Step |
Hyp |
Ref |
Expression |
1 |
|
lt2addrd.1 |
|- ( ph -> A e. RR ) |
2 |
|
lt2addrd.2 |
|- ( ph -> B e. RR ) |
3 |
|
lt2addrd.3 |
|- ( ph -> C e. RR ) |
4 |
|
lt2addrd.4 |
|- ( ph -> A < ( B + C ) ) |
5 |
2 3
|
readdcld |
|- ( ph -> ( B + C ) e. RR ) |
6 |
5 1
|
resubcld |
|- ( ph -> ( ( B + C ) - A ) e. RR ) |
7 |
6
|
rehalfcld |
|- ( ph -> ( ( ( B + C ) - A ) / 2 ) e. RR ) |
8 |
2 7
|
resubcld |
|- ( ph -> ( B - ( ( ( B + C ) - A ) / 2 ) ) e. RR ) |
9 |
3 7
|
resubcld |
|- ( ph -> ( C - ( ( ( B + C ) - A ) / 2 ) ) e. RR ) |
10 |
3
|
recnd |
|- ( ph -> C e. CC ) |
11 |
2
|
recnd |
|- ( ph -> B e. CC ) |
12 |
11 10
|
addcld |
|- ( ph -> ( B + C ) e. CC ) |
13 |
1
|
recnd |
|- ( ph -> A e. CC ) |
14 |
12 13
|
subcld |
|- ( ph -> ( ( B + C ) - A ) e. CC ) |
15 |
14
|
halfcld |
|- ( ph -> ( ( ( B + C ) - A ) / 2 ) e. CC ) |
16 |
10 15 15
|
subsub4d |
|- ( ph -> ( ( C - ( ( ( B + C ) - A ) / 2 ) ) - ( ( ( B + C ) - A ) / 2 ) ) = ( C - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) ) |
17 |
16
|
oveq2d |
|- ( ph -> ( B + ( ( C - ( ( ( B + C ) - A ) / 2 ) ) - ( ( ( B + C ) - A ) / 2 ) ) ) = ( B + ( C - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) ) ) |
18 |
10 15
|
subcld |
|- ( ph -> ( C - ( ( ( B + C ) - A ) / 2 ) ) e. CC ) |
19 |
11 15 18
|
subadd23d |
|- ( ph -> ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) = ( B + ( ( C - ( ( ( B + C ) - A ) / 2 ) ) - ( ( ( B + C ) - A ) / 2 ) ) ) ) |
20 |
14
|
2halvesd |
|- ( ph -> ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) = ( ( B + C ) - A ) ) |
21 |
20 14
|
eqeltrd |
|- ( ph -> ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) e. CC ) |
22 |
11 10 21
|
addsubassd |
|- ( ph -> ( ( B + C ) - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) = ( B + ( C - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) ) ) |
23 |
17 19 22
|
3eqtr4d |
|- ( ph -> ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) = ( ( B + C ) - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) ) |
24 |
20
|
oveq2d |
|- ( ph -> ( ( B + C ) - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) = ( ( B + C ) - ( ( B + C ) - A ) ) ) |
25 |
12 13
|
nncand |
|- ( ph -> ( ( B + C ) - ( ( B + C ) - A ) ) = A ) |
26 |
23 24 25
|
3eqtrrd |
|- ( ph -> A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) ) |
27 |
|
difrp |
|- ( ( A e. RR /\ ( B + C ) e. RR ) -> ( A < ( B + C ) <-> ( ( B + C ) - A ) e. RR+ ) ) |
28 |
1 5 27
|
syl2anc |
|- ( ph -> ( A < ( B + C ) <-> ( ( B + C ) - A ) e. RR+ ) ) |
29 |
4 28
|
mpbid |
|- ( ph -> ( ( B + C ) - A ) e. RR+ ) |
30 |
29
|
rphalfcld |
|- ( ph -> ( ( ( B + C ) - A ) / 2 ) e. RR+ ) |
31 |
2 30
|
ltsubrpd |
|- ( ph -> ( B - ( ( ( B + C ) - A ) / 2 ) ) < B ) |
32 |
3 30
|
ltsubrpd |
|- ( ph -> ( C - ( ( ( B + C ) - A ) / 2 ) ) < C ) |
33 |
|
oveq1 |
|- ( b = ( B - ( ( ( B + C ) - A ) / 2 ) ) -> ( b + c ) = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) ) |
34 |
33
|
eqeq2d |
|- ( b = ( B - ( ( ( B + C ) - A ) / 2 ) ) -> ( A = ( b + c ) <-> A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) ) ) |
35 |
|
breq1 |
|- ( b = ( B - ( ( ( B + C ) - A ) / 2 ) ) -> ( b < B <-> ( B - ( ( ( B + C ) - A ) / 2 ) ) < B ) ) |
36 |
34 35
|
3anbi12d |
|- ( b = ( B - ( ( ( B + C ) - A ) / 2 ) ) -> ( ( A = ( b + c ) /\ b < B /\ c < C ) <-> ( A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) /\ ( B - ( ( ( B + C ) - A ) / 2 ) ) < B /\ c < C ) ) ) |
37 |
|
oveq2 |
|- ( c = ( C - ( ( ( B + C ) - A ) / 2 ) ) -> ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) ) |
38 |
37
|
eqeq2d |
|- ( c = ( C - ( ( ( B + C ) - A ) / 2 ) ) -> ( A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) <-> A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) ) ) |
39 |
|
breq1 |
|- ( c = ( C - ( ( ( B + C ) - A ) / 2 ) ) -> ( c < C <-> ( C - ( ( ( B + C ) - A ) / 2 ) ) < C ) ) |
40 |
38 39
|
3anbi13d |
|- ( c = ( C - ( ( ( B + C ) - A ) / 2 ) ) -> ( ( A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) /\ ( B - ( ( ( B + C ) - A ) / 2 ) ) < B /\ c < C ) <-> ( A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) /\ ( B - ( ( ( B + C ) - A ) / 2 ) ) < B /\ ( C - ( ( ( B + C ) - A ) / 2 ) ) < C ) ) ) |
41 |
36 40
|
rspc2ev |
|- ( ( ( B - ( ( ( B + C ) - A ) / 2 ) ) e. RR /\ ( C - ( ( ( B + C ) - A ) / 2 ) ) e. RR /\ ( A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) /\ ( B - ( ( ( B + C ) - A ) / 2 ) ) < B /\ ( C - ( ( ( B + C ) - A ) / 2 ) ) < C ) ) -> E. b e. RR E. c e. RR ( A = ( b + c ) /\ b < B /\ c < C ) ) |
42 |
8 9 26 31 32 41
|
syl113anc |
|- ( ph -> E. b e. RR E. c e. RR ( A = ( b + c ) /\ b < B /\ c < C ) ) |