Step |
Hyp |
Ref |
Expression |
1 |
|
nn0re |
|- ( A e. NN0 -> A e. RR ) |
2 |
|
nnrp |
|- ( B e. NN -> B e. RR+ ) |
3 |
1 2
|
anim12i |
|- ( ( A e. NN0 /\ B e. NN ) -> ( A e. RR /\ B e. RR+ ) ) |
4 |
3
|
3adant3 |
|- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> ( A e. RR /\ B e. RR+ ) ) |
5 |
|
nn0ge0 |
|- ( A e. NN0 -> 0 <_ A ) |
6 |
5
|
3ad2ant1 |
|- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> 0 <_ A ) |
7 |
6
|
anim1i |
|- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> ( 0 <_ A /\ A < B ) ) |
8 |
7
|
ancoms |
|- ( ( A < B /\ ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) ) -> ( 0 <_ A /\ A < B ) ) |
9 |
|
modid |
|- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A mod B ) = A ) |
10 |
4 8 9
|
syl2an2 |
|- ( ( A < B /\ ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) ) -> ( A mod B ) = A ) |
11 |
|
iftrue |
|- ( A < B -> if ( A < B , A , ( A - B ) ) = A ) |
12 |
11
|
eqcomd |
|- ( A < B -> A = if ( A < B , A , ( A - B ) ) ) |
13 |
12
|
adantr |
|- ( ( A < B /\ ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) ) -> A = if ( A < B , A , ( A - B ) ) ) |
14 |
10 13
|
eqtrd |
|- ( ( A < B /\ ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) ) -> ( A mod B ) = if ( A < B , A , ( A - B ) ) ) |
15 |
14
|
ex |
|- ( A < B -> ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> ( A mod B ) = if ( A < B , A , ( A - B ) ) ) ) |
16 |
4
|
adantr |
|- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> ( A e. RR /\ B e. RR+ ) ) |
17 |
|
nnre |
|- ( B e. NN -> B e. RR ) |
18 |
|
lenlt |
|- ( ( B e. RR /\ A e. RR ) -> ( B <_ A <-> -. A < B ) ) |
19 |
17 1 18
|
syl2anr |
|- ( ( A e. NN0 /\ B e. NN ) -> ( B <_ A <-> -. A < B ) ) |
20 |
19
|
3adant3 |
|- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> ( B <_ A <-> -. A < B ) ) |
21 |
20
|
biimpar |
|- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> B <_ A ) |
22 |
|
simpl3 |
|- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> A < ( 2 x. B ) ) |
23 |
|
2submod |
|- ( ( ( A e. RR /\ B e. RR+ ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( A mod B ) = ( A - B ) ) |
24 |
16 21 22 23
|
syl12anc |
|- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> ( A mod B ) = ( A - B ) ) |
25 |
|
iffalse |
|- ( -. A < B -> if ( A < B , A , ( A - B ) ) = ( A - B ) ) |
26 |
25
|
adantl |
|- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> if ( A < B , A , ( A - B ) ) = ( A - B ) ) |
27 |
26
|
eqcomd |
|- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> ( A - B ) = if ( A < B , A , ( A - B ) ) ) |
28 |
24 27
|
eqtrd |
|- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> ( A mod B ) = if ( A < B , A , ( A - B ) ) ) |
29 |
28
|
expcom |
|- ( -. A < B -> ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> ( A mod B ) = if ( A < B , A , ( A - B ) ) ) ) |
30 |
15 29
|
pm2.61i |
|- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> ( A mod B ) = if ( A < B , A , ( A - B ) ) ) |