Metamath Proof Explorer


Theorem cdleme50rnlem

Description: Part of proof of Lemma D in Crawley p. 113. TODO: fix comment. TODO: can we get rid of G stuff if we show G =`' F ` earlier? (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 ) )
cdlemef50.v 𝑉 = ( ( 𝑄 𝑃 ) 𝑊 )
cdlemef50.n 𝑁 = ( ( 𝑣 𝑉 ) ( 𝑃 ( ( 𝑄 𝑣 ) 𝑊 ) ) )
cdlemefs50.o 𝑂 = ( ( 𝑄 𝑃 ) ( 𝑁 ( ( 𝑢 𝑣 ) 𝑊 ) ) )
cdlemef50.g 𝐺 = ( 𝑎𝐵 ↦ if ( ( 𝑄𝑃 ∧ ¬ 𝑎 𝑊 ) , ( 𝑐𝐵𝑢𝐴 ( ( ¬ 𝑢 𝑊 ∧ ( 𝑢 ( 𝑎 𝑊 ) ) = 𝑎 ) → 𝑐 = ( if ( 𝑢 ( 𝑄 𝑃 ) , ( 𝑏𝐵𝑣𝐴 ( ( ¬ 𝑣 𝑊 ∧ ¬ 𝑣 ( 𝑄 𝑃 ) ) → 𝑏 = 𝑂 ) ) , 𝑢 / 𝑣 𝑁 ) ( 𝑎 𝑊 ) ) ) ) , 𝑎 ) )
Assertion cdleme50rnlem ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ran 𝐹 = 𝐵 )

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 cdlemef50.v 𝑉 = ( ( 𝑄 𝑃 ) 𝑊 )
12 cdlemef50.n 𝑁 = ( ( 𝑣 𝑉 ) ( 𝑃 ( ( 𝑄 𝑣 ) 𝑊 ) ) )
13 cdlemefs50.o 𝑂 = ( ( 𝑄 𝑃 ) ( 𝑁 ( ( 𝑢 𝑣 ) 𝑊 ) ) )
14 cdlemef50.g 𝐺 = ( 𝑎𝐵 ↦ if ( ( 𝑄𝑃 ∧ ¬ 𝑎 𝑊 ) , ( 𝑐𝐵𝑢𝐴 ( ( ¬ 𝑢 𝑊 ∧ ( 𝑢 ( 𝑎 𝑊 ) ) = 𝑎 ) → 𝑐 = ( if ( 𝑢 ( 𝑄 𝑃 ) , ( 𝑏𝐵𝑣𝐴 ( ( ¬ 𝑣 𝑊 ∧ ¬ 𝑣 ( 𝑄 𝑃 ) ) → 𝑏 = 𝑂 ) ) , 𝑢 / 𝑣 𝑁 ) ( 𝑎 𝑊 ) ) ) ) , 𝑎 ) )
15 1 2 3 4 5 6 7 8 9 10 cdleme50f ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → 𝐹 : 𝐵𝐵 )
16 15 frnd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ran 𝐹𝐵 )
17 1 2 3 4 5 6 11 12 13 14 cdlemeg46fvcl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑒𝐵 ) → ( 𝐺𝑒 ) ∈ 𝐵 )
18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdleme48fgv ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑒𝐵 ) → ( 𝐹 ‘ ( 𝐺𝑒 ) ) = 𝑒 )
19 fveqeq2 ( 𝑑 = ( 𝐺𝑒 ) → ( ( 𝐹𝑑 ) = 𝑒 ↔ ( 𝐹 ‘ ( 𝐺𝑒 ) ) = 𝑒 ) )
20 19 rspcev ( ( ( 𝐺𝑒 ) ∈ 𝐵 ∧ ( 𝐹 ‘ ( 𝐺𝑒 ) ) = 𝑒 ) → ∃ 𝑑𝐵 ( 𝐹𝑑 ) = 𝑒 )
21 17 18 20 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑒𝐵 ) → ∃ 𝑑𝐵 ( 𝐹𝑑 ) = 𝑒 )
22 15 adantr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑒𝐵 ) → 𝐹 : 𝐵𝐵 )
23 ffn ( 𝐹 : 𝐵𝐵𝐹 Fn 𝐵 )
24 fvelrnb ( 𝐹 Fn 𝐵 → ( 𝑒 ∈ ran 𝐹 ↔ ∃ 𝑑𝐵 ( 𝐹𝑑 ) = 𝑒 ) )
25 22 23 24 3syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑒𝐵 ) → ( 𝑒 ∈ ran 𝐹 ↔ ∃ 𝑑𝐵 ( 𝐹𝑑 ) = 𝑒 ) )
26 21 25 mpbird ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑒𝐵 ) → 𝑒 ∈ ran 𝐹 )
27 16 26 eqelssd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ran 𝐹 = 𝐵 )