poml4N - Mathbox for Norm Megill

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Theorem poml4N 40616

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 2776	. . 3 ⊢ (( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌 ↔ 𝑌 = ( ⊥ ‘( ⊥ ‘𝑌)))
2		eqid 2769	. . . . . . 7 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		poml4.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
4		eqid 2769	. . . . . . 7 ⊢ (pmap‘𝐾) = (pmap‘𝐾)
5		poml4.p	. . . . . . 7 ⊢ ⊥ = (⊥_𝑃‘𝐾)
6	2, 3, 4, 5	2polvalN 40577	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑌)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))
7	6	3adant2 1147	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑌)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))
8	7	eqeq2d 2780	. . . 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 1208	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝐾 ∈ HL)
12		hloml 40020	. . . . . . . 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 40021	. . . . . . . . 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 1209	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑋 ⊆ 𝐴)
17		eqid 2769	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
18	17, 3	atssbase 39953	. . . . . . . . 9 ⊢ 𝐴 ⊆ (Base‘𝐾)
19	16, 18	sstrdi 3957	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑋 ⊆ (Base‘𝐾))
20	17, 2	clatlubcl 18558	. . . . . . . 8 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾))
21	15, 19, 20	syl2anc 595	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾))
22		simpl3 1210	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑌 ⊆ 𝐴)
23	22, 18	sstrdi 3957	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑌 ⊆ (Base‘𝐾))
24	17, 2	clatlubcl 18558	. . . . . . . 8 ⊢ ((𝐾 ∈ CLat ∧ 𝑌 ⊆ (Base‘𝐾)) → ((lub‘𝐾)‘𝑌) ∈ (Base‘𝐾))
25	15, 23, 24	syl2anc 595	. . . . . . 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 1144	. . . . . 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 782	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝑋 ⊆ 𝑌)
28		eqid 2769	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
29	17, 28, 2	lubss 18568	. . . . . . 7 ⊢ ((𝐾 ∈ CLat ∧ 𝑌 ⊆ (Base‘𝐾) ∧ 𝑋 ⊆ 𝑌) → ((lub‘𝐾)‘𝑋)(le‘𝐾)((lub‘𝐾)‘𝑌))
30	15, 23, 27, 29	syl3anc 1396	. . . . . 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 2769	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
32		eqid 2769	. . . . . . 7 ⊢ (oc‘𝐾) = (oc‘𝐾)
33	17, 28, 31, 32	omllaw4 39909	. . . . . 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 6886	. . . 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 40569	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝑋))))
37	11, 16, 36	syl2anc 595	. . . . . . . . . 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 784	. . . . . . . . . 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 4182	. . . . . . . . 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 40025	. . . . . . . . . . . 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 39857	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ ((lub‘𝐾)‘𝑋) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((lub‘𝐾)‘𝑋)) ∈ (Base‘𝐾))
43	41, 21, 42	syl2anc 595	. . . . . . . . . 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 40436	. . . . . . . . . 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 1396	. . . . . . . . 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 2807	. . . . . . . 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 6886	. . . . . . 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 40027	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → 𝐾 ∈ Lat)
49	17, 31	latmcl 18495	. . . . . . . . 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 1396	. . . . . . . 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 40575	. . . . . . . 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 595	. . . . . . 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 2804	. . . . . 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 4182	. . . . 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 39857	. . . . . . 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 595	. . . . . 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 40436	. . . . . 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 1396	. . . . 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 2807	. . . 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 40577	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑋)))
61	11, 16, 60	syl2anc 595	. . . 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 2814	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌)))) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋)))
63	62	ex 417	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝑋 ⊆ 𝑌 ∧ 𝑌 = ((pmap‘𝐾)‘((lub‘𝐾)‘𝑌))) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))))
64	10, 63	sylan2d 616	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 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∩ cin 3912 ⊆ wss 3913 class class class wbr 5113 ‘cfv 6537 (class class class)co 7411 Basecbs 17268 lecple 17316 occoc 17317 lubclub 18364 meetcmee 18367 Latclat 18486 CLatccla 18553 OPcops 39835 OMLcoml 39838 Atomscatm 39926 HLchlt 40013 pmapcpmap 40160 ⊥_𝑃cpolN 40565

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-id 5557 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-proset 18349 df-poset 18368 df-plt 18383 df-lub 18399 df-glb 18400 df-join 18401 df-meet 18402 df-p0 18478 df-p1 18479 df-lat 18487 df-clat 18554 df-oposet 39839 df-ol 39841 df-oml 39842 df-covers 39929 df-ats 39930 df-atl 39961 df-cvlat 39985 df-hlat 40014 df-pmap 40167 df-polarityN 40566

This theorem is referenced by: poml5N 40617 poml6N 40618 pexmidlem6N 40638

W3C validator