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Metamath Proof Explorer

Theorem cdlemg7fvN

Description: Value of a translation composition in terms of an associated atom. (Contributed by NM, 28-Apr-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemg7fv.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemg7fv.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg7fv.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg7fv.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemg7fv.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg7fv.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg7fv.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemg7fvN ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝑋 ∧ 𝑊 ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg7fv.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemg7fv.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemg7fv.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemg7fv.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdlemg7fv.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemg7fv.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemg7fv.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 simp32 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝐺 ∈ 𝑇 )
10 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
11 2 5 6 7 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
12 8 9 10 11 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
13 simp2r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) )
14 2 5 6 7 1 cdlemg7fvbwN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) → ( ( 𝐺 ‘ 𝑋 ) ∈ 𝐵 ∧ ¬ ( 𝐺 ‘ 𝑋 ) ≤ 𝑊 ) )
15 8 13 9 14 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐺 ‘ 𝑋 ) ∈ 𝐵 ∧ ¬ ( 𝐺 ‘ 𝑋 ) ≤ 𝑊 ) )
16 simp31 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝐹 ∈ 𝑇 )
17 simp33 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 )
18 6 7 2 3 5 4 1 cdlemg2fv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐺 ‘ 𝑋 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑋 ∧ 𝑊 ) ) )
19 8 10 13 9 17 18 syl122anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐺 ‘ 𝑋 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑋 ∧ 𝑊 ) ) )
20 19 oveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐺 ‘ 𝑋 ) ∧ 𝑊 ) = ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑊 ) )
21 simp2rl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑋 ∈ 𝐵 )
22 1 2 3 4 5 6 lhpelim ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑊 ) = ( 𝑋 ∧ 𝑊 ) )
23 8 12 21 22 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑊 ) = ( 𝑋 ∧ 𝑊 ) )
24 20 23 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐺 ‘ 𝑋 ) ∧ 𝑊 ) = ( 𝑋 ∧ 𝑊 ) )
25 24 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( ( 𝐺 ‘ 𝑋 ) ∧ 𝑊 ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑋 ∧ 𝑊 ) ) )
26 25 19 eqtr4d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( ( 𝐺 ‘ 𝑋 ) ∧ 𝑊 ) ) = ( 𝐺 ‘ 𝑋 ) )
27 6 7 2 3 5 4 1 cdlemg2fv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) ∧ ( ( 𝐺 ‘ 𝑋 ) ∈ 𝐵 ∧ ¬ ( 𝐺 ‘ 𝑋 ) ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( ( 𝐺 ‘ 𝑃 ) ∨ ( ( 𝐺 ‘ 𝑋 ) ∧ 𝑊 ) ) = ( 𝐺 ‘ 𝑋 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( ( 𝐺 ‘ 𝑋 ) ∧ 𝑊 ) ) )
28 8 12 15 16 26 27 syl122anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( ( 𝐺 ‘ 𝑋 ) ∧ 𝑊 ) ) )
29 24 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( ( 𝐺 ‘ 𝑋 ) ∧ 𝑊 ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝑋 ∧ 𝑊 ) ) )
30 28 29 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝑋 ∧ 𝑊 ) ) )