Step |
Hyp |
Ref |
Expression |
1 |
|
iundjiunlem.z |
|- Z = ( ZZ>= ` N ) |
2 |
|
iundjiunlem.f |
|- F = ( n e. Z |-> ( ( E ` n ) \ U_ i e. ( N ..^ n ) ( E ` i ) ) ) |
3 |
|
iundjiunlem.j |
|- ( ph -> J e. Z ) |
4 |
|
iundjiunlem.k |
|- ( ph -> K e. Z ) |
5 |
|
iundjiunlem.lt |
|- ( ph -> J < K ) |
6 |
|
incom |
|- ( ( F ` J ) i^i ( F ` K ) ) = ( ( F ` K ) i^i ( F ` J ) ) |
7 |
|
simpl |
|- ( ( ph /\ x e. ( F ` K ) ) -> ph ) |
8 |
|
simpr |
|- ( ( ph /\ x e. ( F ` K ) ) -> x e. ( F ` K ) ) |
9 |
|
fveq2 |
|- ( n = K -> ( E ` n ) = ( E ` K ) ) |
10 |
|
oveq2 |
|- ( n = K -> ( N ..^ n ) = ( N ..^ K ) ) |
11 |
10
|
iuneq1d |
|- ( n = K -> U_ i e. ( N ..^ n ) ( E ` i ) = U_ i e. ( N ..^ K ) ( E ` i ) ) |
12 |
9 11
|
difeq12d |
|- ( n = K -> ( ( E ` n ) \ U_ i e. ( N ..^ n ) ( E ` i ) ) = ( ( E ` K ) \ U_ i e. ( N ..^ K ) ( E ` i ) ) ) |
13 |
|
fvex |
|- ( E ` K ) e. _V |
14 |
13
|
difexi |
|- ( ( E ` K ) \ U_ i e. ( N ..^ K ) ( E ` i ) ) e. _V |
15 |
12 2 14
|
fvmpt |
|- ( K e. Z -> ( F ` K ) = ( ( E ` K ) \ U_ i e. ( N ..^ K ) ( E ` i ) ) ) |
16 |
4 15
|
syl |
|- ( ph -> ( F ` K ) = ( ( E ` K ) \ U_ i e. ( N ..^ K ) ( E ` i ) ) ) |
17 |
16
|
adantr |
|- ( ( ph /\ x e. ( F ` K ) ) -> ( F ` K ) = ( ( E ` K ) \ U_ i e. ( N ..^ K ) ( E ` i ) ) ) |
18 |
8 17
|
eleqtrd |
|- ( ( ph /\ x e. ( F ` K ) ) -> x e. ( ( E ` K ) \ U_ i e. ( N ..^ K ) ( E ` i ) ) ) |
19 |
18
|
eldifbd |
|- ( ( ph /\ x e. ( F ` K ) ) -> -. x e. U_ i e. ( N ..^ K ) ( E ` i ) ) |
20 |
3 1
|
eleqtrdi |
|- ( ph -> J e. ( ZZ>= ` N ) ) |
21 |
1 4
|
eluzelz2d |
|- ( ph -> K e. ZZ ) |
22 |
20 21 5
|
elfzod |
|- ( ph -> J e. ( N ..^ K ) ) |
23 |
|
fveq2 |
|- ( i = J -> ( E ` i ) = ( E ` J ) ) |
24 |
23
|
ssiun2s |
|- ( J e. ( N ..^ K ) -> ( E ` J ) C_ U_ i e. ( N ..^ K ) ( E ` i ) ) |
25 |
22 24
|
syl |
|- ( ph -> ( E ` J ) C_ U_ i e. ( N ..^ K ) ( E ` i ) ) |
26 |
25
|
ssneld |
|- ( ph -> ( -. x e. U_ i e. ( N ..^ K ) ( E ` i ) -> -. x e. ( E ` J ) ) ) |
27 |
7 19 26
|
sylc |
|- ( ( ph /\ x e. ( F ` K ) ) -> -. x e. ( E ` J ) ) |
28 |
|
eldifi |
|- ( x e. ( ( E ` J ) \ U_ i e. ( N ..^ J ) ( E ` i ) ) -> x e. ( E ` J ) ) |
29 |
27 28
|
nsyl |
|- ( ( ph /\ x e. ( F ` K ) ) -> -. x e. ( ( E ` J ) \ U_ i e. ( N ..^ J ) ( E ` i ) ) ) |
30 |
|
fveq2 |
|- ( n = J -> ( E ` n ) = ( E ` J ) ) |
31 |
|
oveq2 |
|- ( n = J -> ( N ..^ n ) = ( N ..^ J ) ) |
32 |
31
|
iuneq1d |
|- ( n = J -> U_ i e. ( N ..^ n ) ( E ` i ) = U_ i e. ( N ..^ J ) ( E ` i ) ) |
33 |
30 32
|
difeq12d |
|- ( n = J -> ( ( E ` n ) \ U_ i e. ( N ..^ n ) ( E ` i ) ) = ( ( E ` J ) \ U_ i e. ( N ..^ J ) ( E ` i ) ) ) |
34 |
|
fvex |
|- ( E ` J ) e. _V |
35 |
34
|
difexi |
|- ( ( E ` J ) \ U_ i e. ( N ..^ J ) ( E ` i ) ) e. _V |
36 |
33 2 35
|
fvmpt |
|- ( J e. Z -> ( F ` J ) = ( ( E ` J ) \ U_ i e. ( N ..^ J ) ( E ` i ) ) ) |
37 |
3 36
|
syl |
|- ( ph -> ( F ` J ) = ( ( E ` J ) \ U_ i e. ( N ..^ J ) ( E ` i ) ) ) |
38 |
37
|
adantr |
|- ( ( ph /\ x e. ( F ` K ) ) -> ( F ` J ) = ( ( E ` J ) \ U_ i e. ( N ..^ J ) ( E ` i ) ) ) |
39 |
29 38
|
neleqtrrd |
|- ( ( ph /\ x e. ( F ` K ) ) -> -. x e. ( F ` J ) ) |
40 |
39
|
ralrimiva |
|- ( ph -> A. x e. ( F ` K ) -. x e. ( F ` J ) ) |
41 |
|
disj |
|- ( ( ( F ` K ) i^i ( F ` J ) ) = (/) <-> A. x e. ( F ` K ) -. x e. ( F ` J ) ) |
42 |
40 41
|
sylibr |
|- ( ph -> ( ( F ` K ) i^i ( F ` J ) ) = (/) ) |
43 |
6 42
|
syl5eq |
|- ( ph -> ( ( F ` J ) i^i ( F ` K ) ) = (/) ) |