Metamath Proof Explorer


Theorem cdlemg7aN

Description: TODO: FIX COMMENT. (Contributed by NM, 28-Apr-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemg7.b 𝐵 = ( Base ‘ 𝐾 )
cdlemg7.l = ( le ‘ 𝐾 )
cdlemg7.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg7.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg7.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemg7aN ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺𝑋 ) ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 cdlemg7.b 𝐵 = ( Base ‘ 𝐾 )
2 cdlemg7.l = ( le ‘ 𝐾 )
3 cdlemg7.a 𝐴 = ( Atoms ‘ 𝐾 )
4 cdlemg7.h 𝐻 = ( LHyp ‘ 𝐾 )
5 cdlemg7.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) → 𝐾 ∈ HL )
7 simp1r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) → 𝑊𝐻 )
8 simp2r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) → ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) )
9 eqid ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
10 eqid ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
11 1 2 9 10 3 4 lhpmcvr2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) → ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) )
12 6 7 8 11 syl21anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) → ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) )
13 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) ∧ 𝑟𝐴 ∧ ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
14 simp2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) ∧ 𝑟𝐴 ∧ ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → 𝑟𝐴 )
15 simp3l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) ∧ 𝑟𝐴 ∧ ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ¬ 𝑟 𝑊 )
16 14 15 jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) ∧ 𝑟𝐴 ∧ ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) )
17 simp12r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) ∧ 𝑟𝐴 ∧ ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) )
18 simp131 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) ∧ 𝑟𝐴 ∧ ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → 𝐹𝑇 )
19 simp132 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) ∧ 𝑟𝐴 ∧ ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → 𝐺𝑇 )
20 simp3r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) ∧ 𝑟𝐴 ∧ ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 )
21 1 2 9 10 3 4 5 cdlemg7fvN ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝐹 ‘ ( 𝐺𝑋 ) ) = ( ( 𝐹 ‘ ( 𝐺𝑟 ) ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) )
22 13 16 17 18 19 20 21 syl123anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) ∧ 𝑟𝐴 ∧ ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝐹 ‘ ( 𝐺𝑋 ) ) = ( ( 𝐹 ‘ ( 𝐺𝑟 ) ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) )
23 simp12l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) ∧ 𝑟𝐴 ∧ ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
24 simp133 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) ∧ 𝑟𝐴 ∧ ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 )
25 2 3 4 5 cdlemg6 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺𝑟 ) ) = 𝑟 )
26 13 23 16 18 19 24 25 syl123anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) ∧ 𝑟𝐴 ∧ ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝐹 ‘ ( 𝐺𝑟 ) ) = 𝑟 )
27 26 oveq1d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) ∧ 𝑟𝐴 ∧ ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐹 ‘ ( 𝐺𝑟 ) ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) )
28 22 27 20 3eqtrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) ∧ 𝑟𝐴 ∧ ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝐹 ‘ ( 𝐺𝑋 ) ) = 𝑋 )
29 28 rexlimdv3a ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) → ( ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) → ( 𝐹 ‘ ( 𝐺𝑋 ) ) = 𝑋 ) )
30 12 29 mpd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺𝑋 ) ) = 𝑋 )