Step |
Hyp |
Ref |
Expression |
1 |
|
ssuzfz.1 |
|- Z = ( ZZ>= ` M ) |
2 |
|
ssuzfz.2 |
|- ( ph -> A C_ Z ) |
3 |
|
ssuzfz.3 |
|- ( ph -> A e. Fin ) |
4 |
2
|
sselda |
|- ( ( ph /\ k e. A ) -> k e. Z ) |
5 |
4 1
|
eleqtrdi |
|- ( ( ph /\ k e. A ) -> k e. ( ZZ>= ` M ) ) |
6 |
|
eluzel2 |
|- ( k e. ( ZZ>= ` M ) -> M e. ZZ ) |
7 |
5 6
|
syl |
|- ( ( ph /\ k e. A ) -> M e. ZZ ) |
8 |
|
uzssz |
|- ( ZZ>= ` M ) C_ ZZ |
9 |
1 8
|
eqsstri |
|- Z C_ ZZ |
10 |
9
|
a1i |
|- ( ph -> Z C_ ZZ ) |
11 |
2 10
|
sstrd |
|- ( ph -> A C_ ZZ ) |
12 |
11
|
adantr |
|- ( ( ph /\ k e. A ) -> A C_ ZZ ) |
13 |
|
ne0i |
|- ( k e. A -> A =/= (/) ) |
14 |
13
|
adantl |
|- ( ( ph /\ k e. A ) -> A =/= (/) ) |
15 |
3
|
adantr |
|- ( ( ph /\ k e. A ) -> A e. Fin ) |
16 |
|
suprfinzcl |
|- ( ( A C_ ZZ /\ A =/= (/) /\ A e. Fin ) -> sup ( A , RR , < ) e. A ) |
17 |
12 14 15 16
|
syl3anc |
|- ( ( ph /\ k e. A ) -> sup ( A , RR , < ) e. A ) |
18 |
12 17
|
sseldd |
|- ( ( ph /\ k e. A ) -> sup ( A , RR , < ) e. ZZ ) |
19 |
11
|
sselda |
|- ( ( ph /\ k e. A ) -> k e. ZZ ) |
20 |
7 18 19
|
3jca |
|- ( ( ph /\ k e. A ) -> ( M e. ZZ /\ sup ( A , RR , < ) e. ZZ /\ k e. ZZ ) ) |
21 |
|
eluzle |
|- ( k e. ( ZZ>= ` M ) -> M <_ k ) |
22 |
5 21
|
syl |
|- ( ( ph /\ k e. A ) -> M <_ k ) |
23 |
|
zssre |
|- ZZ C_ RR |
24 |
23
|
a1i |
|- ( ph -> ZZ C_ RR ) |
25 |
11 24
|
sstrd |
|- ( ph -> A C_ RR ) |
26 |
25
|
adantr |
|- ( ( ph /\ k e. A ) -> A C_ RR ) |
27 |
|
simpr |
|- ( ( ph /\ k e. A ) -> k e. A ) |
28 |
|
eqidd |
|- ( ( ph /\ k e. A ) -> sup ( A , RR , < ) = sup ( A , RR , < ) ) |
29 |
26 15 27 28
|
supfirege |
|- ( ( ph /\ k e. A ) -> k <_ sup ( A , RR , < ) ) |
30 |
20 22 29
|
jca32 |
|- ( ( ph /\ k e. A ) -> ( ( M e. ZZ /\ sup ( A , RR , < ) e. ZZ /\ k e. ZZ ) /\ ( M <_ k /\ k <_ sup ( A , RR , < ) ) ) ) |
31 |
|
elfz2 |
|- ( k e. ( M ... sup ( A , RR , < ) ) <-> ( ( M e. ZZ /\ sup ( A , RR , < ) e. ZZ /\ k e. ZZ ) /\ ( M <_ k /\ k <_ sup ( A , RR , < ) ) ) ) |
32 |
30 31
|
sylibr |
|- ( ( ph /\ k e. A ) -> k e. ( M ... sup ( A , RR , < ) ) ) |
33 |
32
|
ex |
|- ( ph -> ( k e. A -> k e. ( M ... sup ( A , RR , < ) ) ) ) |
34 |
33
|
ralrimiv |
|- ( ph -> A. k e. A k e. ( M ... sup ( A , RR , < ) ) ) |
35 |
|
dfss3 |
|- ( A C_ ( M ... sup ( A , RR , < ) ) <-> A. k e. A k e. ( M ... sup ( A , RR , < ) ) ) |
36 |
34 35
|
sylibr |
|- ( ph -> A C_ ( M ... sup ( A , RR , < ) ) ) |