Step |
Hyp |
Ref |
Expression |
1 |
|
2re |
|- 2 e. RR |
2 |
|
1re |
|- 1 e. RR |
3 |
|
ltsub1 |
|- ( ( 2 e. RR /\ A e. RR /\ 1 e. RR ) -> ( 2 < A <-> ( 2 - 1 ) < ( A - 1 ) ) ) |
4 |
1 2 3
|
mp3an13 |
|- ( A e. RR -> ( 2 < A <-> ( 2 - 1 ) < ( A - 1 ) ) ) |
5 |
|
2m1e1 |
|- ( 2 - 1 ) = 1 |
6 |
5
|
breq1i |
|- ( ( 2 - 1 ) < ( A - 1 ) <-> 1 < ( A - 1 ) ) |
7 |
4 6
|
bitrdi |
|- ( A e. RR -> ( 2 < A <-> 1 < ( A - 1 ) ) ) |
8 |
|
ltsub1 |
|- ( ( 2 e. RR /\ B e. RR /\ 1 e. RR ) -> ( 2 < B <-> ( 2 - 1 ) < ( B - 1 ) ) ) |
9 |
1 2 8
|
mp3an13 |
|- ( B e. RR -> ( 2 < B <-> ( 2 - 1 ) < ( B - 1 ) ) ) |
10 |
5
|
breq1i |
|- ( ( 2 - 1 ) < ( B - 1 ) <-> 1 < ( B - 1 ) ) |
11 |
9 10
|
bitrdi |
|- ( B e. RR -> ( 2 < B <-> 1 < ( B - 1 ) ) ) |
12 |
7 11
|
bi2anan9 |
|- ( ( A e. RR /\ B e. RR ) -> ( ( 2 < A /\ 2 < B ) <-> ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) ) ) |
13 |
|
peano2rem |
|- ( A e. RR -> ( A - 1 ) e. RR ) |
14 |
|
peano2rem |
|- ( B e. RR -> ( B - 1 ) e. RR ) |
15 |
|
mulgt1 |
|- ( ( ( ( A - 1 ) e. RR /\ ( B - 1 ) e. RR ) /\ ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) ) -> 1 < ( ( A - 1 ) x. ( B - 1 ) ) ) |
16 |
15
|
ex |
|- ( ( ( A - 1 ) e. RR /\ ( B - 1 ) e. RR ) -> ( ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) -> 1 < ( ( A - 1 ) x. ( B - 1 ) ) ) ) |
17 |
13 14 16
|
syl2an |
|- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) -> 1 < ( ( A - 1 ) x. ( B - 1 ) ) ) ) |
18 |
12 17
|
sylbid |
|- ( ( A e. RR /\ B e. RR ) -> ( ( 2 < A /\ 2 < B ) -> 1 < ( ( A - 1 ) x. ( B - 1 ) ) ) ) |
19 |
|
recn |
|- ( A e. RR -> A e. CC ) |
20 |
|
recn |
|- ( B e. RR -> B e. CC ) |
21 |
|
ax-1cn |
|- 1 e. CC |
22 |
|
mulsub |
|- ( ( ( A e. CC /\ 1 e. CC ) /\ ( B e. CC /\ 1 e. CC ) ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) |
23 |
21 22
|
mpanl2 |
|- ( ( A e. CC /\ ( B e. CC /\ 1 e. CC ) ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) |
24 |
21 23
|
mpanr2 |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) |
25 |
19 20 24
|
syl2an |
|- ( ( A e. RR /\ B e. RR ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) |
26 |
25
|
breq2d |
|- ( ( A e. RR /\ B e. RR ) -> ( 1 < ( ( A - 1 ) x. ( B - 1 ) ) <-> 1 < ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) ) |
27 |
|
remulcl |
|- ( ( A e. RR /\ 1 e. RR ) -> ( A x. 1 ) e. RR ) |
28 |
2 27
|
mpan2 |
|- ( A e. RR -> ( A x. 1 ) e. RR ) |
29 |
|
remulcl |
|- ( ( B e. RR /\ 1 e. RR ) -> ( B x. 1 ) e. RR ) |
30 |
2 29
|
mpan2 |
|- ( B e. RR -> ( B x. 1 ) e. RR ) |
31 |
|
readdcl |
|- ( ( ( A x. 1 ) e. RR /\ ( B x. 1 ) e. RR ) -> ( ( A x. 1 ) + ( B x. 1 ) ) e. RR ) |
32 |
28 30 31
|
syl2an |
|- ( ( A e. RR /\ B e. RR ) -> ( ( A x. 1 ) + ( B x. 1 ) ) e. RR ) |
33 |
|
remulcl |
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR ) |
34 |
2 2
|
remulcli |
|- ( 1 x. 1 ) e. RR |
35 |
|
readdcl |
|- ( ( ( A x. B ) e. RR /\ ( 1 x. 1 ) e. RR ) -> ( ( A x. B ) + ( 1 x. 1 ) ) e. RR ) |
36 |
33 34 35
|
sylancl |
|- ( ( A e. RR /\ B e. RR ) -> ( ( A x. B ) + ( 1 x. 1 ) ) e. RR ) |
37 |
|
ltaddsub2 |
|- ( ( ( ( A x. 1 ) + ( B x. 1 ) ) e. RR /\ 1 e. RR /\ ( ( A x. B ) + ( 1 x. 1 ) ) e. RR ) -> ( ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + ( 1 x. 1 ) ) <-> 1 < ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) ) |
38 |
2 37
|
mp3an2 |
|- ( ( ( ( A x. 1 ) + ( B x. 1 ) ) e. RR /\ ( ( A x. B ) + ( 1 x. 1 ) ) e. RR ) -> ( ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + ( 1 x. 1 ) ) <-> 1 < ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) ) |
39 |
32 36 38
|
syl2anc |
|- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + ( 1 x. 1 ) ) <-> 1 < ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) ) |
40 |
|
1t1e1 |
|- ( 1 x. 1 ) = 1 |
41 |
40
|
oveq2i |
|- ( ( A x. B ) + ( 1 x. 1 ) ) = ( ( A x. B ) + 1 ) |
42 |
41
|
breq2i |
|- ( ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + ( 1 x. 1 ) ) <-> ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + 1 ) ) |
43 |
39 42
|
bitr3di |
|- ( ( A e. RR /\ B e. RR ) -> ( 1 < ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) <-> ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + 1 ) ) ) |
44 |
|
ltadd1 |
|- ( ( ( ( A x. 1 ) + ( B x. 1 ) ) e. RR /\ ( A x. B ) e. RR /\ 1 e. RR ) -> ( ( ( A x. 1 ) + ( B x. 1 ) ) < ( A x. B ) <-> ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + 1 ) ) ) |
45 |
2 44
|
mp3an3 |
|- ( ( ( ( A x. 1 ) + ( B x. 1 ) ) e. RR /\ ( A x. B ) e. RR ) -> ( ( ( A x. 1 ) + ( B x. 1 ) ) < ( A x. B ) <-> ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + 1 ) ) ) |
46 |
32 33 45
|
syl2anc |
|- ( ( A e. RR /\ B e. RR ) -> ( ( ( A x. 1 ) + ( B x. 1 ) ) < ( A x. B ) <-> ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + 1 ) ) ) |
47 |
|
ax-1rid |
|- ( A e. RR -> ( A x. 1 ) = A ) |
48 |
|
ax-1rid |
|- ( B e. RR -> ( B x. 1 ) = B ) |
49 |
47 48
|
oveqan12d |
|- ( ( A e. RR /\ B e. RR ) -> ( ( A x. 1 ) + ( B x. 1 ) ) = ( A + B ) ) |
50 |
49
|
breq1d |
|- ( ( A e. RR /\ B e. RR ) -> ( ( ( A x. 1 ) + ( B x. 1 ) ) < ( A x. B ) <-> ( A + B ) < ( A x. B ) ) ) |
51 |
46 50
|
bitr3d |
|- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + 1 ) <-> ( A + B ) < ( A x. B ) ) ) |
52 |
26 43 51
|
3bitrd |
|- ( ( A e. RR /\ B e. RR ) -> ( 1 < ( ( A - 1 ) x. ( B - 1 ) ) <-> ( A + B ) < ( A x. B ) ) ) |
53 |
18 52
|
sylibd |
|- ( ( A e. RR /\ B e. RR ) -> ( ( 2 < A /\ 2 < B ) -> ( A + B ) < ( A x. B ) ) ) |
54 |
53
|
imp |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 2 < A /\ 2 < B ) ) -> ( A + B ) < ( A x. B ) ) |