Step |
Hyp |
Ref |
Expression |
1 |
|
uzfissfz.m |
|- ( ph -> M e. ZZ ) |
2 |
|
uzfissfz.z |
|- Z = ( ZZ>= ` M ) |
3 |
|
uzfissfz.a |
|- ( ph -> A C_ Z ) |
4 |
|
uzfissfz.fi |
|- ( ph -> A e. Fin ) |
5 |
|
uzid |
|- ( M e. ZZ -> M e. ( ZZ>= ` M ) ) |
6 |
1 5
|
syl |
|- ( ph -> M e. ( ZZ>= ` M ) ) |
7 |
2
|
a1i |
|- ( ph -> Z = ( ZZ>= ` M ) ) |
8 |
7
|
eqcomd |
|- ( ph -> ( ZZ>= ` M ) = Z ) |
9 |
6 8
|
eleqtrd |
|- ( ph -> M e. Z ) |
10 |
9
|
adantr |
|- ( ( ph /\ A = (/) ) -> M e. Z ) |
11 |
|
id |
|- ( A = (/) -> A = (/) ) |
12 |
|
0ss |
|- (/) C_ ( M ... M ) |
13 |
12
|
a1i |
|- ( A = (/) -> (/) C_ ( M ... M ) ) |
14 |
11 13
|
eqsstrd |
|- ( A = (/) -> A C_ ( M ... M ) ) |
15 |
14
|
adantl |
|- ( ( ph /\ A = (/) ) -> A C_ ( M ... M ) ) |
16 |
|
oveq2 |
|- ( k = M -> ( M ... k ) = ( M ... M ) ) |
17 |
16
|
sseq2d |
|- ( k = M -> ( A C_ ( M ... k ) <-> A C_ ( M ... M ) ) ) |
18 |
17
|
rspcev |
|- ( ( M e. Z /\ A C_ ( M ... M ) ) -> E. k e. Z A C_ ( M ... k ) ) |
19 |
10 15 18
|
syl2anc |
|- ( ( ph /\ A = (/) ) -> E. k e. Z A C_ ( M ... k ) ) |
20 |
3
|
adantr |
|- ( ( ph /\ -. A = (/) ) -> A C_ Z ) |
21 |
|
uzssz |
|- ( ZZ>= ` M ) C_ ZZ |
22 |
2 21
|
eqsstri |
|- Z C_ ZZ |
23 |
22
|
a1i |
|- ( ph -> Z C_ ZZ ) |
24 |
3 23
|
sstrd |
|- ( ph -> A C_ ZZ ) |
25 |
24
|
adantr |
|- ( ( ph /\ -. A = (/) ) -> A C_ ZZ ) |
26 |
11
|
necon3bi |
|- ( -. A = (/) -> A =/= (/) ) |
27 |
26
|
adantl |
|- ( ( ph /\ -. A = (/) ) -> A =/= (/) ) |
28 |
4
|
adantr |
|- ( ( ph /\ -. A = (/) ) -> A e. Fin ) |
29 |
|
suprfinzcl |
|- ( ( A C_ ZZ /\ A =/= (/) /\ A e. Fin ) -> sup ( A , RR , < ) e. A ) |
30 |
25 27 28 29
|
syl3anc |
|- ( ( ph /\ -. A = (/) ) -> sup ( A , RR , < ) e. A ) |
31 |
20 30
|
sseldd |
|- ( ( ph /\ -. A = (/) ) -> sup ( A , RR , < ) e. Z ) |
32 |
1
|
ad2antrr |
|- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> M e. ZZ ) |
33 |
22 31
|
sseldi |
|- ( ( ph /\ -. A = (/) ) -> sup ( A , RR , < ) e. ZZ ) |
34 |
33
|
adantr |
|- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> sup ( A , RR , < ) e. ZZ ) |
35 |
25
|
sselda |
|- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> j e. ZZ ) |
36 |
3
|
sselda |
|- ( ( ph /\ j e. A ) -> j e. Z ) |
37 |
2
|
a1i |
|- ( ( ph /\ j e. A ) -> Z = ( ZZ>= ` M ) ) |
38 |
36 37
|
eleqtrd |
|- ( ( ph /\ j e. A ) -> j e. ( ZZ>= ` M ) ) |
39 |
|
eluzle |
|- ( j e. ( ZZ>= ` M ) -> M <_ j ) |
40 |
38 39
|
syl |
|- ( ( ph /\ j e. A ) -> M <_ j ) |
41 |
40
|
adantlr |
|- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> M <_ j ) |
42 |
|
zssre |
|- ZZ C_ RR |
43 |
24 42
|
sstrdi |
|- ( ph -> A C_ RR ) |
44 |
43
|
ad2antrr |
|- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> A C_ RR ) |
45 |
27
|
adantr |
|- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> A =/= (/) ) |
46 |
|
fimaxre2 |
|- ( ( A C_ RR /\ A e. Fin ) -> E. x e. RR A. y e. A y <_ x ) |
47 |
43 4 46
|
syl2anc |
|- ( ph -> E. x e. RR A. y e. A y <_ x ) |
48 |
47
|
ad2antrr |
|- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> E. x e. RR A. y e. A y <_ x ) |
49 |
|
simpr |
|- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> j e. A ) |
50 |
|
suprub |
|- ( ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) /\ j e. A ) -> j <_ sup ( A , RR , < ) ) |
51 |
44 45 48 49 50
|
syl31anc |
|- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> j <_ sup ( A , RR , < ) ) |
52 |
32 34 35 41 51
|
elfzd |
|- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> j e. ( M ... sup ( A , RR , < ) ) ) |
53 |
52
|
ralrimiva |
|- ( ( ph /\ -. A = (/) ) -> A. j e. A j e. ( M ... sup ( A , RR , < ) ) ) |
54 |
|
dfss3 |
|- ( A C_ ( M ... sup ( A , RR , < ) ) <-> A. j e. A j e. ( M ... sup ( A , RR , < ) ) ) |
55 |
53 54
|
sylibr |
|- ( ( ph /\ -. A = (/) ) -> A C_ ( M ... sup ( A , RR , < ) ) ) |
56 |
|
oveq2 |
|- ( k = sup ( A , RR , < ) -> ( M ... k ) = ( M ... sup ( A , RR , < ) ) ) |
57 |
56
|
sseq2d |
|- ( k = sup ( A , RR , < ) -> ( A C_ ( M ... k ) <-> A C_ ( M ... sup ( A , RR , < ) ) ) ) |
58 |
57
|
rspcev |
|- ( ( sup ( A , RR , < ) e. Z /\ A C_ ( M ... sup ( A , RR , < ) ) ) -> E. k e. Z A C_ ( M ... k ) ) |
59 |
31 55 58
|
syl2anc |
|- ( ( ph /\ -. A = (/) ) -> E. k e. Z A C_ ( M ... k ) ) |
60 |
19 59
|
pm2.61dan |
|- ( ph -> E. k e. Z A C_ ( M ... k ) ) |