Step |
Hyp |
Ref |
Expression |
1 |
|
supsubc.a1 |
|- ( ph -> A C_ RR ) |
2 |
|
supsubc.a2 |
|- ( ph -> A =/= (/) ) |
3 |
|
supsubc.a3 |
|- ( ph -> E. x e. RR A. y e. A y <_ x ) |
4 |
|
supsubc.b |
|- ( ph -> B e. RR ) |
5 |
|
supsubc.c |
|- C = { z | E. v e. A z = ( v - B ) } |
6 |
5
|
a1i |
|- ( ph -> C = { z | E. v e. A z = ( v - B ) } ) |
7 |
1
|
sselda |
|- ( ( ph /\ v e. A ) -> v e. RR ) |
8 |
7
|
recnd |
|- ( ( ph /\ v e. A ) -> v e. CC ) |
9 |
4
|
recnd |
|- ( ph -> B e. CC ) |
10 |
9
|
adantr |
|- ( ( ph /\ v e. A ) -> B e. CC ) |
11 |
8 10
|
negsubd |
|- ( ( ph /\ v e. A ) -> ( v + -u B ) = ( v - B ) ) |
12 |
11
|
eqcomd |
|- ( ( ph /\ v e. A ) -> ( v - B ) = ( v + -u B ) ) |
13 |
12
|
eqeq2d |
|- ( ( ph /\ v e. A ) -> ( z = ( v - B ) <-> z = ( v + -u B ) ) ) |
14 |
13
|
rexbidva |
|- ( ph -> ( E. v e. A z = ( v - B ) <-> E. v e. A z = ( v + -u B ) ) ) |
15 |
14
|
abbidv |
|- ( ph -> { z | E. v e. A z = ( v - B ) } = { z | E. v e. A z = ( v + -u B ) } ) |
16 |
|
eqidd |
|- ( ph -> { z | E. v e. A z = ( v + -u B ) } = { z | E. v e. A z = ( v + -u B ) } ) |
17 |
6 15 16
|
3eqtrd |
|- ( ph -> C = { z | E. v e. A z = ( v + -u B ) } ) |
18 |
17
|
supeq1d |
|- ( ph -> sup ( C , RR , < ) = sup ( { z | E. v e. A z = ( v + -u B ) } , RR , < ) ) |
19 |
4
|
renegcld |
|- ( ph -> -u B e. RR ) |
20 |
|
eqid |
|- { z | E. v e. A z = ( v + -u B ) } = { z | E. v e. A z = ( v + -u B ) } |
21 |
1 2 3 19 20
|
supaddc |
|- ( ph -> ( sup ( A , RR , < ) + -u B ) = sup ( { z | E. v e. A z = ( v + -u B ) } , RR , < ) ) |
22 |
21
|
eqcomd |
|- ( ph -> sup ( { z | E. v e. A z = ( v + -u B ) } , RR , < ) = ( sup ( A , RR , < ) + -u B ) ) |
23 |
|
suprcl |
|- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> sup ( A , RR , < ) e. RR ) |
24 |
1 2 3 23
|
syl3anc |
|- ( ph -> sup ( A , RR , < ) e. RR ) |
25 |
24
|
recnd |
|- ( ph -> sup ( A , RR , < ) e. CC ) |
26 |
25 9
|
negsubd |
|- ( ph -> ( sup ( A , RR , < ) + -u B ) = ( sup ( A , RR , < ) - B ) ) |
27 |
18 22 26
|
3eqtrrd |
|- ( ph -> ( sup ( A , RR , < ) - B ) = sup ( C , RR , < ) ) |