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Metamath Proof Explorer

Theorem cdlemg2idN

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

Ref Expression
Hypotheses 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 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

Step Hyp Ref Expression
1 cdlemg2id.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg2id.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 cdlemg2id.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 cdlemg2id.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 cdlemg2id.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
6 simp111 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = 𝑄 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑃 = 𝑄 ) → 𝐾 ∈ HL )
7 simp112 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = 𝑄 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑃 = 𝑄 ) → 𝑊 ∈ 𝐻 )
8 simp12 ⊢ ( ( ( ( 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 ) )
9 simp13 ⊢ ( ( ( ( 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 ) )
10 simp113 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = 𝑄 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑃 = 𝑄 ) → 𝐹 ∈ 𝑇 )
11 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = 𝑄 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑃 = 𝑄 ) → ( 𝐹 ‘ 𝑃 ) = 𝑄 )
12 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
13 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
14 eqid ⊢ ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 )
15 eqid ⊢ ( ( 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 ⊢ ( ( 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 ⊢ ( 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 5 1 12 13 2 3 4 14 15 16 17 cdlemg2dN ⊢ ( ( ( 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 6 7 8 9 10 11 18 syl222anc ⊢ ( ( ( ( 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 ⊢ ( ( ( ( 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 ⊢ ( ( ( ( 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 ⊢ ( ( ( ( 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 ⊢ ( ( 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 ⊢ ( ( ( ( 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 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = 𝑄 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑃 = 𝑄 ) → ( 𝐹 ‘ 𝑋 ) = 𝑋 )