Step |
Hyp |
Ref |
Expression |
1 |
|
lefldiveq.a |
|- ( ph -> A e. RR ) |
2 |
|
lefldiveq.b |
|- ( ph -> B e. RR+ ) |
3 |
|
lefldiveq.c |
|- ( ph -> C e. ( ( A - ( A mod B ) ) [,] A ) ) |
4 |
|
moddiffl |
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A - ( A mod B ) ) / B ) = ( |_ ` ( A / B ) ) ) |
5 |
1 2 4
|
syl2anc |
|- ( ph -> ( ( A - ( A mod B ) ) / B ) = ( |_ ` ( A / B ) ) ) |
6 |
1 2
|
rerpdivcld |
|- ( ph -> ( A / B ) e. RR ) |
7 |
6
|
flcld |
|- ( ph -> ( |_ ` ( A / B ) ) e. ZZ ) |
8 |
5 7
|
eqeltrd |
|- ( ph -> ( ( A - ( A mod B ) ) / B ) e. ZZ ) |
9 |
|
flid |
|- ( ( ( A - ( A mod B ) ) / B ) e. ZZ -> ( |_ ` ( ( A - ( A mod B ) ) / B ) ) = ( ( A - ( A mod B ) ) / B ) ) |
10 |
8 9
|
syl |
|- ( ph -> ( |_ ` ( ( A - ( A mod B ) ) / B ) ) = ( ( A - ( A mod B ) ) / B ) ) |
11 |
10 5
|
eqtr2d |
|- ( ph -> ( |_ ` ( A / B ) ) = ( |_ ` ( ( A - ( A mod B ) ) / B ) ) ) |
12 |
1 2
|
modcld |
|- ( ph -> ( A mod B ) e. RR ) |
13 |
1 12
|
resubcld |
|- ( ph -> ( A - ( A mod B ) ) e. RR ) |
14 |
13 2
|
rerpdivcld |
|- ( ph -> ( ( A - ( A mod B ) ) / B ) e. RR ) |
15 |
|
iccssre |
|- ( ( ( A - ( A mod B ) ) e. RR /\ A e. RR ) -> ( ( A - ( A mod B ) ) [,] A ) C_ RR ) |
16 |
13 1 15
|
syl2anc |
|- ( ph -> ( ( A - ( A mod B ) ) [,] A ) C_ RR ) |
17 |
16 3
|
sseldd |
|- ( ph -> C e. RR ) |
18 |
17 2
|
rerpdivcld |
|- ( ph -> ( C / B ) e. RR ) |
19 |
13
|
rexrd |
|- ( ph -> ( A - ( A mod B ) ) e. RR* ) |
20 |
1
|
rexrd |
|- ( ph -> A e. RR* ) |
21 |
|
iccgelb |
|- ( ( ( A - ( A mod B ) ) e. RR* /\ A e. RR* /\ C e. ( ( A - ( A mod B ) ) [,] A ) ) -> ( A - ( A mod B ) ) <_ C ) |
22 |
19 20 3 21
|
syl3anc |
|- ( ph -> ( A - ( A mod B ) ) <_ C ) |
23 |
13 17 2 22
|
lediv1dd |
|- ( ph -> ( ( A - ( A mod B ) ) / B ) <_ ( C / B ) ) |
24 |
|
flwordi |
|- ( ( ( ( A - ( A mod B ) ) / B ) e. RR /\ ( C / B ) e. RR /\ ( ( A - ( A mod B ) ) / B ) <_ ( C / B ) ) -> ( |_ ` ( ( A - ( A mod B ) ) / B ) ) <_ ( |_ ` ( C / B ) ) ) |
25 |
14 18 23 24
|
syl3anc |
|- ( ph -> ( |_ ` ( ( A - ( A mod B ) ) / B ) ) <_ ( |_ ` ( C / B ) ) ) |
26 |
11 25
|
eqbrtrd |
|- ( ph -> ( |_ ` ( A / B ) ) <_ ( |_ ` ( C / B ) ) ) |
27 |
|
iccleub |
|- ( ( ( A - ( A mod B ) ) e. RR* /\ A e. RR* /\ C e. ( ( A - ( A mod B ) ) [,] A ) ) -> C <_ A ) |
28 |
19 20 3 27
|
syl3anc |
|- ( ph -> C <_ A ) |
29 |
17 1 2 28
|
lediv1dd |
|- ( ph -> ( C / B ) <_ ( A / B ) ) |
30 |
|
flwordi |
|- ( ( ( C / B ) e. RR /\ ( A / B ) e. RR /\ ( C / B ) <_ ( A / B ) ) -> ( |_ ` ( C / B ) ) <_ ( |_ ` ( A / B ) ) ) |
31 |
18 6 29 30
|
syl3anc |
|- ( ph -> ( |_ ` ( C / B ) ) <_ ( |_ ` ( A / B ) ) ) |
32 |
|
reflcl |
|- ( ( A / B ) e. RR -> ( |_ ` ( A / B ) ) e. RR ) |
33 |
6 32
|
syl |
|- ( ph -> ( |_ ` ( A / B ) ) e. RR ) |
34 |
|
reflcl |
|- ( ( C / B ) e. RR -> ( |_ ` ( C / B ) ) e. RR ) |
35 |
18 34
|
syl |
|- ( ph -> ( |_ ` ( C / B ) ) e. RR ) |
36 |
33 35
|
letri3d |
|- ( ph -> ( ( |_ ` ( A / B ) ) = ( |_ ` ( C / B ) ) <-> ( ( |_ ` ( A / B ) ) <_ ( |_ ` ( C / B ) ) /\ ( |_ ` ( C / B ) ) <_ ( |_ ` ( A / B ) ) ) ) ) |
37 |
26 31 36
|
mpbir2and |
|- ( ph -> ( |_ ` ( A / B ) ) = ( |_ ` ( C / B ) ) ) |