Metamath Proof Explorer


Theorem zlmodzxzldeplem4

Description: Lemma 4 for zlmodzxzldep . (Contributed by AV, 24-May-2019) (Revised by AV, 10-Jun-2019)

Ref Expression
Hypotheses zlmodzxzldep.z 𝑍 = ( ℤring freeLMod { 0 , 1 } )
zlmodzxzldep.a 𝐴 = { ⟨ 0 , 3 ⟩ , ⟨ 1 , 6 ⟩ }
zlmodzxzldep.b 𝐵 = { ⟨ 0 , 2 ⟩ , ⟨ 1 , 4 ⟩ }
zlmodzxzldeplem.f 𝐹 = { ⟨ 𝐴 , 2 ⟩ , ⟨ 𝐵 , - 3 ⟩ }
Assertion zlmodzxzldeplem4 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝐹𝑦 ) ≠ 0

Proof

Step Hyp Ref Expression
1 zlmodzxzldep.z 𝑍 = ( ℤring freeLMod { 0 , 1 } )
2 zlmodzxzldep.a 𝐴 = { ⟨ 0 , 3 ⟩ , ⟨ 1 , 6 ⟩ }
3 zlmodzxzldep.b 𝐵 = { ⟨ 0 , 2 ⟩ , ⟨ 1 , 4 ⟩ }
4 zlmodzxzldeplem.f 𝐹 = { ⟨ 𝐴 , 2 ⟩ , ⟨ 𝐵 , - 3 ⟩ }
5 prex { ⟨ 0 , 3 ⟩ , ⟨ 1 , 6 ⟩ } ∈ V
6 2 5 eqeltri 𝐴 ∈ V
7 prex { ⟨ 0 , 2 ⟩ , ⟨ 1 , 4 ⟩ } ∈ V
8 3 7 eqeltri 𝐵 ∈ V
9 2ne0 2 ≠ 0
10 4 fveq1i ( 𝐹𝐴 ) = ( { ⟨ 𝐴 , 2 ⟩ , ⟨ 𝐵 , - 3 ⟩ } ‘ 𝐴 )
11 1 2 3 zlmodzxzldeplem 𝐴𝐵
12 2ex 2 ∈ V
13 6 12 fvpr1 ( 𝐴𝐵 → ( { ⟨ 𝐴 , 2 ⟩ , ⟨ 𝐵 , - 3 ⟩ } ‘ 𝐴 ) = 2 )
14 11 13 mp1i ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( { ⟨ 𝐴 , 2 ⟩ , ⟨ 𝐵 , - 3 ⟩ } ‘ 𝐴 ) = 2 )
15 10 14 syl5eq ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐹𝐴 ) = 2 )
16 15 neeq1d ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ( 𝐹𝐴 ) ≠ 0 ↔ 2 ≠ 0 ) )
17 9 16 mpbiri ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐹𝐴 ) ≠ 0 )
18 17 orcd ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ( 𝐹𝐴 ) ≠ 0 ∨ ( 𝐹𝐵 ) ≠ 0 ) )
19 fveq2 ( 𝑦 = 𝐴 → ( 𝐹𝑦 ) = ( 𝐹𝐴 ) )
20 19 neeq1d ( 𝑦 = 𝐴 → ( ( 𝐹𝑦 ) ≠ 0 ↔ ( 𝐹𝐴 ) ≠ 0 ) )
21 fveq2 ( 𝑦 = 𝐵 → ( 𝐹𝑦 ) = ( 𝐹𝐵 ) )
22 21 neeq1d ( 𝑦 = 𝐵 → ( ( 𝐹𝑦 ) ≠ 0 ↔ ( 𝐹𝐵 ) ≠ 0 ) )
23 20 22 rexprg ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝐹𝑦 ) ≠ 0 ↔ ( ( 𝐹𝐴 ) ≠ 0 ∨ ( 𝐹𝐵 ) ≠ 0 ) ) )
24 18 23 mpbird ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝐹𝑦 ) ≠ 0 )
25 6 8 24 mp2an 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝐹𝑦 ) ≠ 0