Step |
Hyp |
Ref |
Expression |
1 |
|
eldif |
|- ( ( A / B ) e. ( RR \ QQ ) <-> ( ( A / B ) e. RR /\ -. ( A / B ) e. QQ ) ) |
2 |
|
modval |
|- ( ( A e. RR /\ B e. RR+ ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) ) |
3 |
2
|
eqeq1d |
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A mod B ) = 0 <-> ( A - ( B x. ( |_ ` ( A / B ) ) ) ) = 0 ) ) |
4 |
|
recn |
|- ( A e. RR -> A e. CC ) |
5 |
4
|
adantr |
|- ( ( A e. RR /\ B e. RR+ ) -> A e. CC ) |
6 |
|
rpre |
|- ( B e. RR+ -> B e. RR ) |
7 |
6
|
adantl |
|- ( ( A e. RR /\ B e. RR+ ) -> B e. RR ) |
8 |
|
refldivcl |
|- ( ( A e. RR /\ B e. RR+ ) -> ( |_ ` ( A / B ) ) e. RR ) |
9 |
7 8
|
remulcld |
|- ( ( A e. RR /\ B e. RR+ ) -> ( B x. ( |_ ` ( A / B ) ) ) e. RR ) |
10 |
9
|
recnd |
|- ( ( A e. RR /\ B e. RR+ ) -> ( B x. ( |_ ` ( A / B ) ) ) e. CC ) |
11 |
5 10
|
subeq0ad |
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A - ( B x. ( |_ ` ( A / B ) ) ) ) = 0 <-> A = ( B x. ( |_ ` ( A / B ) ) ) ) ) |
12 |
|
rerpdivcl |
|- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. RR ) |
13 |
|
reflcl |
|- ( ( A / B ) e. RR -> ( |_ ` ( A / B ) ) e. RR ) |
14 |
13
|
recnd |
|- ( ( A / B ) e. RR -> ( |_ ` ( A / B ) ) e. CC ) |
15 |
12 14
|
syl |
|- ( ( A e. RR /\ B e. RR+ ) -> ( |_ ` ( A / B ) ) e. CC ) |
16 |
|
rpcnne0 |
|- ( B e. RR+ -> ( B e. CC /\ B =/= 0 ) ) |
17 |
16
|
adantl |
|- ( ( A e. RR /\ B e. RR+ ) -> ( B e. CC /\ B =/= 0 ) ) |
18 |
|
divmul2 |
|- ( ( A e. CC /\ ( |_ ` ( A / B ) ) e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( A / B ) = ( |_ ` ( A / B ) ) <-> A = ( B x. ( |_ ` ( A / B ) ) ) ) ) |
19 |
5 15 17 18
|
syl3anc |
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A / B ) = ( |_ ` ( A / B ) ) <-> A = ( B x. ( |_ ` ( A / B ) ) ) ) ) |
20 |
|
eqcom |
|- ( ( A / B ) = ( |_ ` ( A / B ) ) <-> ( |_ ` ( A / B ) ) = ( A / B ) ) |
21 |
19 20
|
bitr3di |
|- ( ( A e. RR /\ B e. RR+ ) -> ( A = ( B x. ( |_ ` ( A / B ) ) ) <-> ( |_ ` ( A / B ) ) = ( A / B ) ) ) |
22 |
3 11 21
|
3bitrd |
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A mod B ) = 0 <-> ( |_ ` ( A / B ) ) = ( A / B ) ) ) |
23 |
|
flidz |
|- ( ( A / B ) e. RR -> ( ( |_ ` ( A / B ) ) = ( A / B ) <-> ( A / B ) e. ZZ ) ) |
24 |
|
zq |
|- ( ( A / B ) e. ZZ -> ( A / B ) e. QQ ) |
25 |
23 24
|
syl6bi |
|- ( ( A / B ) e. RR -> ( ( |_ ` ( A / B ) ) = ( A / B ) -> ( A / B ) e. QQ ) ) |
26 |
12 25
|
syl |
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( |_ ` ( A / B ) ) = ( A / B ) -> ( A / B ) e. QQ ) ) |
27 |
22 26
|
sylbid |
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A mod B ) = 0 -> ( A / B ) e. QQ ) ) |
28 |
27
|
necon3bd |
|- ( ( A e. RR /\ B e. RR+ ) -> ( -. ( A / B ) e. QQ -> ( A mod B ) =/= 0 ) ) |
29 |
28
|
adantld |
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( ( A / B ) e. RR /\ -. ( A / B ) e. QQ ) -> ( A mod B ) =/= 0 ) ) |
30 |
1 29
|
syl5bi |
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A / B ) e. ( RR \ QQ ) -> ( A mod B ) =/= 0 ) ) |
31 |
30
|
3impia |
|- ( ( A e. RR /\ B e. RR+ /\ ( A / B ) e. ( RR \ QQ ) ) -> ( A mod B ) =/= 0 ) |