Metamath Proof Explorer


Theorem cdlemg1idlemN

Description: Lemma for cdlemg1idN . (Contributed by NM, 18-Apr-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemg1.b 𝐵 = ( Base ‘ 𝐾 )
cdlemg1.l = ( le ‘ 𝐾 )
cdlemg1.j = ( join ‘ 𝐾 )
cdlemg1.m = ( meet ‘ 𝐾 )
cdlemg1.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg1.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg1.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdlemg1.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
cdlemg1.e 𝐸 = ( ( 𝑃 𝑄 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
cdlemg1.g 𝐺 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) ) , 𝑥 ) )
cdlemg1.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg1.f 𝐹 = ( 𝑓𝑇 ( 𝑓𝑃 ) = 𝑄 )
Assertion cdlemg1idlemN ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑋𝐵 ) ∧ 𝑃 = 𝑄 ) → ( 𝐹𝑋 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 cdlemg1.b 𝐵 = ( Base ‘ 𝐾 )
2 cdlemg1.l = ( le ‘ 𝐾 )
3 cdlemg1.j = ( join ‘ 𝐾 )
4 cdlemg1.m = ( meet ‘ 𝐾 )
5 cdlemg1.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemg1.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemg1.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 cdlemg1.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
9 cdlemg1.e 𝐸 = ( ( 𝑃 𝑄 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
10 cdlemg1.g 𝐺 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) ) , 𝑥 ) )
11 cdlemg1.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
12 cdlemg1.f 𝐹 = ( 𝑓𝑇 ( 𝑓𝑃 ) = 𝑄 )
13 1 2 3 4 5 6 7 8 9 10 11 12 cdlemg1b2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → 𝐹 = 𝐺 )
14 13 fveq1d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ( 𝐹𝑋 ) = ( 𝐺𝑋 ) )
15 10 cdleme31id ( ( 𝑋𝐵𝑃 = 𝑄 ) → ( 𝐺𝑋 ) = 𝑋 )
16 14 15 sylan9eq ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑋𝐵𝑃 = 𝑄 ) ) → ( 𝐹𝑋 ) = 𝑋 )
17 16 anassrs ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑋𝐵 ) ∧ 𝑃 = 𝑄 ) → ( 𝐹𝑋 ) = 𝑋 )