Metamath Proof Explorer


Theorem prslem

Description: Lemma for prsref and prstr . (Contributed by Mario Carneiro, 1-Feb-2015)

Ref Expression
Hypotheses isprs.b 𝐵 = ( Base ‘ 𝐾 )
isprs.l = ( le ‘ 𝐾 )
Assertion prslem ( ( 𝐾 ∈ Proset ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( 𝑋 𝑋 ∧ ( ( 𝑋 𝑌𝑌 𝑍 ) → 𝑋 𝑍 ) ) )

Proof

Step Hyp Ref Expression
1 isprs.b 𝐵 = ( Base ‘ 𝐾 )
2 isprs.l = ( le ‘ 𝐾 )
3 1 2 isprs ( 𝐾 ∈ Proset ↔ ( 𝐾 ∈ V ∧ ∀ 𝑥𝐵𝑦𝐵𝑧𝐵 ( 𝑥 𝑥 ∧ ( ( 𝑥 𝑦𝑦 𝑧 ) → 𝑥 𝑧 ) ) ) )
4 3 simprbi ( 𝐾 ∈ Proset → ∀ 𝑥𝐵𝑦𝐵𝑧𝐵 ( 𝑥 𝑥 ∧ ( ( 𝑥 𝑦𝑦 𝑧 ) → 𝑥 𝑧 ) ) )
5 breq12 ( ( 𝑥 = 𝑋𝑥 = 𝑋 ) → ( 𝑥 𝑥𝑋 𝑋 ) )
6 5 anidms ( 𝑥 = 𝑋 → ( 𝑥 𝑥𝑋 𝑋 ) )
7 breq1 ( 𝑥 = 𝑋 → ( 𝑥 𝑦𝑋 𝑦 ) )
8 7 anbi1d ( 𝑥 = 𝑋 → ( ( 𝑥 𝑦𝑦 𝑧 ) ↔ ( 𝑋 𝑦𝑦 𝑧 ) ) )
9 breq1 ( 𝑥 = 𝑋 → ( 𝑥 𝑧𝑋 𝑧 ) )
10 8 9 imbi12d ( 𝑥 = 𝑋 → ( ( ( 𝑥 𝑦𝑦 𝑧 ) → 𝑥 𝑧 ) ↔ ( ( 𝑋 𝑦𝑦 𝑧 ) → 𝑋 𝑧 ) ) )
11 6 10 anbi12d ( 𝑥 = 𝑋 → ( ( 𝑥 𝑥 ∧ ( ( 𝑥 𝑦𝑦 𝑧 ) → 𝑥 𝑧 ) ) ↔ ( 𝑋 𝑋 ∧ ( ( 𝑋 𝑦𝑦 𝑧 ) → 𝑋 𝑧 ) ) ) )
12 breq2 ( 𝑦 = 𝑌 → ( 𝑋 𝑦𝑋 𝑌 ) )
13 breq1 ( 𝑦 = 𝑌 → ( 𝑦 𝑧𝑌 𝑧 ) )
14 12 13 anbi12d ( 𝑦 = 𝑌 → ( ( 𝑋 𝑦𝑦 𝑧 ) ↔ ( 𝑋 𝑌𝑌 𝑧 ) ) )
15 14 imbi1d ( 𝑦 = 𝑌 → ( ( ( 𝑋 𝑦𝑦 𝑧 ) → 𝑋 𝑧 ) ↔ ( ( 𝑋 𝑌𝑌 𝑧 ) → 𝑋 𝑧 ) ) )
16 15 anbi2d ( 𝑦 = 𝑌 → ( ( 𝑋 𝑋 ∧ ( ( 𝑋 𝑦𝑦 𝑧 ) → 𝑋 𝑧 ) ) ↔ ( 𝑋 𝑋 ∧ ( ( 𝑋 𝑌𝑌 𝑧 ) → 𝑋 𝑧 ) ) ) )
17 breq2 ( 𝑧 = 𝑍 → ( 𝑌 𝑧𝑌 𝑍 ) )
18 17 anbi2d ( 𝑧 = 𝑍 → ( ( 𝑋 𝑌𝑌 𝑧 ) ↔ ( 𝑋 𝑌𝑌 𝑍 ) ) )
19 breq2 ( 𝑧 = 𝑍 → ( 𝑋 𝑧𝑋 𝑍 ) )
20 18 19 imbi12d ( 𝑧 = 𝑍 → ( ( ( 𝑋 𝑌𝑌 𝑧 ) → 𝑋 𝑧 ) ↔ ( ( 𝑋 𝑌𝑌 𝑍 ) → 𝑋 𝑍 ) ) )
21 20 anbi2d ( 𝑧 = 𝑍 → ( ( 𝑋 𝑋 ∧ ( ( 𝑋 𝑌𝑌 𝑧 ) → 𝑋 𝑧 ) ) ↔ ( 𝑋 𝑋 ∧ ( ( 𝑋 𝑌𝑌 𝑍 ) → 𝑋 𝑍 ) ) ) )
22 11 16 21 rspc3v ( ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) → ( ∀ 𝑥𝐵𝑦𝐵𝑧𝐵 ( 𝑥 𝑥 ∧ ( ( 𝑥 𝑦𝑦 𝑧 ) → 𝑥 𝑧 ) ) → ( 𝑋 𝑋 ∧ ( ( 𝑋 𝑌𝑌 𝑍 ) → 𝑋 𝑍 ) ) ) )
23 4 22 mpan9 ( ( 𝐾 ∈ Proset ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( 𝑋 𝑋 ∧ ( ( 𝑋 𝑌𝑌 𝑍 ) → 𝑋 𝑍 ) ) )