Step |
Hyp |
Ref |
Expression |
1 |
|
ntrivcvg.1 |
|- Z = ( ZZ>= ` M ) |
2 |
|
ntrivcvg.2 |
|- ( ph -> E. n e. Z E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) ) |
3 |
|
ntrivcvg.3 |
|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC ) |
4 |
|
uzm1 |
|- ( n e. ( ZZ>= ` M ) -> ( n = M \/ ( n - 1 ) e. ( ZZ>= ` M ) ) ) |
5 |
4 1
|
eleq2s |
|- ( n e. Z -> ( n = M \/ ( n - 1 ) e. ( ZZ>= ` M ) ) ) |
6 |
5
|
ad2antlr |
|- ( ( ( ph /\ n e. Z ) /\ seq n ( x. , F ) ~~> y ) -> ( n = M \/ ( n - 1 ) e. ( ZZ>= ` M ) ) ) |
7 |
|
seqeq1 |
|- ( n = M -> seq n ( x. , F ) = seq M ( x. , F ) ) |
8 |
7
|
breq1d |
|- ( n = M -> ( seq n ( x. , F ) ~~> y <-> seq M ( x. , F ) ~~> y ) ) |
9 |
|
seqex |
|- seq M ( x. , F ) e. _V |
10 |
|
vex |
|- y e. _V |
11 |
9 10
|
breldm |
|- ( seq M ( x. , F ) ~~> y -> seq M ( x. , F ) e. dom ~~> ) |
12 |
8 11
|
syl6bi |
|- ( n = M -> ( seq n ( x. , F ) ~~> y -> seq M ( x. , F ) e. dom ~~> ) ) |
13 |
12
|
adantld |
|- ( n = M -> ( ( ( ph /\ n e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) ) |
14 |
|
simplr |
|- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> ( n - 1 ) e. Z ) |
15 |
3
|
ad5ant15 |
|- ( ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) /\ k e. Z ) -> ( F ` k ) e. CC ) |
16 |
|
uzssz |
|- ( ZZ>= ` M ) C_ ZZ |
17 |
1 16
|
eqsstri |
|- Z C_ ZZ |
18 |
|
simplr |
|- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. Z ) |
19 |
17 18
|
sselid |
|- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. ZZ ) |
20 |
19
|
zcnd |
|- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> n e. CC ) |
21 |
|
1cnd |
|- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> 1 e. CC ) |
22 |
20 21
|
npcand |
|- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> ( ( n - 1 ) + 1 ) = n ) |
23 |
22
|
seqeq1d |
|- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> seq ( ( n - 1 ) + 1 ) ( x. , F ) = seq n ( x. , F ) ) |
24 |
23
|
breq1d |
|- ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) -> ( seq ( ( n - 1 ) + 1 ) ( x. , F ) ~~> y <-> seq n ( x. , F ) ~~> y ) ) |
25 |
24
|
biimpar |
|- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq ( ( n - 1 ) + 1 ) ( x. , F ) ~~> y ) |
26 |
1 14 15 25
|
clim2prod |
|- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) ~~> ( ( seq M ( x. , F ) ` ( n - 1 ) ) x. y ) ) |
27 |
|
ovex |
|- ( ( seq M ( x. , F ) ` ( n - 1 ) ) x. y ) e. _V |
28 |
9 27
|
breldm |
|- ( seq M ( x. , F ) ~~> ( ( seq M ( x. , F ) ` ( n - 1 ) ) x. y ) -> seq M ( x. , F ) e. dom ~~> ) |
29 |
26 28
|
syl |
|- ( ( ( ( ph /\ n e. Z ) /\ ( n - 1 ) e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) |
30 |
29
|
an32s |
|- ( ( ( ( ph /\ n e. Z ) /\ seq n ( x. , F ) ~~> y ) /\ ( n - 1 ) e. Z ) -> seq M ( x. , F ) e. dom ~~> ) |
31 |
30
|
expcom |
|- ( ( n - 1 ) e. Z -> ( ( ( ph /\ n e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) ) |
32 |
1
|
eqcomi |
|- ( ZZ>= ` M ) = Z |
33 |
31 32
|
eleq2s |
|- ( ( n - 1 ) e. ( ZZ>= ` M ) -> ( ( ( ph /\ n e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) ) |
34 |
13 33
|
jaoi |
|- ( ( n = M \/ ( n - 1 ) e. ( ZZ>= ` M ) ) -> ( ( ( ph /\ n e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) ) |
35 |
6 34
|
mpcom |
|- ( ( ( ph /\ n e. Z ) /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) |
36 |
35
|
ex |
|- ( ( ph /\ n e. Z ) -> ( seq n ( x. , F ) ~~> y -> seq M ( x. , F ) e. dom ~~> ) ) |
37 |
36
|
adantld |
|- ( ( ph /\ n e. Z ) -> ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) ) |
38 |
37
|
exlimdv |
|- ( ( ph /\ n e. Z ) -> ( E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) ) |
39 |
38
|
rexlimdva |
|- ( ph -> ( E. n e. Z E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) -> seq M ( x. , F ) e. dom ~~> ) ) |
40 |
2 39
|
mpd |
|- ( ph -> seq M ( x. , F ) e. dom ~~> ) |