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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝐾 ∈ HL )
7 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝑊 ∈ 𝐻 )
8 simp2r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) )
9 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
10 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
11 1 2 9 10 3 4 lhpmcvr2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) )
12 6 7 8 11 syl21anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) )
13 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
14 simp2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → 𝑟 ∈ 𝐴 )
15 simp3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ¬ 𝑟 ≤ 𝑊 )
16 14 15 jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) )
17 simp12r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) )
18 simp131 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → 𝐹 ∈ 𝑇 )
19 simp132 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → 𝐺 ∈ 𝑇 )
20 simp3r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑟 ) ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) )
22 13 16 17 18 19 20 21 syl123anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑟 ) ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) )
23 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
24 simp133 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 )
25 2 3 4 5 cdlemg6 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑟 ) ) = 𝑟 )
26 13 23 16 18 19 24 25 syl123anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑟 ) ) = 𝑟 )
27 26 oveq1d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑟 ) ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) )
28 22 27 20 3eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = 𝑋 )
29 28 rexlimdv3a ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑋 ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = 𝑋 ) )
30 12 29 mpd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = 𝑋 )