))
2		eqid 2736	. . . . . . 7 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		poml4.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
4		eqid 2736	. . . . . . 7 ⊢ (pmap‘𝐾) = (pmap‘𝐾)
5		poml4.p	. . . . . . 7 ⊢ ⊥ = (⊥_𝑃‘𝐾)
6	2, 3, 4, 5	2polvalN 38377	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑌)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))
7	6	3adant2 1131	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑌)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))
8	7	eqeq2d 2747	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑌 = ( ⊥ ‘( ⊥ ‘𝑌)) ↔ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))))
9	8	biimpd 228	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑌 = ( ⊥ ‘( ⊥ ‘𝑌)) → 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))))
10	1, 9	biimtrid 241	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌 → 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))))
11		simpl1 1191	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝐾 ∈ HL)
12		hloml 37819	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)
13	11, 12	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝐾 ∈ OML)
14		hlclat 37820	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
15	11, 14	syl 17	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝐾 ∈ CLat)
16		simpl2 1192	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑋 ⊆ 𝐴)
17		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
18	17, 3	atssbase 37752	. . . . . . . . 9 ⊢ 𝐴 ⊆ (Base‘𝐾)
19	16, 18	sstrdi 3956	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑋 ⊆ (Base‘𝐾))
20	17, 2	clatlubcl 18392	. . . . . . . 8 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾))
21	15, 19, 20	syl2anc 584	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾))
22		simpl3 1193	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑌 ⊆ 𝐴)
23	22, 18	sstrdi 3956	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑌 ⊆ (Base‘𝐾))
24	17, 2	clatlubcl 18392	. . . . . . . 8 ⊢ ((𝐾 ∈ CLat ∧ 𝑌 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑌) ∈ (Base‘𝐾))
25	15, 23, 24	syl2anc 584	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → ((lub‘𝐾)‘𝑌) ∈ (Base‘𝐾))
26	13, 21, 25	3jca 1128	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → (𝐾 ∈ OML ∧ ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾) ∧ ((lub‘𝐾)‘𝑌) ∈ (Base‘𝐾)))
27		simprl 769	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑋 ⊆ 𝑌)
28		eqid 2736	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
29	17, 28, 2	lubss 18402	. . . . . . 7 ⊢ ((𝐾 ∈ CLat ∧ 𝑌 ⊆ (Base‘𝐾) ∧ 𝑋 ⊆ 𝑌) → ((lub‘𝐾)‘𝑋)(le‘𝐾)((lub‘𝐾)‘𝑌))
30	15, 23, 27, 29	syl3anc 1371	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → ((lub‘𝐾)‘𝑋)(le‘𝐾)((lub‘𝐾)‘𝑌))
31		eqid 2736	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
32		eqid 2736	. . . . . . 7 ⊢ (oc‘𝐾) = (oc‘𝐾)
33	17, 28, 31, 32	omllaw4 37708	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾) ∧ ((lub‘𝐾)‘𝑌) ∈ (Base‘𝐾)) → (((lub‘𝐾)‘𝑋)(le‘𝐾)((lub‘𝐾)‘𝑌) → (((oc‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)))(meet‘𝐾)((lub‘𝐾)‘𝑌)) = ((lub‘𝐾)‘𝑋)))
34	26, 30, 33	sylc 65	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → (((oc‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)))(meet‘𝐾)((lub‘𝐾)‘𝑌)) = ((lub‘𝐾)‘𝑋))
35	34	fveq2d 6846	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → ((pmap‘𝐾)‘(((oc‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)))(meet‘𝐾)((lub‘𝐾)‘𝑌))) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑋)))
36	2, 32, 3, 4, 5	polval2N 38369	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))))
37	11, 16, 36	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → ( ⊥ ‘𝑋) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))))
38		simprr 771	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))
39	37, 38	ineq12d 4173	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → (( ⊥ ‘𝑋) ∩ 𝑌) = (((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∩ ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))))
40		hlop 37824	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
41	11, 40	syl 17	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝐾 ∈ OP)
42	17, 32	opoccl 37656	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾))
43	41, 21, 42	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾))
44	17, 31, 3, 4	pmapmeet 38236	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾) ∧ ((lub‘𝐾)‘𝑌) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌))) = (((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∩ ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))))
45	11, 43, 25, 44	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → ((pmap‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌))) = (((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))) ∩ ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))))
46	39, 45	eqtr4d 2779	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → (( ⊥ ‘𝑋) ∩ 𝑌) = ((pmap‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌))))
47	46	fveq2d 6846	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) = ( ⊥ ‘((pmap‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)))))
48	11	hllatd 37826	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝐾 ∈ Lat)
49	17, 31	latmcl 18329	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾) ∧ ((lub‘𝐾)‘𝑌) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)) ∈ (Base‘𝐾))
50	48, 43, 25, 49	syl3anc 1371	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → (((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)) ∈ (Base‘𝐾))
51	17, 32, 4, 5	polpmapN 38375	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)) ∈ (Base‘𝐾)) → ( ⊥ ‘((pmap‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)))) = ((pmap‘𝐾)‘((oc‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)))))
52	11, 50, 51	syl2anc 584	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → ( ⊥ ‘((pmap‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)))) = ((pmap‘𝐾)‘((oc‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)))))
53	47, 52	eqtrd 2776	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) = ((pmap‘𝐾)‘((oc‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)))))
54	53, 38	ineq12d 4173	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = (((pmap‘𝐾)‘((oc‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)))) ∩ ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))))
55	17, 32	opoccl 37656	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ (((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌))) ∈ (Base‘𝐾))
56	41, 50, 55	syl2anc 584	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → ((oc‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌))) ∈ (Base‘𝐾))
57	17, 31, 3, 4	pmapmeet 38236	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ ((oc‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌))) ∈ (Base‘𝐾) ∧ ((lub‘𝐾)‘𝑌) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘(((oc‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)))(meet‘𝐾)((lub‘𝐾)‘𝑌))) = (((pmap‘𝐾)‘((oc‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)))) ∩ ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))))
58	11, 56, 25, 57	syl3anc 1371	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → ((pmap‘𝐾)‘(((oc‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)))(meet‘𝐾)((lub‘𝐾)‘𝑌))) = (((pmap‘𝐾)‘((oc‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)))) ∩ ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))))
59	54, 58	eqtr4d 2779	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ((pmap‘𝐾)‘(((oc‘𝐾)‘(((oc‘𝐾)‘((lub‘𝐾)‘𝑋))(meet‘𝐾)((lub‘𝐾)‘𝑌)))(meet‘𝐾)((lub‘𝐾)‘𝑌))))
60	2, 3, 4, 5	2polvalN 38377	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑋)))
61	11, 16, 60	syl2anc 584	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → ( ⊥ ‘( ⊥ ‘𝑋)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑋)))
62	35, 59, 61	3eqtr4d 2786	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋)))
63	62	ex 413	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))))
64	10, 63	sylan2d 605	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝑋 ⊆ 𝑌 ∧ ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∩ cin 3909 ⊆ wss 3910 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 lecple 17140 occoc 17141 lubclub 18198 meetcmee 18201 Latclat 18320 CLatccla 18387 OPcops 37634 OMLcoml 37637 Atomscatm 37725 HLchlt 37812 pmapcpmap 37960 ⊥_𝑃cpolN 38365

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-proset 18184 df-poset 18202 df-plt 18219 df-lub 18235 df-glb 18236 df-join 18237 df-meet 18238 df-p0 18314 df-p1 18315 df-lat 18321 df-clat 18388 df-oposet 37638 df-ol 37640 df-oml 37641 df-covers 37728 df-ats 37729 df-atl 37760 df-cvlat 37784 df-hlat 37813 df-pmap 37967 df-polarityN 38366

This theorem is referenced by: poml5N 38417 poml6N 38418 pexmidlem6N 38438

W3C validator