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Metamath Proof Explorer

Theorem poltletr

Description: Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015)

Ref Expression
Assertion poltletr ( ( 𝑅 Po 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝑅 𝐵 ∧ 𝐵 ( 𝑅 ∪ I ) 𝐶 ) → 𝐴 𝑅 𝐶 ) )

Proof

Step Hyp Ref Expression
1 poleloe ⊢ ( 𝐶 ∈ 𝑋 → ( 𝐵 ( 𝑅 ∪ I ) 𝐶 ↔ ( 𝐵 𝑅 𝐶 ∨ 𝐵 = 𝐶 ) ) )
2 1 3ad2ant3 ⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐵 ( 𝑅 ∪ I ) 𝐶 ↔ ( 𝐵 𝑅 𝐶 ∨ 𝐵 = 𝐶 ) ) )
3 2 adantl ⊢ ( ( 𝑅 Po 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐵 ( 𝑅 ∪ I ) 𝐶 ↔ ( 𝐵 𝑅 𝐶 ∨ 𝐵 = 𝐶 ) ) )
4 3 anbi2d ⊢ ( ( 𝑅 Po 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝑅 𝐵 ∧ 𝐵 ( 𝑅 ∪ I ) 𝐶 ) ↔ ( 𝐴 𝑅 𝐵 ∧ ( 𝐵 𝑅 𝐶 ∨ 𝐵 = 𝐶 ) ) ) )
5 potr ⊢ ( ( 𝑅 Po 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝑅 𝐵 ∧ 𝐵 𝑅 𝐶 ) → 𝐴 𝑅 𝐶 ) )
6 5 com12 ⊢ ( ( 𝐴 𝑅 𝐵 ∧ 𝐵 𝑅 𝐶 ) → ( ( 𝑅 Po 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → 𝐴 𝑅 𝐶 ) )
7 breq2 ⊢ ( 𝐵 = 𝐶 → ( 𝐴 𝑅 𝐵 ↔ 𝐴 𝑅 𝐶 ) )
8 7 biimpac ⊢ ( ( 𝐴 𝑅 𝐵 ∧ 𝐵 = 𝐶 ) → 𝐴 𝑅 𝐶 )
9 8 a1d ⊢ ( ( 𝐴 𝑅 𝐵 ∧ 𝐵 = 𝐶 ) → ( ( 𝑅 Po 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → 𝐴 𝑅 𝐶 ) )
10 6 9 jaodan ⊢ ( ( 𝐴 𝑅 𝐵 ∧ ( 𝐵 𝑅 𝐶 ∨ 𝐵 = 𝐶 ) ) → ( ( 𝑅 Po 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → 𝐴 𝑅 𝐶 ) )
11 10 com12 ⊢ ( ( 𝑅 Po 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝑅 𝐵 ∧ ( 𝐵 𝑅 𝐶 ∨ 𝐵 = 𝐶 ) ) → 𝐴 𝑅 𝐶 ) )
12 4 11 sylbid ⊢ ( ( 𝑅 Po 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝑅 𝐵 ∧ 𝐵 ( 𝑅 ∪ I ) 𝐶 ) → 𝐴 𝑅 𝐶 ) )