Step |
Hyp |
Ref |
Expression |
1 |
|
negscl |
⊢ ( 𝐴 ∈ No → ( -us ‘ 𝐴 ) ∈ No ) |
2 |
1
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( abss ‘ 𝐴 ) <s 𝐵 ) → ( -us ‘ 𝐴 ) ∈ No ) |
3 |
|
absscl |
⊢ ( ( -us ‘ 𝐴 ) ∈ No → ( abss ‘ ( -us ‘ 𝐴 ) ) ∈ No ) |
4 |
1 3
|
syl |
⊢ ( 𝐴 ∈ No → ( abss ‘ ( -us ‘ 𝐴 ) ) ∈ No ) |
5 |
4
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( abss ‘ 𝐴 ) <s 𝐵 ) → ( abss ‘ ( -us ‘ 𝐴 ) ) ∈ No ) |
6 |
|
simplr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( abss ‘ 𝐴 ) <s 𝐵 ) → 𝐵 ∈ No ) |
7 |
|
sleabs |
⊢ ( ( -us ‘ 𝐴 ) ∈ No → ( -us ‘ 𝐴 ) ≤s ( abss ‘ ( -us ‘ 𝐴 ) ) ) |
8 |
1 7
|
syl |
⊢ ( 𝐴 ∈ No → ( -us ‘ 𝐴 ) ≤s ( abss ‘ ( -us ‘ 𝐴 ) ) ) |
9 |
8
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( abss ‘ 𝐴 ) <s 𝐵 ) → ( -us ‘ 𝐴 ) ≤s ( abss ‘ ( -us ‘ 𝐴 ) ) ) |
10 |
|
abssneg |
⊢ ( 𝐴 ∈ No → ( abss ‘ ( -us ‘ 𝐴 ) ) = ( abss ‘ 𝐴 ) ) |
11 |
10
|
adantr |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( abss ‘ ( -us ‘ 𝐴 ) ) = ( abss ‘ 𝐴 ) ) |
12 |
11
|
breq1d |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( abss ‘ ( -us ‘ 𝐴 ) ) <s 𝐵 ↔ ( abss ‘ 𝐴 ) <s 𝐵 ) ) |
13 |
12
|
biimpar |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( abss ‘ 𝐴 ) <s 𝐵 ) → ( abss ‘ ( -us ‘ 𝐴 ) ) <s 𝐵 ) |
14 |
2 5 6 9 13
|
slelttrd |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( abss ‘ 𝐴 ) <s 𝐵 ) → ( -us ‘ 𝐴 ) <s 𝐵 ) |
15 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( abss ‘ 𝐴 ) <s 𝐵 ) → 𝐴 ∈ No ) |
16 |
|
absscl |
⊢ ( 𝐴 ∈ No → ( abss ‘ 𝐴 ) ∈ No ) |
17 |
16
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( abss ‘ 𝐴 ) <s 𝐵 ) → ( abss ‘ 𝐴 ) ∈ No ) |
18 |
|
sleabs |
⊢ ( 𝐴 ∈ No → 𝐴 ≤s ( abss ‘ 𝐴 ) ) |
19 |
18
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( abss ‘ 𝐴 ) <s 𝐵 ) → 𝐴 ≤s ( abss ‘ 𝐴 ) ) |
20 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( abss ‘ 𝐴 ) <s 𝐵 ) → ( abss ‘ 𝐴 ) <s 𝐵 ) |
21 |
15 17 6 19 20
|
slelttrd |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( abss ‘ 𝐴 ) <s 𝐵 ) → 𝐴 <s 𝐵 ) |
22 |
14 21
|
jca |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( abss ‘ 𝐴 ) <s 𝐵 ) → ( ( -us ‘ 𝐴 ) <s 𝐵 ∧ 𝐴 <s 𝐵 ) ) |
23 |
22
|
ex |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( abss ‘ 𝐴 ) <s 𝐵 → ( ( -us ‘ 𝐴 ) <s 𝐵 ∧ 𝐴 <s 𝐵 ) ) ) |
24 |
|
abssor |
⊢ ( 𝐴 ∈ No → ( ( abss ‘ 𝐴 ) = 𝐴 ∨ ( abss ‘ 𝐴 ) = ( -us ‘ 𝐴 ) ) ) |
25 |
24
|
adantr |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( abss ‘ 𝐴 ) = 𝐴 ∨ ( abss ‘ 𝐴 ) = ( -us ‘ 𝐴 ) ) ) |
26 |
|
breq1 |
⊢ ( ( abss ‘ 𝐴 ) = 𝐴 → ( ( abss ‘ 𝐴 ) <s 𝐵 ↔ 𝐴 <s 𝐵 ) ) |
27 |
26
|
biimprd |
⊢ ( ( abss ‘ 𝐴 ) = 𝐴 → ( 𝐴 <s 𝐵 → ( abss ‘ 𝐴 ) <s 𝐵 ) ) |
28 |
|
breq1 |
⊢ ( ( abss ‘ 𝐴 ) = ( -us ‘ 𝐴 ) → ( ( abss ‘ 𝐴 ) <s 𝐵 ↔ ( -us ‘ 𝐴 ) <s 𝐵 ) ) |
29 |
28
|
biimprd |
⊢ ( ( abss ‘ 𝐴 ) = ( -us ‘ 𝐴 ) → ( ( -us ‘ 𝐴 ) <s 𝐵 → ( abss ‘ 𝐴 ) <s 𝐵 ) ) |
30 |
27 29
|
jaoa |
⊢ ( ( ( abss ‘ 𝐴 ) = 𝐴 ∨ ( abss ‘ 𝐴 ) = ( -us ‘ 𝐴 ) ) → ( ( 𝐴 <s 𝐵 ∧ ( -us ‘ 𝐴 ) <s 𝐵 ) → ( abss ‘ 𝐴 ) <s 𝐵 ) ) |
31 |
30
|
ancomsd |
⊢ ( ( ( abss ‘ 𝐴 ) = 𝐴 ∨ ( abss ‘ 𝐴 ) = ( -us ‘ 𝐴 ) ) → ( ( ( -us ‘ 𝐴 ) <s 𝐵 ∧ 𝐴 <s 𝐵 ) → ( abss ‘ 𝐴 ) <s 𝐵 ) ) |
32 |
25 31
|
syl |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( ( -us ‘ 𝐴 ) <s 𝐵 ∧ 𝐴 <s 𝐵 ) → ( abss ‘ 𝐴 ) <s 𝐵 ) ) |
33 |
23 32
|
impbid |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( abss ‘ 𝐴 ) <s 𝐵 ↔ ( ( -us ‘ 𝐴 ) <s 𝐵 ∧ 𝐴 <s 𝐵 ) ) ) |
34 |
1
|
adantr |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘ 𝐴 ) ∈ No ) |
35 |
|
simpr |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ No ) |
36 |
34 35
|
sltnegd |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( -us ‘ 𝐴 ) <s 𝐵 ↔ ( -us ‘ 𝐵 ) <s ( -us ‘ ( -us ‘ 𝐴 ) ) ) ) |
37 |
|
negnegs |
⊢ ( 𝐴 ∈ No → ( -us ‘ ( -us ‘ 𝐴 ) ) = 𝐴 ) |
38 |
37
|
adantr |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘ ( -us ‘ 𝐴 ) ) = 𝐴 ) |
39 |
38
|
breq2d |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( -us ‘ 𝐵 ) <s ( -us ‘ ( -us ‘ 𝐴 ) ) ↔ ( -us ‘ 𝐵 ) <s 𝐴 ) ) |
40 |
36 39
|
bitrd |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( -us ‘ 𝐴 ) <s 𝐵 ↔ ( -us ‘ 𝐵 ) <s 𝐴 ) ) |
41 |
40
|
anbi1d |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( ( -us ‘ 𝐴 ) <s 𝐵 ∧ 𝐴 <s 𝐵 ) ↔ ( ( -us ‘ 𝐵 ) <s 𝐴 ∧ 𝐴 <s 𝐵 ) ) ) |
42 |
33 41
|
bitrd |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( abss ‘ 𝐴 ) <s 𝐵 ↔ ( ( -us ‘ 𝐵 ) <s 𝐴 ∧ 𝐴 <s 𝐵 ) ) ) |