Theorem cdlemg2idN 41654

Description: Version of cdleme31id 41452 with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 21-Apr-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemg2id.l	⊢ ≤ = (le‘𝐾)
cdlemg2id.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg2id.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg2id.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg2id.b	⊢ 𝐵 = (Base‘𝐾)

Assertion

Ref	Expression
cdlemg2idN	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹‘𝑃) = 𝑄 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 = 𝑄) → (𝐹‘𝑋) = 𝑋)

Proof of Theorem cdlemg2idN

Dummy variables 𝑡 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp111 1321	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹‘𝑃) = 𝑄 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 = 𝑄) → 𝐾 ∈ HL)
2		simp112 1322	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹‘𝑃) = 𝑄 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 = 𝑄) → 𝑊 ∈ 𝐻)
3		simp12 1223	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹‘𝑃) = 𝑄 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 = 𝑄) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
4		simp13 1224	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹‘𝑃) = 𝑄 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 = 𝑄) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
5		simp113 1323	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹‘𝑃) = 𝑄 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 = 𝑄) → 𝐹 ∈ 𝑇)
6		simp2l 1218	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹‘𝑃) = 𝑄 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 = 𝑄) → (𝐹‘𝑃) = 𝑄)
7		cdlemg2id.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
8		cdlemg2id.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
9		eqid 2760	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
10		eqid 2760	. . . . 5 ⊢ (meet‘𝐾) = (meet‘𝐾)
11		cdlemg2id.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
12		cdlemg2id.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
13		cdlemg2id.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
14		eqid 2760	. . . . 5 ⊢ ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊) = ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊)
15		eqid 2760	. . . . 5 ⊢ ((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊))) = ((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))
16		eqid 2760	. . . . 5 ⊢ ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)(((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))(join‘𝐾)((𝑠(join‘𝐾)𝑡)(meet‘𝐾)𝑊))) = ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)(((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))(join‘𝐾)((𝑠(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))
17		eqid 2760	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠(join‘𝐾)(𝑥(meet‘𝐾)𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃(join‘𝐾)𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃(join‘𝐾)𝑄)) → 𝑦 = ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)(((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))(join‘𝐾)((𝑠(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))), ⦋𝑠 / 𝑡⦌((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))(join‘𝐾)(𝑥(meet‘𝐾)𝑊)))), 𝑥)) = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠(join‘𝐾)(𝑥(meet‘𝐾)𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃(join‘𝐾)𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃(join‘𝐾)𝑄)) → 𝑦 = ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)(((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))(join‘𝐾)((𝑠(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))), ⦋𝑠 / 𝑡⦌((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))(join‘𝐾)(𝑥(meet‘𝐾)𝑊)))), 𝑥))
18	7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17	cdlemg2dN 41648	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑄)) → 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠(join‘𝐾)(𝑥(meet‘𝐾)𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃(join‘𝐾)𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃(join‘𝐾)𝑄)) → 𝑦 = ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)(((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))(join‘𝐾)((𝑠(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))), ⦋𝑠 / 𝑡⦌((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))(join‘𝐾)(𝑥(meet‘𝐾)𝑊)))), 𝑥)))
19	1, 2, 3, 4, 5, 6, 18	syl222anc 1413	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹‘𝑃) = 𝑄 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 = 𝑄) → 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠(join‘𝐾)(𝑥(meet‘𝐾)𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃(join‘𝐾)𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃(join‘𝐾)𝑄)) → 𝑦 = ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)(((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))(join‘𝐾)((𝑠(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))), ⦋𝑠 / 𝑡⦌((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))(join‘𝐾)(𝑥(meet‘𝐾)𝑊)))), 𝑥)))
20	19	fveq1d 6887	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹‘𝑃) = 𝑄 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 = 𝑄) → (𝐹‘𝑋) = ((𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠(join‘𝐾)(𝑥(meet‘𝐾)𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃(join‘𝐾)𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃(join‘𝐾)𝑄)) → 𝑦 = ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)(((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))(join‘𝐾)((𝑠(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))), ⦋𝑠 / 𝑡⦌((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))(join‘𝐾)(𝑥(meet‘𝐾)𝑊)))), 𝑥))‘𝑋))
21		simp2r 1219	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹‘𝑃) = 𝑄 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 = 𝑄) → 𝑋 ∈ 𝐵)
22		simp3 1156	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹‘𝑃) = 𝑄 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 = 𝑄) → 𝑃 = 𝑄)
23	17	cdleme31id 41452	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑃 = 𝑄) → ((𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠(join‘𝐾)(𝑥(meet‘𝐾)𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃(join‘𝐾)𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃(join‘𝐾)𝑄)) → 𝑦 = ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)(((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))(join‘𝐾)((𝑠(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))), ⦋𝑠 / 𝑡⦌((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))(join‘𝐾)(𝑥(meet‘𝐾)𝑊)))), 𝑥))‘𝑋) = 𝑋)
24	21, 22, 23	syl2anc 596	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹‘𝑃) = 𝑄 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 = 𝑄) → ((𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠(join‘𝐾)(𝑥(meet‘𝐾)𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃(join‘𝐾)𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃(join‘𝐾)𝑄)) → 𝑦 = ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)(((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))(join‘𝐾)((𝑠(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))), ⦋𝑠 / 𝑡⦌((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))(join‘𝐾)(𝑥(meet‘𝐾)𝑊)))), 𝑥))‘𝑋) = 𝑋)
25	20, 24	eqtrd 2795	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹‘𝑃) = 𝑄 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 = 𝑄) → (𝐹‘𝑋) = 𝑋)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ⦋csb 3846  ifcif 4481   class class class wbr 5102   ↦ cmpt 5185  ‘cfv 6538  ℩crio 7376  (class class class)co 7420  Basecbs 17387  lecple 17435  joincjn 18485  meetcmee 18486  Atomscatm 40321  HLchlt 40408  LHypclh 41042  LTrncltrn 41159

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-riotaBAD 40011

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-undef 8290  df-map 8849  df-proset 18468  df-poset 18487  df-plt 18502  df-lub 18518  df-glb 18519  df-join 18520  df-meet 18521  df-p0 18597  df-p1 18598  df-lat 18606  df-clat 18673  df-oposet 40234  df-ol 40236  df-oml 40237  df-covers 40324  df-ats 40325  df-atl 40356  df-cvlat 40380  df-hlat 40409  df-llines 40556  df-lplanes 40557  df-lvols 40558  df-lines 40559  df-psubsp 40561  df-pmap 40562  df-padd 40854  df-lhyp 41046  df-laut 41047  df-ldil 41162  df-ltrn 41163  df-trl 41217

This theorem is used by: (None)