Metamath Proof Explorer


Theorem ltmpi

Description: Ordering property of multiplication for positive integers. (Contributed by NM, 8-Feb-1996) (New usage is discouraged.)

Ref Expression
Assertion ltmpi ( 𝐶N → ( 𝐴 <N 𝐵 ↔ ( 𝐶 ·N 𝐴 ) <N ( 𝐶 ·N 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 dmmulpi dom ·N = ( N × N )
2 ltrelpi <N ⊆ ( N × N )
3 0npi ¬ ∅ ∈ N
4 pinn ( 𝐴N𝐴 ∈ ω )
5 pinn ( 𝐵N𝐵 ∈ ω )
6 elni2 ( 𝐶N ↔ ( 𝐶 ∈ ω ∧ ∅ ∈ 𝐶 ) )
7 iba ( ∅ ∈ 𝐶 → ( 𝐴𝐵 ↔ ( 𝐴𝐵 ∧ ∅ ∈ 𝐶 ) ) )
8 nnmord ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( ( 𝐴𝐵 ∧ ∅ ∈ 𝐶 ) ↔ ( 𝐶 ·o 𝐴 ) ∈ ( 𝐶 ·o 𝐵 ) ) )
9 7 8 sylan9bbr ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) ∧ ∅ ∈ 𝐶 ) → ( 𝐴𝐵 ↔ ( 𝐶 ·o 𝐴 ) ∈ ( 𝐶 ·o 𝐵 ) ) )
10 9 3exp1 ( 𝐴 ∈ ω → ( 𝐵 ∈ ω → ( 𝐶 ∈ ω → ( ∅ ∈ 𝐶 → ( 𝐴𝐵 ↔ ( 𝐶 ·o 𝐴 ) ∈ ( 𝐶 ·o 𝐵 ) ) ) ) ) )
11 10 imp4b ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐶 ∈ ω ∧ ∅ ∈ 𝐶 ) → ( 𝐴𝐵 ↔ ( 𝐶 ·o 𝐴 ) ∈ ( 𝐶 ·o 𝐵 ) ) ) )
12 6 11 syl5bi ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐶N → ( 𝐴𝐵 ↔ ( 𝐶 ·o 𝐴 ) ∈ ( 𝐶 ·o 𝐵 ) ) ) )
13 4 5 12 syl2an ( ( 𝐴N𝐵N ) → ( 𝐶N → ( 𝐴𝐵 ↔ ( 𝐶 ·o 𝐴 ) ∈ ( 𝐶 ·o 𝐵 ) ) ) )
14 13 imp ( ( ( 𝐴N𝐵N ) ∧ 𝐶N ) → ( 𝐴𝐵 ↔ ( 𝐶 ·o 𝐴 ) ∈ ( 𝐶 ·o 𝐵 ) ) )
15 ltpiord ( ( 𝐴N𝐵N ) → ( 𝐴 <N 𝐵𝐴𝐵 ) )
16 15 adantr ( ( ( 𝐴N𝐵N ) ∧ 𝐶N ) → ( 𝐴 <N 𝐵𝐴𝐵 ) )
17 mulclpi ( ( 𝐶N𝐴N ) → ( 𝐶 ·N 𝐴 ) ∈ N )
18 mulclpi ( ( 𝐶N𝐵N ) → ( 𝐶 ·N 𝐵 ) ∈ N )
19 ltpiord ( ( ( 𝐶 ·N 𝐴 ) ∈ N ∧ ( 𝐶 ·N 𝐵 ) ∈ N ) → ( ( 𝐶 ·N 𝐴 ) <N ( 𝐶 ·N 𝐵 ) ↔ ( 𝐶 ·N 𝐴 ) ∈ ( 𝐶 ·N 𝐵 ) ) )
20 17 18 19 syl2an ( ( ( 𝐶N𝐴N ) ∧ ( 𝐶N𝐵N ) ) → ( ( 𝐶 ·N 𝐴 ) <N ( 𝐶 ·N 𝐵 ) ↔ ( 𝐶 ·N 𝐴 ) ∈ ( 𝐶 ·N 𝐵 ) ) )
21 mulpiord ( ( 𝐶N𝐴N ) → ( 𝐶 ·N 𝐴 ) = ( 𝐶 ·o 𝐴 ) )
22 21 adantr ( ( ( 𝐶N𝐴N ) ∧ ( 𝐶N𝐵N ) ) → ( 𝐶 ·N 𝐴 ) = ( 𝐶 ·o 𝐴 ) )
23 mulpiord ( ( 𝐶N𝐵N ) → ( 𝐶 ·N 𝐵 ) = ( 𝐶 ·o 𝐵 ) )
24 23 adantl ( ( ( 𝐶N𝐴N ) ∧ ( 𝐶N𝐵N ) ) → ( 𝐶 ·N 𝐵 ) = ( 𝐶 ·o 𝐵 ) )
25 22 24 eleq12d ( ( ( 𝐶N𝐴N ) ∧ ( 𝐶N𝐵N ) ) → ( ( 𝐶 ·N 𝐴 ) ∈ ( 𝐶 ·N 𝐵 ) ↔ ( 𝐶 ·o 𝐴 ) ∈ ( 𝐶 ·o 𝐵 ) ) )
26 20 25 bitrd ( ( ( 𝐶N𝐴N ) ∧ ( 𝐶N𝐵N ) ) → ( ( 𝐶 ·N 𝐴 ) <N ( 𝐶 ·N 𝐵 ) ↔ ( 𝐶 ·o 𝐴 ) ∈ ( 𝐶 ·o 𝐵 ) ) )
27 26 anandis ( ( 𝐶N ∧ ( 𝐴N𝐵N ) ) → ( ( 𝐶 ·N 𝐴 ) <N ( 𝐶 ·N 𝐵 ) ↔ ( 𝐶 ·o 𝐴 ) ∈ ( 𝐶 ·o 𝐵 ) ) )
28 27 ancoms ( ( ( 𝐴N𝐵N ) ∧ 𝐶N ) → ( ( 𝐶 ·N 𝐴 ) <N ( 𝐶 ·N 𝐵 ) ↔ ( 𝐶 ·o 𝐴 ) ∈ ( 𝐶 ·o 𝐵 ) ) )
29 14 16 28 3bitr4d ( ( ( 𝐴N𝐵N ) ∧ 𝐶N ) → ( 𝐴 <N 𝐵 ↔ ( 𝐶 ·N 𝐴 ) <N ( 𝐶 ·N 𝐵 ) ) )
30 29 3impa ( ( 𝐴N𝐵N𝐶N ) → ( 𝐴 <N 𝐵 ↔ ( 𝐶 ·N 𝐴 ) <N ( 𝐶 ·N 𝐵 ) ) )
31 1 2 3 30 ndmovord ( 𝐶N → ( 𝐴 <N 𝐵 ↔ ( 𝐶 ·N 𝐴 ) <N ( 𝐶 ·N 𝐵 ) ) )