Step |
Hyp |
Ref |
Expression |
1 |
|
dvdsadd2b |
|- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) ) |
2 |
1
|
a1d |
|- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> ( B e. RR -> ( A || B <-> A || ( C + B ) ) ) ) |
3 |
2
|
3exp |
|- ( A e. ZZ -> ( B e. ZZ -> ( ( C e. ZZ /\ A || C ) -> ( B e. RR -> ( A || B <-> A || ( C + B ) ) ) ) ) ) |
4 |
3
|
com24 |
|- ( A e. ZZ -> ( B e. RR -> ( ( C e. ZZ /\ A || C ) -> ( B e. ZZ -> ( A || B <-> A || ( C + B ) ) ) ) ) ) |
5 |
4
|
3imp |
|- ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( B e. ZZ -> ( A || B <-> A || ( C + B ) ) ) ) |
6 |
5
|
com12 |
|- ( B e. ZZ -> ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) ) ) |
7 |
|
dvdszrcl |
|- ( A || B -> ( A e. ZZ /\ B e. ZZ ) ) |
8 |
|
pm2.24 |
|- ( B e. ZZ -> ( -. B e. ZZ -> A || ( C + B ) ) ) |
9 |
7 8
|
simpl2im |
|- ( A || B -> ( -. B e. ZZ -> A || ( C + B ) ) ) |
10 |
9
|
com12 |
|- ( -. B e. ZZ -> ( A || B -> A || ( C + B ) ) ) |
11 |
10
|
adantr |
|- ( ( -. B e. ZZ /\ ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) ) -> ( A || B -> A || ( C + B ) ) ) |
12 |
|
dvdszrcl |
|- ( A || ( C + B ) -> ( A e. ZZ /\ ( C + B ) e. ZZ ) ) |
13 |
|
zcn |
|- ( C e. ZZ -> C e. CC ) |
14 |
13
|
adantr |
|- ( ( C e. ZZ /\ ( B e. RR /\ -. B e. ZZ ) ) -> C e. CC ) |
15 |
|
recn |
|- ( B e. RR -> B e. CC ) |
16 |
15
|
ad2antrl |
|- ( ( C e. ZZ /\ ( B e. RR /\ -. B e. ZZ ) ) -> B e. CC ) |
17 |
14 16
|
addcomd |
|- ( ( C e. ZZ /\ ( B e. RR /\ -. B e. ZZ ) ) -> ( C + B ) = ( B + C ) ) |
18 |
|
eldif |
|- ( B e. ( RR \ ZZ ) <-> ( B e. RR /\ -. B e. ZZ ) ) |
19 |
|
nzadd |
|- ( ( B e. ( RR \ ZZ ) /\ C e. ZZ ) -> ( B + C ) e. ( RR \ ZZ ) ) |
20 |
19
|
eldifbd |
|- ( ( B e. ( RR \ ZZ ) /\ C e. ZZ ) -> -. ( B + C ) e. ZZ ) |
21 |
20
|
expcom |
|- ( C e. ZZ -> ( B e. ( RR \ ZZ ) -> -. ( B + C ) e. ZZ ) ) |
22 |
18 21
|
syl5bir |
|- ( C e. ZZ -> ( ( B e. RR /\ -. B e. ZZ ) -> -. ( B + C ) e. ZZ ) ) |
23 |
22
|
imp |
|- ( ( C e. ZZ /\ ( B e. RR /\ -. B e. ZZ ) ) -> -. ( B + C ) e. ZZ ) |
24 |
17 23
|
eqneltrd |
|- ( ( C e. ZZ /\ ( B e. RR /\ -. B e. ZZ ) ) -> -. ( C + B ) e. ZZ ) |
25 |
24
|
exp32 |
|- ( C e. ZZ -> ( B e. RR -> ( -. B e. ZZ -> -. ( C + B ) e. ZZ ) ) ) |
26 |
|
pm2.21 |
|- ( -. ( C + B ) e. ZZ -> ( ( C + B ) e. ZZ -> A || B ) ) |
27 |
25 26
|
syl8 |
|- ( C e. ZZ -> ( B e. RR -> ( -. B e. ZZ -> ( ( C + B ) e. ZZ -> A || B ) ) ) ) |
28 |
27
|
adantr |
|- ( ( C e. ZZ /\ A || C ) -> ( B e. RR -> ( -. B e. ZZ -> ( ( C + B ) e. ZZ -> A || B ) ) ) ) |
29 |
28
|
com12 |
|- ( B e. RR -> ( ( C e. ZZ /\ A || C ) -> ( -. B e. ZZ -> ( ( C + B ) e. ZZ -> A || B ) ) ) ) |
30 |
29
|
a1i |
|- ( A e. ZZ -> ( B e. RR -> ( ( C e. ZZ /\ A || C ) -> ( -. B e. ZZ -> ( ( C + B ) e. ZZ -> A || B ) ) ) ) ) |
31 |
30
|
3imp |
|- ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( -. B e. ZZ -> ( ( C + B ) e. ZZ -> A || B ) ) ) |
32 |
31
|
impcom |
|- ( ( -. B e. ZZ /\ ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) ) -> ( ( C + B ) e. ZZ -> A || B ) ) |
33 |
32
|
com12 |
|- ( ( C + B ) e. ZZ -> ( ( -. B e. ZZ /\ ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) ) -> A || B ) ) |
34 |
12 33
|
simpl2im |
|- ( A || ( C + B ) -> ( ( -. B e. ZZ /\ ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) ) -> A || B ) ) |
35 |
34
|
com12 |
|- ( ( -. B e. ZZ /\ ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) ) -> ( A || ( C + B ) -> A || B ) ) |
36 |
11 35
|
impbid |
|- ( ( -. B e. ZZ /\ ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) ) -> ( A || B <-> A || ( C + B ) ) ) |
37 |
36
|
ex |
|- ( -. B e. ZZ -> ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) ) ) |
38 |
6 37
|
pm2.61i |
|- ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) ) |