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