Proof of Theorem absslt
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | negscl 28068 | . . . . . . 7
⊢ (𝐴 ∈ 
No  → ( -us ‘𝐴) ∈  No
) | 
| 2 | 1 | ad2antrr 726 | . . . . . 6
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (abss‘𝐴) <s 𝐵) → ( -us ‘𝐴) ∈ 
No ) | 
| 3 |  | absscl 28264 | . . . . . . . 8
⊢ ((
-us ‘𝐴)
∈  No  → (abss‘(
-us ‘𝐴))
∈  No ) | 
| 4 | 1, 3 | syl 17 | . . . . . . 7
⊢ (𝐴 ∈ 
No  → (abss‘( -us ‘𝐴)) ∈ 
No ) | 
| 5 | 4 | ad2antrr 726 | . . . . . 6
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (abss‘𝐴) <s 𝐵) → (abss‘(
-us ‘𝐴))
∈  No ) | 
| 6 |  | simplr 769 | . . . . . 6
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (abss‘𝐴) <s 𝐵) → 𝐵 ∈  No
) | 
| 7 |  | sleabs 28272 | . . . . . . . 8
⊢ ((
-us ‘𝐴)
∈  No  → ( -us ‘𝐴) ≤s (abss‘(
-us ‘𝐴))) | 
| 8 | 1, 7 | syl 17 | . . . . . . 7
⊢ (𝐴 ∈ 
No  → ( -us ‘𝐴) ≤s (abss‘(
-us ‘𝐴))) | 
| 9 | 8 | ad2antrr 726 | . . . . . 6
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (abss‘𝐴) <s 𝐵) → ( -us ‘𝐴) ≤s (abss‘(
-us ‘𝐴))) | 
| 10 |  | abssneg 28271 | . . . . . . . . 9
⊢ (𝐴 ∈ 
No  → (abss‘( -us ‘𝐴)) =
(abss‘𝐴)) | 
| 11 | 10 | adantr 480 | . . . . . . . 8
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (abss‘(
-us ‘𝐴)) =
(abss‘𝐴)) | 
| 12 | 11 | breq1d 5153 | . . . . . . 7
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ((abss‘(
-us ‘𝐴))
<s 𝐵 ↔
(abss‘𝐴)
<s 𝐵)) | 
| 13 | 12 | biimpar 477 | . . . . . 6
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (abss‘𝐴) <s 𝐵) → (abss‘(
-us ‘𝐴))
<s 𝐵) | 
| 14 | 2, 5, 6, 9, 13 | slelttrd 27806 | . . . . 5
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (abss‘𝐴) <s 𝐵) → ( -us ‘𝐴) <s 𝐵) | 
| 15 |  | simpll 767 | . . . . . 6
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (abss‘𝐴) <s 𝐵) → 𝐴 ∈  No
) | 
| 16 |  | absscl 28264 | . . . . . . 7
⊢ (𝐴 ∈ 
No  → (abss‘𝐴) ∈  No
) | 
| 17 | 16 | ad2antrr 726 | . . . . . 6
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (abss‘𝐴) <s 𝐵) → (abss‘𝐴) ∈ 
No ) | 
| 18 |  | sleabs 28272 | . . . . . . 7
⊢ (𝐴 ∈ 
No  → 𝐴 ≤s
(abss‘𝐴)) | 
| 19 | 18 | ad2antrr 726 | . . . . . 6
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (abss‘𝐴) <s 𝐵) → 𝐴 ≤s (abss‘𝐴)) | 
| 20 |  | simpr 484 | . . . . . 6
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (abss‘𝐴) <s 𝐵) → (abss‘𝐴) <s 𝐵) | 
| 21 | 15, 17, 6, 19, 20 | slelttrd 27806 | . . . . 5
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (abss‘𝐴) <s 𝐵) → 𝐴 <s 𝐵) | 
| 22 | 14, 21 | jca 511 | . . . 4
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) ∧ (abss‘𝐴) <s 𝐵) → (( -us ‘𝐴) <s 𝐵 ∧ 𝐴 <s 𝐵)) | 
| 23 | 22 | ex 412 | . . 3
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ((abss‘𝐴) <s 𝐵 → (( -us ‘𝐴) <s 𝐵 ∧ 𝐴 <s 𝐵))) | 
| 24 |  | abssor 28270 | . . . . 5
⊢ (𝐴 ∈ 
No  → ((abss‘𝐴) = 𝐴 ∨ (abss‘𝐴) = ( -us
‘𝐴))) | 
| 25 | 24 | adantr 480 | . . . 4
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ((abss‘𝐴) = 𝐴 ∨ (abss‘𝐴) = ( -us
‘𝐴))) | 
| 26 |  | breq1 5146 | . . . . . . 7
⊢
((abss‘𝐴) = 𝐴 → ((abss‘𝐴) <s 𝐵 ↔ 𝐴 <s 𝐵)) | 
| 27 | 26 | biimprd 248 | . . . . . 6
⊢
((abss‘𝐴) = 𝐴 → (𝐴 <s 𝐵 → (abss‘𝐴) <s 𝐵)) | 
| 28 |  | breq1 5146 | . . . . . . 7
⊢
((abss‘𝐴) = ( -us ‘𝐴) →
((abss‘𝐴)
<s 𝐵 ↔ (
-us ‘𝐴)
<s 𝐵)) | 
| 29 | 28 | biimprd 248 | . . . . . 6
⊢
((abss‘𝐴) = ( -us ‘𝐴) → (( -us
‘𝐴) <s 𝐵 →
(abss‘𝐴)
<s 𝐵)) | 
| 30 | 27, 29 | jaoa 958 | . . . . 5
⊢
(((abss‘𝐴) = 𝐴 ∨ (abss‘𝐴) = ( -us
‘𝐴)) → ((𝐴 <s 𝐵 ∧ ( -us ‘𝐴) <s 𝐵) → (abss‘𝐴) <s 𝐵)) | 
| 31 | 30 | ancomsd 465 | . . . 4
⊢
(((abss‘𝐴) = 𝐴 ∨ (abss‘𝐴) = ( -us
‘𝐴)) → (((
-us ‘𝐴)
<s 𝐵 ∧ 𝐴 <s 𝐵) → (abss‘𝐴) <s 𝐵)) | 
| 32 | 25, 31 | syl 17 | . . 3
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ((( -us ‘𝐴) <s 𝐵 ∧ 𝐴 <s 𝐵) → (abss‘𝐴) <s 𝐵)) | 
| 33 | 23, 32 | impbid 212 | . 2
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ((abss‘𝐴) <s 𝐵 ↔ (( -us ‘𝐴) <s 𝐵 ∧ 𝐴 <s 𝐵))) | 
| 34 | 1 | adantr 480 | . . . . 5
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ( -us ‘𝐴) ∈ 
No ) | 
| 35 |  | simpr 484 | . . . . 5
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → 𝐵 ∈  No
) | 
| 36 | 34, 35 | sltnegd 28079 | . . . 4
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (( -us ‘𝐴) <s 𝐵 ↔ ( -us ‘𝐵) <s ( -us
‘( -us ‘𝐴)))) | 
| 37 |  | negnegs 28076 | . . . . . 6
⊢ (𝐴 ∈ 
No  → ( -us ‘( -us ‘𝐴)) = 𝐴) | 
| 38 | 37 | adantr 480 | . . . . 5
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ( -us ‘(
-us ‘𝐴)) =
𝐴) | 
| 39 | 38 | breq2d 5155 | . . . 4
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (( -us ‘𝐵) <s ( -us
‘( -us ‘𝐴)) ↔ ( -us ‘𝐵) <s 𝐴)) | 
| 40 | 36, 39 | bitrd 279 | . . 3
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (( -us ‘𝐴) <s 𝐵 ↔ ( -us ‘𝐵) <s 𝐴)) | 
| 41 | 40 | anbi1d 631 | . 2
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ((( -us ‘𝐴) <s 𝐵 ∧ 𝐴 <s 𝐵) ↔ (( -us ‘𝐵) <s 𝐴 ∧ 𝐴 <s 𝐵))) | 
| 42 | 33, 41 | bitrd 279 | 1
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → ((abss‘𝐴) <s 𝐵 ↔ (( -us ‘𝐵) <s 𝐴 ∧ 𝐴 <s 𝐵))) |