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Metamath Proof Explorer

Theorem cdlemd6

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 31-May-2012)

Ref Expression
Hypotheses cdlemd4.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemd4.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemd4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemd4.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemd4.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemd6 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) )

Proof

Step Hyp Ref Expression
1 cdlemd4.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemd4.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemd4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 cdlemd4.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 cdlemd4.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 simp3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) )
7 6 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) )
8 7 oveq1d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
9 simp1l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10 simp1rl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → 𝐹 ∈ 𝑇 )
11 simp21 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
12 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
13 eqid ⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
14 1 2 12 3 4 5 13 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
15 9 10 11 14 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
16 simp1rr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → 𝐺 ∈ 𝑇 )
17 1 2 12 3 4 5 13 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
18 9 16 11 17 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
19 8 15 18 3eqtr4d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) = ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) )
20 19 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝑄 ∨ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) ) = ( 𝑄 ∨ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) )
21 6 oveq1d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) )
22 20 21 oveq12d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( ( 𝑄 ∨ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) ) ( meet ‘ 𝐾 ) ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) = ( ( 𝑄 ∨ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ( meet ‘ 𝐾 ) ( ( 𝐺 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) )
23 simp22 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
24 simp23 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
25 1 2 12 3 4 5 13 cdlemc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → ( 𝐹 ‘ 𝑄 ) = ( ( 𝑄 ∨ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) ) ( meet ‘ 𝐾 ) ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) )
26 9 10 11 23 24 25 syl131anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝐹 ‘ 𝑄 ) = ( ( 𝑄 ∨ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) ) ( meet ‘ 𝐾 ) ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) )
27 oveq2 ⊢ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) )
28 27 breq2d ⊢ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) → ( 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ↔ 𝑄 ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ) )
29 28 notbid ⊢ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) → ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ↔ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ) )
30 29 biimpd ⊢ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) → ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) → ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ) )
31 6 24 30 sylc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) )
32 1 2 12 3 4 5 13 cdlemc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ) → ( 𝐺 ‘ 𝑄 ) = ( ( 𝑄 ∨ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ( meet ‘ 𝐾 ) ( ( 𝐺 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) )
33 9 16 11 23 31 32 syl131anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝐺 ‘ 𝑄 ) = ( ( 𝑄 ∨ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ( meet ‘ 𝐾 ) ( ( 𝐺 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) )
34 22 26 33 3eqtr4d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) )