Metamath Proof Explorer

Theorem ltrneq

Description: The equality of two translations is determined by their equality at atoms not under co-atom W . (Contributed by NM, 20-Jun-2013)

Ref Expression
Hypotheses ltrne.l ⊢ ≤ = ( le ‘ 𝐾 )
ltrne.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
ltrne.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ltrne.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion ltrneq ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ∀ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) ↔ 𝐹 = 𝐺 ) )

Proof

Step Hyp Ref Expression
1 ltrne.l ⊢ ≤ = ( le ‘ 𝐾 )
2 ltrne.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 ltrne.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 ltrne.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊 ) → 𝐹 ∈ 𝑇 )
7 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
8 7 2 atbase ⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ ( Base ‘ 𝐾 ) )
9 8 3ad2ant2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊 ) → 𝑝 ∈ ( Base ‘ 𝐾 ) )
10 simp3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊 ) → 𝑝 ≤ 𝑊 )
11 7 1 3 4 ltrnval1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Base ‘ 𝐾 ) ∧ 𝑝 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑝 ) = 𝑝 )
12 5 6 9 10 11 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊 ) → ( 𝐹 ‘ 𝑝 ) = 𝑝 )
13 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊 ) → 𝐺 ∈ 𝑇 )
14 7 1 3 4 ltrnval1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Base ‘ 𝐾 ) ∧ 𝑝 ≤ 𝑊 ) ) → ( 𝐺 ‘ 𝑝 ) = 𝑝 )
15 5 13 9 10 14 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊 ) → ( 𝐺 ‘ 𝑝 ) = 𝑝 )
16 12 15 eqtr4d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊 ) → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) )
17 16 3expia ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) )
18 pm2.61 ⊢ ( ( 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) → ( ( ¬ 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) )
19 17 18 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( ¬ 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) )
20 re1tbw2 ⊢ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) → ( ¬ 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) )
21 19 20 impbid1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( ¬ 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) ↔ ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) )
22 21 ralbidva ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ∀ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) ↔ ∀ 𝑝 ∈ 𝐴 ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) )
23 2 3 4 ltrneq2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ∀ 𝑝 ∈ 𝐴 ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ↔ 𝐹 = 𝐺 ) )
24 22 23 bitrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ∀ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) ↔ 𝐹 = 𝐺 ) )