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 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 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 = ( 𝑎 ∈ 𝐵 ↦ if ( ( 𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊 ) , ( ℩ 𝑐 ∈ 𝐵 ∀ 𝑢 ∈ 𝐴 ( ( ¬ 𝑢 ≤ 𝑊 ∧ ( 𝑢 ∨ ( 𝑎 ∧ 𝑊 ) ) = 𝑎 ) → 𝑐 = ( if ( 𝑢 ≤ ( 𝑄 ∨ 𝑃 ) , ( ℩ 𝑏 ∈ 𝐵 ∀ 𝑣 ∈ 𝐴 ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑄 ∨ 𝑃 ) ) → 𝑏 = 𝑂 ) ) , ⦋ 𝑢 / 𝑣 ⦌ 𝑁 ) ∨ ( 𝑎 ∧ 𝑊 ) ) ) ) , 𝑎 ) )
Assertion cdleme50rnlem ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 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 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 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 = ( 𝑎 ∈ 𝐵 ↦ if ( ( 𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊 ) , ( ℩ 𝑐 ∈ 𝐵 ∀ 𝑢 ∈ 𝐴 ( ( ¬ 𝑢 ≤ 𝑊 ∧ ( 𝑢 ∨ ( 𝑎 ∧ 𝑊 ) ) = 𝑎 ) → 𝑐 = ( if ( 𝑢 ≤ ( 𝑄 ∨ 𝑃 ) , ( ℩ 𝑏 ∈ 𝐵 ∀ 𝑣 ∈ 𝐴 ( ( ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 : 𝐵 ⟶ 𝐵 )
16 15 frnd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ran 𝐹 ⊆ 𝐵 )
17 1 2 3 4 5 6 11 12 13 14 cdlemeg46fvcl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑒 ∈ 𝐵 ) → ( 𝐺 ‘ 𝑒 ) ∈ 𝐵 )
18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdleme48fgv ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑒 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑒 ) ) = 𝑒 )
19 fveqeq2 ⊢ ( 𝑑 = ( 𝐺 ‘ 𝑒 ) → ( ( 𝐹 ‘ 𝑑 ) = 𝑒 ↔ ( 𝐹 ‘ ( 𝐺 ‘ 𝑒 ) ) = 𝑒 ) )
20 19 rspcev ⊢ ( ( ( 𝐺 ‘ 𝑒 ) ∈ 𝐵 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑒 ) ) = 𝑒 ) → ∃ 𝑑 ∈ 𝐵 ( 𝐹 ‘ 𝑑 ) = 𝑒 )
21 17 18 20 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑒 ∈ 𝐵 ) → ∃ 𝑑 ∈ 𝐵 ( 𝐹 ‘ 𝑑 ) = 𝑒 )
22 15 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑒 ∈ 𝐵 ) → 𝐹 : 𝐵 ⟶ 𝐵 )
23 ffn ⊢ ( 𝐹 : 𝐵 ⟶ 𝐵 → 𝐹 Fn 𝐵 )
24 fvelrnb ⊢ ( 𝐹 Fn 𝐵 → ( 𝑒 ∈ ran 𝐹 ↔ ∃ 𝑑 ∈ 𝐵 ( 𝐹 ‘ 𝑑 ) = 𝑒 ) )
25 22 23 24 3syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑒 ∈ 𝐵 ) → ( 𝑒 ∈ ran 𝐹 ↔ ∃ 𝑑 ∈ 𝐵 ( 𝐹 ‘ 𝑑 ) = 𝑒 ) )
26 21 25 mpbird ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑒 ∈ 𝐵 ) → 𝑒 ∈ ran 𝐹 )
27 16 26 eqelssd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ran 𝐹 = 𝐵 )