Metamath Proof Explorer


Theorem cdleme50f

Description: Part of proof of Lemma D in Crawley p. 113. TODO: fix comment. TODO: can we use just F Fn B since range is computed in cdleme50rn ? (Contributed by NM, 9-Apr-2013)

Ref Expression
Hypotheses cdlemef50.b 𝐵 = ( Base ‘ 𝐾 )
cdlemef50.l = ( le ‘ 𝐾 )
cdlemef50.j = ( join ‘ 𝐾 )
cdlemef50.m = ( meet ‘ 𝐾 )
cdlemef50.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemef50.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemef50.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdlemef50.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
cdlemefs50.e 𝐸 = ( ( 𝑃 𝑄 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
cdlemef50.f 𝐹 = ( 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 ) )
Assertion cdleme50f ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → 𝐹 : 𝐵𝐵 )

Proof

Step Hyp Ref Expression
1 cdlemef50.b 𝐵 = ( Base ‘ 𝐾 )
2 cdlemef50.l = ( le ‘ 𝐾 )
3 cdlemef50.j = ( join ‘ 𝐾 )
4 cdlemef50.m = ( meet ‘ 𝐾 )
5 cdlemef50.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemef50.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemef50.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 cdlemef50.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
9 cdlemefs50.e 𝐸 = ( ( 𝑃 𝑄 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
10 cdlemef50.f 𝐹 = ( 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 riotaex ( 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 ) ) ) ) ∈ V
12 vex 𝑥 ∈ V
13 11 12 ifex 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 ) ∈ V
14 13 a1i ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 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 ) ∈ V )
15 10 a1i ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → 𝐹 = ( 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 ) ) )
16 vex 𝑠 ∈ V
17 eqid ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) ) = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
18 8 17 cdleme31sc ( 𝑠 ∈ V → 𝑠 / 𝑡 𝐷 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) ) )
19 16 18 ax-mp 𝑠 / 𝑡 𝐷 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
20 eqid ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) = ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) )
21 eqid if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) = if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 )
22 eqid ( 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 ) ) ) ) = ( 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 ) ) ) )
23 1 2 3 4 5 6 7 19 8 9 20 21 22 10 cdleme32fvcl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑒𝐵 ) → ( 𝐹𝑒 ) ∈ 𝐵 )
24 14 15 23 fmpt2d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → 𝐹 : 𝐵𝐵 )