Theorem cdlemg2jlemOLDN

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. f preserves join: f(r \/ s) = f(r) \/ s, p. 115 10th line from bottom. TODO: Combine with cdlemg2jOLDN ? (Contributed by NM, 22-Apr-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemg2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cdlemg2.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdlemg2.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdlemg2.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdlemg2.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdlemg2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdlemg2.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	cdlemg2ex.u	⊢ 𝑈 = ( ( 𝑝 ∨ 𝑞 ) ∧ 𝑊 )
	cdlemg2ex.d	⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑡 ) ∧ 𝑊 ) ) )
	cdlemg2ex.e	⊢ 𝐸 = ( ( 𝑝 ∨ 𝑞 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
	cdlemg2ex.g	⊢ 𝐺 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑝 ∨ 𝑞 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑝 ∨ 𝑞 ) ) → 𝑦 = 𝐸 ) ) , ⦋ 𝑠 / 𝑡 ⦌ 𝐷 ) ∨ ( 𝑥 ∧ 𝑊 ) ) ) ) , 𝑥 ) )
Assertion	cdlemg2jlemOLDN	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemg2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cdlemg2.l	⊢ ≤ = ( le ‘ 𝐾 )
3		cdlemg2.j	⊢ ∨ = ( join ‘ 𝐾 )
4		cdlemg2.m	⊢ ∧ = ( meet ‘ 𝐾 )
5		cdlemg2.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6		cdlemg2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7		cdlemg2.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8		cdlemg2ex.u	⊢ 𝑈 = ( ( 𝑝 ∨ 𝑞 ) ∧ 𝑊 )
9		cdlemg2ex.d	⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑡 ) ∧ 𝑊 ) ) )
10		cdlemg2ex.e	⊢ 𝐸 = ( ( 𝑝 ∨ 𝑞 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
11		cdlemg2ex.g	⊢ 𝐺 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑝 ∨ 𝑞 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑝 ∨ 𝑞 ) ) → 𝑦 = 𝐸 ) ) , ⦋ 𝑠 / 𝑡 ⦌ 𝐷 ) ∨ ( 𝑥 ∧ 𝑊 ) ) ) ) , 𝑥 ) )
12		fveq1	⊢ ( 𝐹 = 𝐺 → ( 𝐹 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( 𝐺 ‘ ( 𝑃 ∨ 𝑄 ) ) )
13		fveq1	⊢ ( 𝐹 = 𝐺 → ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) )
14		fveq1	⊢ ( 𝐹 = 𝐺 → ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) )
15	13 14	oveq12d	⊢ ( 𝐹 = 𝐺 → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) )
16	12 15	eqeq12d	⊢ ( 𝐹 = 𝐺 → ( ( 𝐹 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) ↔ ( 𝐺 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) )
17		vex	⊢ 𝑠 ∈ V
18		eqid	⊢ ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑠 ) ∧ 𝑊 ) ) ) = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑠 ) ∧ 𝑊 ) ) )
19	9 18	cdleme31sc	⊢ ( 𝑠 ∈ V → ⦋ 𝑠 / 𝑡 ⦌ 𝐷 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑠 ) ∧ 𝑊 ) ) ) )
20	17 19	ax-mp	⊢ ⦋ 𝑠 / 𝑡 ⦌ 𝐷 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑠 ) ∧ 𝑊 ) ) )
21		eqid	⊢ ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑝 ∨ 𝑞 ) ) → 𝑦 = 𝐸 ) ) = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑝 ∨ 𝑞 ) ) → 𝑦 = 𝐸 ) )
22		eqid	⊢ if ( 𝑠 ≤ ( 𝑝 ∨ 𝑞 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑝 ∨ 𝑞 ) ) → 𝑦 = 𝐸 ) ) , ⦋ 𝑠 / 𝑡 ⦌ 𝐷 ) = if ( 𝑠 ≤ ( 𝑝 ∨ 𝑞 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑝 ∨ 𝑞 ) ) → 𝑦 = 𝐸 ) ) , ⦋ 𝑠 / 𝑡 ⦌ 𝐷 )
23		eqid	⊢ ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑝 ∨ 𝑞 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑝 ∨ 𝑞 ) ) → 𝑦 = 𝐸 ) ) , ⦋ 𝑠 / 𝑡 ⦌ 𝐷 ) ∨ ( 𝑥 ∧ 𝑊 ) ) ) ) = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑝 ∨ 𝑞 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑝 ∨ 𝑞 ) ) → 𝑦 = 𝐸 ) ) , ⦋ 𝑠 / 𝑡 ⦌ 𝐷 ) ∨ ( 𝑥 ∧ 𝑊 ) ) ) )
24	1 2 3 4 5 6 8 20 9 10 21 22 23 11	cdleme42mgN	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( 𝐺 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) )
25	1 2 3 4 5 6 7 8 9 10 11 16 24	cdlemg2ce	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( 𝐹 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) )
26	25	3com23	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) )