Step |
Hyp |
Ref |
Expression |
1 |
|
sltadd1im |
⊢ ( ( 𝐵 ∈ No ∧ 𝐴 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 <s 𝐴 → ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) ) ) |
2 |
1
|
3com12 |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 <s 𝐴 → ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) ) ) |
3 |
|
sltnle |
⊢ ( ( 𝐵 ∈ No ∧ 𝐴 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵 ) ) |
4 |
3
|
ancoms |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵 ) ) |
5 |
4
|
3adant3 |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵 ) ) |
6 |
|
addscl |
⊢ ( ( 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 +s 𝐶 ) ∈ No ) |
7 |
6
|
3adant1 |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 +s 𝐶 ) ∈ No ) |
8 |
|
addscl |
⊢ ( ( 𝐴 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 +s 𝐶 ) ∈ No ) |
9 |
|
sltnle |
⊢ ( ( ( 𝐵 +s 𝐶 ) ∈ No ∧ ( 𝐴 +s 𝐶 ) ∈ No ) → ( ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) ↔ ¬ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) ) |
10 |
7 8 9
|
3imp3i2an |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) ↔ ¬ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) ) |
11 |
2 5 10
|
3imtr3d |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ¬ 𝐴 ≤s 𝐵 → ¬ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) ) |
12 |
11
|
con4d |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) → 𝐴 ≤s 𝐵 ) ) |