Theorem cdlemg2idN 37724

Description: Version of cdleme31id 37522 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 1297	. . . 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 1298	. . . 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 1199	. . . 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 1200	. . . 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 1299	. . . 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 1194	. . . 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 2819	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
10		eqid 2819	. . . . 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 2819	. . . . 5 ⊢ ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊) = ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊)
15		eqid 2819	. . . . 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 2819	. . . . 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 2819	. . . . 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 37718	. . . 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 1381	. . 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 6665	. 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 1195	. . 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 1133	. . 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 37522	. . 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 586	. 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 2854	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)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1082   = wceq 1531   ∈ wcel 2108   ≠ wne 3014  ∀wral 3136  ⦋csb 3881  ifcif 4465   class class class wbr 5057   ↦ cmpt 5137  ‘cfv 6348  ℩crio 7105  (class class class)co 7148  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  Atomscatm 36391  HLchlt 36478  LHypclh 37112  LTrncltrn 37229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-riotaBAD 36081

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-undef 7931  df-map 8400  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36304  df-ol 36306  df-oml 36307  df-covers 36394  df-ats 36395  df-atl 36426  df-cvlat 36450  df-hlat 36479  df-llines 36626  df-lplanes 36627  df-lvols 36628  df-lines 36629  df-psubsp 36631  df-pmap 36632  df-padd 36924  df-lhyp 37116  df-laut 37117  df-ldil 37232  df-ltrn 37233  df-trl 37287

This theorem is referenced by: (None)