Step |
Hyp |
Ref |
Expression |
1 |
|
quorem.1 |
|- Q = ( |_ ` ( A / B ) ) |
2 |
|
quorem.2 |
|- R = ( A - ( B x. Q ) ) |
3 |
|
fldivnn0 |
|- ( ( A e. NN0 /\ B e. NN ) -> ( |_ ` ( A / B ) ) e. NN0 ) |
4 |
1 3
|
eqeltrid |
|- ( ( A e. NN0 /\ B e. NN ) -> Q e. NN0 ) |
5 |
|
nn0z |
|- ( A e. NN0 -> A e. ZZ ) |
6 |
1 2
|
quoremz |
|- ( ( A e. ZZ /\ B e. NN ) -> ( ( Q e. ZZ /\ R e. NN0 ) /\ ( R < B /\ A = ( ( B x. Q ) + R ) ) ) ) |
7 |
5 6
|
sylan |
|- ( ( A e. NN0 /\ B e. NN ) -> ( ( Q e. ZZ /\ R e. NN0 ) /\ ( R < B /\ A = ( ( B x. Q ) + R ) ) ) ) |
8 |
|
simpl |
|- ( ( Q e. NN0 /\ Q e. ZZ ) -> Q e. NN0 ) |
9 |
8
|
anim1i |
|- ( ( ( Q e. NN0 /\ Q e. ZZ ) /\ R e. NN0 ) -> ( Q e. NN0 /\ R e. NN0 ) ) |
10 |
9
|
anasss |
|- ( ( Q e. NN0 /\ ( Q e. ZZ /\ R e. NN0 ) ) -> ( Q e. NN0 /\ R e. NN0 ) ) |
11 |
10
|
anim1i |
|- ( ( ( Q e. NN0 /\ ( Q e. ZZ /\ R e. NN0 ) ) /\ ( R < B /\ A = ( ( B x. Q ) + R ) ) ) -> ( ( Q e. NN0 /\ R e. NN0 ) /\ ( R < B /\ A = ( ( B x. Q ) + R ) ) ) ) |
12 |
11
|
anasss |
|- ( ( Q e. NN0 /\ ( ( Q e. ZZ /\ R e. NN0 ) /\ ( R < B /\ A = ( ( B x. Q ) + R ) ) ) ) -> ( ( Q e. NN0 /\ R e. NN0 ) /\ ( R < B /\ A = ( ( B x. Q ) + R ) ) ) ) |
13 |
4 7 12
|
syl2anc |
|- ( ( A e. NN0 /\ B e. NN ) -> ( ( Q e. NN0 /\ R e. NN0 ) /\ ( R < B /\ A = ( ( B x. Q ) + R ) ) ) ) |