Metamath Proof Explorer


Theorem cdlemkuv2-2

Description: Part of proof of Lemma K of Crawley p. 118. Line 16 on p. 119 for i = 2, where sigma_2 (p) is V , f_2 is C , and k_2 is Q . (Contributed by NM, 2-Jul-2013)

Ref Expression
Hypotheses cdlemk2.b 𝐵 = ( Base ‘ 𝐾 )
cdlemk2.l = ( le ‘ 𝐾 )
cdlemk2.j = ( join ‘ 𝐾 )
cdlemk2.m = ( meet ‘ 𝐾 )
cdlemk2.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemk2.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemk2.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemk2.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemk2.s 𝑆 = ( 𝑓𝑇 ↦ ( 𝑖𝑇 ( 𝑖𝑃 ) = ( ( 𝑃 ( 𝑅𝑓 ) ) ( ( 𝑁𝑃 ) ( 𝑅 ‘ ( 𝑓 𝐹 ) ) ) ) ) )
cdlemk2.q 𝑄 = ( 𝑆𝐶 )
cdlemk2.v 𝑉 = ( 𝑑𝑇 ↦ ( 𝑘𝑇 ( 𝑘𝑃 ) = ( ( 𝑃 ( 𝑅𝑑 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝑑 𝐶 ) ) ) ) ) )
Assertion cdlemkuv2-2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ∧ 𝐺𝑇 ) ∧ ( 𝐹𝑇𝐶𝑇𝑁𝑇 ) ∧ ( ( ( 𝑅𝐶 ) ≠ ( 𝑅𝐹 ) ∧ ( 𝑅𝐶 ) ≠ ( 𝑅𝐺 ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ 𝐶 ≠ ( I ↾ 𝐵 ) ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ) → ( ( 𝑉𝐺 ) ‘ 𝑃 ) = ( ( 𝑃 ( 𝑅𝐺 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝐺 𝐶 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemk2.b 𝐵 = ( Base ‘ 𝐾 )
2 cdlemk2.l = ( le ‘ 𝐾 )
3 cdlemk2.j = ( join ‘ 𝐾 )
4 cdlemk2.m = ( meet ‘ 𝐾 )
5 cdlemk2.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemk2.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemk2.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8 cdlemk2.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
9 cdlemk2.s 𝑆 = ( 𝑓𝑇 ↦ ( 𝑖𝑇 ( 𝑖𝑃 ) = ( ( 𝑃 ( 𝑅𝑓 ) ) ( ( 𝑁𝑃 ) ( 𝑅 ‘ ( 𝑓 𝐹 ) ) ) ) ) )
10 cdlemk2.q 𝑄 = ( 𝑆𝐶 )
11 cdlemk2.v 𝑉 = ( 𝑑𝑇 ↦ ( 𝑘𝑇 ( 𝑘𝑃 ) = ( ( 𝑃 ( 𝑅𝑑 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝑑 𝐶 ) ) ) ) ) )
12 1 2 3 4 5 6 7 8 9 10 11 cdlemkuv2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ∧ 𝐺𝑇 ) ∧ ( 𝐹𝑇𝐶𝑇𝑁𝑇 ) ∧ ( ( ( 𝑅𝐶 ) ≠ ( 𝑅𝐹 ) ∧ ( 𝑅𝐶 ) ≠ ( 𝑅𝐺 ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ 𝐶 ≠ ( I ↾ 𝐵 ) ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ) → ( ( 𝑉𝐺 ) ‘ 𝑃 ) = ( ( 𝑃 ( 𝑅𝐺 ) ) ( ( 𝑄𝑃 ) ( 𝑅 ‘ ( 𝐺 𝐶 ) ) ) ) )