Metamath Proof Explorer

Theorem cdlemkid

Description: The value of the tau function (in Lemma K of Crawley p. 118) on the identity relation. (Contributed by NM, 25-Jul-2013)

Ref Expression
Hypotheses cdlemk5.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemk5.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemk5.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemk5.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemk5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemk5.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemk5.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemk5.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemk5.z ⊢ 𝑍 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑏 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) )
cdlemk5.y ⊢ 𝑌 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑔 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) )
cdlemk5.x ⊢ 𝑋 = ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑔 ) ) → ( 𝑧 ‘ 𝑃 ) = 𝑌 ) )
Assertion cdlemkid ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 = ( I ↾ 𝐵 ) ) ) → ⦋ 𝐺 / 𝑔 ⦌ 𝑋 = ( I ↾ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 cdlemk5.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemk5.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemk5.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemk5.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdlemk5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemk5.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemk5.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8 cdlemk5.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
9 cdlemk5.z ⊢ 𝑍 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑏 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) )
10 cdlemk5.y ⊢ 𝑌 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑔 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) )
11 cdlemk5.x ⊢ 𝑋 = ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑔 ) ) → ( 𝑧 ‘ 𝑃 ) = 𝑌 ) )
12 7 fvexi ⊢ 𝑇 ∈ V
13 nfv ⊢ Ⅎ 𝑏 ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 = ( I ↾ 𝐵 ) ) )
14 nfcv ⊢ Ⅎ 𝑏 𝐺
15 nfra1 ⊢ Ⅎ 𝑏 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑔 ) ) → ( 𝑧 ‘ 𝑃 ) = 𝑌 )
16 nfcv ⊢ Ⅎ 𝑏 𝑇
17 15 16 nfriota ⊢ Ⅎ 𝑏 ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑔 ) ) → ( 𝑧 ‘ 𝑃 ) = 𝑌 ) )
18 11 17 nfcxfr ⊢ Ⅎ 𝑏 𝑋
19 14 18 nfcsbw ⊢ Ⅎ 𝑏 ⦋ 𝐺 / 𝑔 ⦌ 𝑋
20 19 nfeq1 ⊢ Ⅎ 𝑏 ⦋ 𝐺 / 𝑔 ⦌ 𝑋 = ( I ↾ 𝐵 )
21 20 a1i ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 = ( I ↾ 𝐵 ) ) ) → Ⅎ 𝑏 ⦋ 𝐺 / 𝑔 ⦌ 𝑋 = ( I ↾ 𝐵 ) )
22 1 2 3 4 5 6 7 8 9 10 11 cdlemkid4 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 = ( I ↾ 𝐵 ) ) ) → ⦋ 𝐺 / 𝑔 ⦌ 𝑋 = ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → 𝑧 = ( I ↾ 𝐵 ) ) ) )
23 eqeq1 ⊢ ( ( I ↾ 𝐵 ) = ⦋ 𝐺 / 𝑔 ⦌ 𝑋 → ( ( I ↾ 𝐵 ) = ( I ↾ 𝐵 ) ↔ ⦋ 𝐺 / 𝑔 ⦌ 𝑋 = ( I ↾ 𝐵 ) ) )
24 23 adantl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 = ( I ↾ 𝐵 ) ) ) ∧ ( I ↾ 𝐵 ) = ⦋ 𝐺 / 𝑔 ⦌ 𝑋 ) → ( ( I ↾ 𝐵 ) = ( I ↾ 𝐵 ) ↔ ⦋ 𝐺 / 𝑔 ⦌ 𝑋 = ( I ↾ 𝐵 ) ) )
25 eqidd ⊢ ( ( 𝑏 ∈ 𝑇 ∧ ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( I ↾ 𝐵 ) = ( I ↾ 𝐵 ) )
26 25 a1i ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 = ( I ↾ 𝐵 ) ) ) → ( ( 𝑏 ∈ 𝑇 ∧ ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( I ↾ 𝐵 ) = ( I ↾ 𝐵 ) ) )
27 1 2 3 4 5 6 7 8 9 10 11 cdlemkid5 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 = ( I ↾ 𝐵 ) ) ) → ⦋ 𝐺 / 𝑔 ⦌ 𝑋 ∈ 𝑇 )
28 1 6 7 8 cdlemftr2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑏 ∈ 𝑇 ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐺 ) ) )
29 28 3ad2ant1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 = ( I ↾ 𝐵 ) ) ) → ∃ 𝑏 ∈ 𝑇 ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐺 ) ) )
30 13 21 22 24 26 27 29 riotasv3d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 = ( I ↾ 𝐵 ) ) ) ∧ 𝑇 ∈ V ) → ⦋ 𝐺 / 𝑔 ⦌ 𝑋 = ( I ↾ 𝐵 ) )
31 12 30 mpan2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 = ( I ↾ 𝐵 ) ) ) → ⦋ 𝐺 / 𝑔 ⦌ 𝑋 = ( I ↾ 𝐵 ) )