Step |
Hyp |
Ref |
Expression |
1 |
|
suppssfz.z |
|- ( ph -> Z e. V ) |
2 |
|
suppssfz.f |
|- ( ph -> F e. ( B ^m NN0 ) ) |
3 |
|
suppssfz.s |
|- ( ph -> S e. NN0 ) |
4 |
|
suppssfz.b |
|- ( ph -> A. x e. NN0 ( S < x -> ( F ` x ) = Z ) ) |
5 |
|
elmapfn |
|- ( F e. ( B ^m NN0 ) -> F Fn NN0 ) |
6 |
2 5
|
syl |
|- ( ph -> F Fn NN0 ) |
7 |
|
nn0ex |
|- NN0 e. _V |
8 |
7
|
a1i |
|- ( ph -> NN0 e. _V ) |
9 |
6 8 1
|
3jca |
|- ( ph -> ( F Fn NN0 /\ NN0 e. _V /\ Z e. V ) ) |
10 |
9
|
adantr |
|- ( ( ph /\ A. x e. NN0 ( S < x -> ( F ` x ) = Z ) ) -> ( F Fn NN0 /\ NN0 e. _V /\ Z e. V ) ) |
11 |
|
elsuppfn |
|- ( ( F Fn NN0 /\ NN0 e. _V /\ Z e. V ) -> ( n e. ( F supp Z ) <-> ( n e. NN0 /\ ( F ` n ) =/= Z ) ) ) |
12 |
10 11
|
syl |
|- ( ( ph /\ A. x e. NN0 ( S < x -> ( F ` x ) = Z ) ) -> ( n e. ( F supp Z ) <-> ( n e. NN0 /\ ( F ` n ) =/= Z ) ) ) |
13 |
|
breq2 |
|- ( x = n -> ( S < x <-> S < n ) ) |
14 |
|
fveqeq2 |
|- ( x = n -> ( ( F ` x ) = Z <-> ( F ` n ) = Z ) ) |
15 |
13 14
|
imbi12d |
|- ( x = n -> ( ( S < x -> ( F ` x ) = Z ) <-> ( S < n -> ( F ` n ) = Z ) ) ) |
16 |
15
|
rspcva |
|- ( ( n e. NN0 /\ A. x e. NN0 ( S < x -> ( F ` x ) = Z ) ) -> ( S < n -> ( F ` n ) = Z ) ) |
17 |
|
simplr |
|- ( ( ( ph /\ n e. NN0 ) /\ -. S < n ) -> n e. NN0 ) |
18 |
3
|
adantr |
|- ( ( ph /\ n e. NN0 ) -> S e. NN0 ) |
19 |
18
|
adantr |
|- ( ( ( ph /\ n e. NN0 ) /\ -. S < n ) -> S e. NN0 ) |
20 |
|
nn0re |
|- ( n e. NN0 -> n e. RR ) |
21 |
|
nn0re |
|- ( S e. NN0 -> S e. RR ) |
22 |
3 21
|
syl |
|- ( ph -> S e. RR ) |
23 |
|
lenlt |
|- ( ( n e. RR /\ S e. RR ) -> ( n <_ S <-> -. S < n ) ) |
24 |
20 22 23
|
syl2anr |
|- ( ( ph /\ n e. NN0 ) -> ( n <_ S <-> -. S < n ) ) |
25 |
24
|
biimpar |
|- ( ( ( ph /\ n e. NN0 ) /\ -. S < n ) -> n <_ S ) |
26 |
|
elfz2nn0 |
|- ( n e. ( 0 ... S ) <-> ( n e. NN0 /\ S e. NN0 /\ n <_ S ) ) |
27 |
17 19 25 26
|
syl3anbrc |
|- ( ( ( ph /\ n e. NN0 ) /\ -. S < n ) -> n e. ( 0 ... S ) ) |
28 |
27
|
a1d |
|- ( ( ( ph /\ n e. NN0 ) /\ -. S < n ) -> ( ( F ` n ) =/= Z -> n e. ( 0 ... S ) ) ) |
29 |
28
|
ex |
|- ( ( ph /\ n e. NN0 ) -> ( -. S < n -> ( ( F ` n ) =/= Z -> n e. ( 0 ... S ) ) ) ) |
30 |
|
eqneqall |
|- ( ( F ` n ) = Z -> ( ( F ` n ) =/= Z -> n e. ( 0 ... S ) ) ) |
31 |
30
|
a1i |
|- ( ( ph /\ n e. NN0 ) -> ( ( F ` n ) = Z -> ( ( F ` n ) =/= Z -> n e. ( 0 ... S ) ) ) ) |
32 |
29 31
|
jad |
|- ( ( ph /\ n e. NN0 ) -> ( ( S < n -> ( F ` n ) = Z ) -> ( ( F ` n ) =/= Z -> n e. ( 0 ... S ) ) ) ) |
33 |
32
|
com23 |
|- ( ( ph /\ n e. NN0 ) -> ( ( F ` n ) =/= Z -> ( ( S < n -> ( F ` n ) = Z ) -> n e. ( 0 ... S ) ) ) ) |
34 |
33
|
ex |
|- ( ph -> ( n e. NN0 -> ( ( F ` n ) =/= Z -> ( ( S < n -> ( F ` n ) = Z ) -> n e. ( 0 ... S ) ) ) ) ) |
35 |
34
|
com14 |
|- ( ( S < n -> ( F ` n ) = Z ) -> ( n e. NN0 -> ( ( F ` n ) =/= Z -> ( ph -> n e. ( 0 ... S ) ) ) ) ) |
36 |
16 35
|
syl |
|- ( ( n e. NN0 /\ A. x e. NN0 ( S < x -> ( F ` x ) = Z ) ) -> ( n e. NN0 -> ( ( F ` n ) =/= Z -> ( ph -> n e. ( 0 ... S ) ) ) ) ) |
37 |
36
|
ex |
|- ( n e. NN0 -> ( A. x e. NN0 ( S < x -> ( F ` x ) = Z ) -> ( n e. NN0 -> ( ( F ` n ) =/= Z -> ( ph -> n e. ( 0 ... S ) ) ) ) ) ) |
38 |
37
|
pm2.43a |
|- ( n e. NN0 -> ( A. x e. NN0 ( S < x -> ( F ` x ) = Z ) -> ( ( F ` n ) =/= Z -> ( ph -> n e. ( 0 ... S ) ) ) ) ) |
39 |
38
|
com23 |
|- ( n e. NN0 -> ( ( F ` n ) =/= Z -> ( A. x e. NN0 ( S < x -> ( F ` x ) = Z ) -> ( ph -> n e. ( 0 ... S ) ) ) ) ) |
40 |
39
|
imp |
|- ( ( n e. NN0 /\ ( F ` n ) =/= Z ) -> ( A. x e. NN0 ( S < x -> ( F ` x ) = Z ) -> ( ph -> n e. ( 0 ... S ) ) ) ) |
41 |
40
|
com13 |
|- ( ph -> ( A. x e. NN0 ( S < x -> ( F ` x ) = Z ) -> ( ( n e. NN0 /\ ( F ` n ) =/= Z ) -> n e. ( 0 ... S ) ) ) ) |
42 |
41
|
imp |
|- ( ( ph /\ A. x e. NN0 ( S < x -> ( F ` x ) = Z ) ) -> ( ( n e. NN0 /\ ( F ` n ) =/= Z ) -> n e. ( 0 ... S ) ) ) |
43 |
12 42
|
sylbid |
|- ( ( ph /\ A. x e. NN0 ( S < x -> ( F ` x ) = Z ) ) -> ( n e. ( F supp Z ) -> n e. ( 0 ... S ) ) ) |
44 |
43
|
ssrdv |
|- ( ( ph /\ A. x e. NN0 ( S < x -> ( F ` x ) = Z ) ) -> ( F supp Z ) C_ ( 0 ... S ) ) |
45 |
4 44
|
mpdan |
|- ( ph -> ( F supp Z ) C_ ( 0 ... S ) ) |