Theorem osumcllem3N

Description: Lemma for osumclN . (Contributed by NM, 23-Mar-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	osumcllem.l	⊢ ≤ = ( le ‘ 𝐾 )
	osumcllem.j	⊢ ∨ = ( join ‘ 𝐾 )
	osumcllem.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	osumcllem.p	⊢ + = ( +𝑃 ‘ 𝐾 )
	osumcllem.o	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
	osumcllem.c	⊢ 𝐶 = ( PSubCl ‘ 𝐾 )
	osumcllem.m	⊢ 𝑀 = ( 𝑋 + { 𝑝 } )
	osumcllem.u	⊢ 𝑈 = ( ⊥ ‘ ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) )
Assertion	osumcllem3N	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → ( ( ⊥ ‘ 𝑋 ) ∩ 𝑈 ) = 𝑌 )

Proof

Step	Hyp	Ref	Expression
1		osumcllem.l	⊢ ≤ = ( le ‘ 𝐾 )
2		osumcllem.j	⊢ ∨ = ( join ‘ 𝐾 )
3		osumcllem.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		osumcllem.p	⊢ + = ( +𝑃 ‘ 𝐾 )
5		osumcllem.o	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
6		osumcllem.c	⊢ 𝐶 = ( PSubCl ‘ 𝐾 )
7		osumcllem.m	⊢ 𝑀 = ( 𝑋 + { 𝑝 } )
8		osumcllem.u	⊢ 𝑈 = ( ⊥ ‘ ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) )
9		incom	⊢ ( ( ⊥ ‘ 𝑋 ) ∩ 𝑈 ) = ( 𝑈 ∩ ( ⊥ ‘ 𝑋 ) )
10		simp1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → 𝐾 ∈ HL )
11		simp3	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) )
12	3 6	psubclssatN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ) → 𝑌 ⊆ 𝐴 )
13	12	3adant3	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → 𝑌 ⊆ 𝐴 )
14	3 5	polssatN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ) → ( ⊥ ‘ 𝑌 ) ⊆ 𝐴 )
15	10 13 14	syl2anc	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → ( ⊥ ‘ 𝑌 ) ⊆ 𝐴 )
16	11 15	sstrd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → 𝑋 ⊆ 𝐴 )
17	3 4 5	poldmj1N	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) = ( ( ⊥ ‘ 𝑋 ) ∩ ( ⊥ ‘ 𝑌 ) ) )
18	10 16 13 17	syl3anc	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) = ( ( ⊥ ‘ 𝑋 ) ∩ ( ⊥ ‘ 𝑌 ) ) )
19		incom	⊢ ( ( ⊥ ‘ 𝑋 ) ∩ ( ⊥ ‘ 𝑌 ) ) = ( ( ⊥ ‘ 𝑌 ) ∩ ( ⊥ ‘ 𝑋 ) )
20	18 19	eqtrdi	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) = ( ( ⊥ ‘ 𝑌 ) ∩ ( ⊥ ‘ 𝑋 ) ) )
21	20	fveq2d	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) ) = ( ⊥ ‘ ( ( ⊥ ‘ 𝑌 ) ∩ ( ⊥ ‘ 𝑋 ) ) ) )
22	8 21	eqtrid	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → 𝑈 = ( ⊥ ‘ ( ( ⊥ ‘ 𝑌 ) ∩ ( ⊥ ‘ 𝑋 ) ) ) )
23	22	ineq1d	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → ( 𝑈 ∩ ( ⊥ ‘ 𝑋 ) ) = ( ( ⊥ ‘ ( ( ⊥ ‘ 𝑌 ) ∩ ( ⊥ ‘ 𝑋 ) ) ) ∩ ( ⊥ ‘ 𝑋 ) ) )
24	3 5	polcon2N	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → 𝑌 ⊆ ( ⊥ ‘ 𝑋 ) )
25	13 24	syld3an2	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → 𝑌 ⊆ ( ⊥ ‘ 𝑋 ) )
26	3 5	poml5N	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ ( ⊥ ‘ 𝑋 ) ) → ( ( ⊥ ‘ ( ( ⊥ ‘ 𝑌 ) ∩ ( ⊥ ‘ 𝑋 ) ) ) ∩ ( ⊥ ‘ 𝑋 ) ) = ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) )
27	10 16 25 26	syl3anc	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → ( ( ⊥ ‘ ( ( ⊥ ‘ 𝑌 ) ∩ ( ⊥ ‘ 𝑋 ) ) ) ∩ ( ⊥ ‘ 𝑋 ) ) = ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) )
28	5 6	psubcli2N	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 )
29	28	3adant3	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 )
30	23 27 29	3eqtrd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → ( 𝑈 ∩ ( ⊥ ‘ 𝑋 ) ) = 𝑌 )
31	9 30	eqtrid	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → ( ( ⊥ ‘ 𝑋 ) ∩ 𝑈 ) = 𝑌 )