poml4N - Mathbox for Norm Megill

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Theorem poml4N 39910

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 2747	. . 3 ⊢ (( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌 ↔ 𝑌 = ( ⊥ ‘( ⊥ ‘𝑌)))
2		eqid 2740	. . . . . . 7 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		poml4.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
4		eqid 2740	. . . . . . 7 ⊢ (pmap‘𝐾) = (pmap‘𝐾)
5		poml4.p	. . . . . . 7 ⊢ ⊥ = (⊥_𝑃‘𝐾)
6	2, 3, 4, 5	2polvalN 39871	. . . . . 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 2751	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑌 = ( ⊥ ‘( ⊥ ‘𝑌)) ↔ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))))
9	8	biimpd 229	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑌 = ( ⊥ ‘( ⊥ ‘𝑌)) → 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))))
10	1, 9	biimtrid 242	. 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 39313	. . . . . . . 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 39314	. . . . . . . . 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 2740	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
18	17, 3	atssbase 39246	. . . . . . . . 9 ⊢ 𝐴 ⊆ (Base‘𝐾)
19	16, 18	sstrdi 4021	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑋 ⊆ (Base‘𝐾))
20	17, 2	clatlubcl 18573	. . . . . . . 8 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾))
21	15, 19, 20	syl2anc 583	. . . . . . 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 4021	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑌 ⊆ (Base‘𝐾))
24	17, 2	clatlubcl 18573	. . . . . . . 8 ⊢ ((𝐾 ∈ CLat ∧ 𝑌 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑌) ∈ (Base‘𝐾))
25	15, 23, 24	syl2anc 583	. . . . . . 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 770	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑋 ⊆ 𝑌)
28		eqid 2740	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
29	17, 28, 2	lubss 18583	. . . . . . 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 2740	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
32		eqid 2740	. . . . . . 7 ⊢ (oc‘𝐾) = (oc‘𝐾)
33	17, 28, 31, 32	omllaw4 39202	. . . . . 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 6924	. . . 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 39863	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))))
37	11, 16, 36	syl2anc 583	. . . . . . . . . 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 772	. . . . . . . . . 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 4242	. . . . . . . . 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 39318	. . . . . . . . . . . 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 39150	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾))
43	41, 21, 42	syl2anc 583	. . . . . . . . . 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 39730	. . . . . . . . . 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 2783	. . . . . . . 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 6924	. . . . . . 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 39320	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝐾 ∈ Lat)
49	17, 31	latmcl 18510	. . . . . . . . 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 39869	. . . . . . . 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 583	. . . . . . 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 2780	. . . . . 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 4242	. . . . 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 39150	. . . . . . 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 583	. . . . . 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 39730	. . . . . 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 2783	. . . 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 39871	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑋)))
61	11, 16, 60	syl2anc 583	. . . 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 2790	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋)))
63	62	ex 412	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))))
64	10, 63	sylan2d 604	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 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∩ cin 3975 ⊆ wss 3976 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 lecple 17318 occoc 17319 lubclub 18379 meetcmee 18382 Latclat 18501 CLatccla 18568 OPcops 39128 OMLcoml 39131 Atomscatm 39219 HLchlt 39306 pmapcpmap 39454 ⊥_𝑃cpolN 39859

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-proset 18365 df-poset 18383 df-plt 18400 df-lub 18416 df-glb 18417 df-join 18418 df-meet 18419 df-p0 18495 df-p1 18496 df-lat 18502 df-clat 18569 df-oposet 39132 df-ol 39134 df-oml 39135 df-covers 39222 df-ats 39223 df-atl 39254 df-cvlat 39278 df-hlat 39307 df-pmap 39461 df-polarityN 39860

This theorem is referenced by: poml5N 39911 poml6N 39912 pexmidlem6N 39932

W3C validator