poml4N - Mathbox for Norm Megill

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Theorem poml4N 40827

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 2767	. . 3 ⊢ (( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌 ↔ 𝑌 = ( ⊥ ‘( ⊥ ‘𝑌)))
2		eqid 2760	. . . . . . 7 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		poml4.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
4		eqid 2760	. . . . . . 7 ⊢ (pmap‘𝐾) = (pmap‘𝐾)
5		poml4.p	. . . . . . 7 ⊢ ⊥ = (⊥_𝑃‘𝐾)
6	2, 3, 4, 5	2polvalN 40788	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑌)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))
7	6	3adant2 1149	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑌)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))
8	7	eqeq2d 2771	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑌 = ( ⊥ ‘( ⊥ ‘𝑌)) ↔ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))))
9	8	biimpd 232	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑌 = ( ⊥ ‘( ⊥ ‘𝑌)) → 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))))
10	1, 9	biimtrid 245	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌 → 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))))
11		simpl1 1210	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝐾 ∈ HL)
12		hloml 40231	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)
13	11, 12	syl 18	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝐾 ∈ OML)
14		hlclat 40232	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
15	11, 14	syl 18	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝐾 ∈ CLat)
16		simpl2 1211	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑋 ⊆ 𝐴)
17		eqid 2760	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
18	17, 3	atssbase 40164	. . . . . . . . 9 ⊢ 𝐴 ⊆ (Base‘𝐾)
19	16, 18	sstrdi 3943	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑋 ⊆ (Base‘𝐾))
20	17, 2	clatlubcl 18592	. . . . . . . 8 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾))
21	15, 19, 20	syl2anc 596	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾))
22		simpl3 1212	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑌 ⊆ 𝐴)
23	22, 18	sstrdi 3943	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑌 ⊆ (Base‘𝐾))
24	17, 2	clatlubcl 18592	. . . . . . . 8 ⊢ ((𝐾 ∈ CLat ∧ 𝑌 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑌) ∈ (Base‘𝐾))
25	15, 23, 24	syl2anc 596	. . . . . . 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 1146	. . . . . 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 783	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑋 ⊆ 𝑌)
28		eqid 2760	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
29	17, 28, 2	lubss 18602	. . . . . . 7 ⊢ ((𝐾 ∈ CLat ∧ 𝑌 ⊆ (Base‘𝐾) ∧ 𝑋 ⊆ 𝑌) → ((lub‘𝐾)‘𝑋)(le‘𝐾)((lub‘𝐾)‘𝑌))
30	15, 23, 27, 29	syl3anc 1398	. . . . . 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 2760	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
32		eqid 2760	. . . . . . 7 ⊢ (oc‘𝐾) = (oc‘𝐾)
33	17, 28, 31, 32	omllaw4 40120	. . . . . 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 66	. . . . 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 6883	. . . 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 40780	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))))
37	11, 16, 36	syl2anc 596	. . . . . . . . . 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 785	. . . . . . . . . 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 4167	. . . . . . . . 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 40236	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
41	11, 40	syl 18	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝐾 ∈ OP)
42	17, 32	opoccl 40068	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾))
43	41, 21, 42	syl2anc 596	. . . . . . . . . 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 40647	. . . . . . . . . 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 1398	. . . . . . . . 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 2798	. . . . . . . 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 6883	. . . . . . 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 40238	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝐾 ∈ Lat)
49	17, 31	latmcl 18529	. . . . . . . . 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 1398	. . . . . . . 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 40786	. . . . . . . 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 596	. . . . . . 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 2795	. . . . . 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 4167	. . . . 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 40068	. . . . . . 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 596	. . . . . 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 40647	. . . . . 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 1398	. . . . 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 2798	. . . 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 40788	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑋)))
61	11, 16, 60	syl2anc 596	. . . 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 2805	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋)))
63	62	ex 418	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))))
64	10, 63	sylan2d 617	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝑋 ⊆ 𝑌 ∧ ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∩ cin 3898 ⊆ wss 3899 class class class wbr 5103 ‘cfv 6533 (class class class)co 7414 Basecbs 17302 lecple 17350 occoc 17351 lubclub 18398 meetcmee 18401 Latclat 18520 CLatccla 18587 OPcops 40046 OMLcoml 40049 Atomscatm 40137 HLchlt 40224 pmapcpmap 40371 ⊥_𝑃cpolN 40776

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 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 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-proset 18383 df-poset 18402 df-plt 18417 df-lub 18433 df-glb 18434 df-join 18435 df-meet 18436 df-p0 18512 df-p1 18513 df-lat 18521 df-clat 18588 df-oposet 40050 df-ol 40052 df-oml 40053 df-covers 40140 df-ats 40141 df-atl 40172 df-cvlat 40196 df-hlat 40225 df-pmap 40378 df-polarityN 40777

This theorem is used by: poml5N 40828 poml6N 40829 pexmidlem6N 40849

W3C validator