Step |
Hyp |
Ref |
Expression |
1 |
|
ntrivcvgmul.1 |
|- Z = ( ZZ>= ` M ) |
2 |
|
ntrivcvgmul.3 |
|- ( ph -> E. n e. Z E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) ) |
3 |
|
ntrivcvgmul.4 |
|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC ) |
4 |
|
ntrivcvgmul.5 |
|- ( ph -> E. m e. Z E. z ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) |
5 |
|
ntrivcvgmul.6 |
|- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC ) |
6 |
|
ntrivcvgmul.7 |
|- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( ( F ` k ) x. ( G ` k ) ) ) |
7 |
|
exdistrv |
|- ( E. y E. z ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) <-> ( E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ E. z ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) |
8 |
7
|
2rexbii |
|- ( E. n e. Z E. m e. Z E. y E. z ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) <-> E. n e. Z E. m e. Z ( E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ E. z ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) |
9 |
|
reeanv |
|- ( E. n e. Z E. m e. Z ( E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ E. z ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) <-> ( E. n e. Z E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ E. m e. Z E. z ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) |
10 |
8 9
|
bitri |
|- ( E. n e. Z E. m e. Z E. y E. z ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) <-> ( E. n e. Z E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ E. m e. Z E. z ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) |
11 |
2 4 10
|
sylanbrc |
|- ( ph -> E. n e. Z E. m e. Z E. y E. z ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) |
12 |
|
uzssz |
|- ( ZZ>= ` M ) C_ ZZ |
13 |
1 12
|
eqsstri |
|- Z C_ ZZ |
14 |
|
simp2l |
|- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> n e. Z ) |
15 |
13 14
|
sselid |
|- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> n e. ZZ ) |
16 |
15
|
zred |
|- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> n e. RR ) |
17 |
|
simp2r |
|- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> m e. Z ) |
18 |
13 17
|
sselid |
|- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> m e. ZZ ) |
19 |
18
|
zred |
|- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> m e. RR ) |
20 |
|
simpl2l |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> n e. Z ) |
21 |
|
simpl2r |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> m e. Z ) |
22 |
|
simp3ll |
|- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> y =/= 0 ) |
23 |
22
|
adantr |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> y =/= 0 ) |
24 |
|
simp3rl |
|- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> z =/= 0 ) |
25 |
24
|
adantr |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> z =/= 0 ) |
26 |
|
simp3lr |
|- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> seq n ( x. , F ) ~~> y ) |
27 |
26
|
adantr |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> seq n ( x. , F ) ~~> y ) |
28 |
|
simp3rr |
|- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> seq m ( x. , G ) ~~> z ) |
29 |
28
|
adantr |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> seq m ( x. , G ) ~~> z ) |
30 |
|
simpl1 |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> ph ) |
31 |
30 3
|
sylan |
|- ( ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) /\ k e. Z ) -> ( F ` k ) e. CC ) |
32 |
30 5
|
sylan |
|- ( ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) /\ k e. Z ) -> ( G ` k ) e. CC ) |
33 |
|
simpr |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> n <_ m ) |
34 |
30 6
|
sylan |
|- ( ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) /\ k e. Z ) -> ( H ` k ) = ( ( F ` k ) x. ( G ` k ) ) ) |
35 |
1 20 21 23 25 27 29 31 32 33 34
|
ntrivcvgmullem |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , H ) ~~> w ) ) |
36 |
|
simpl2r |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> m e. Z ) |
37 |
|
simpl2l |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> n e. Z ) |
38 |
24
|
adantr |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> z =/= 0 ) |
39 |
22
|
adantr |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> y =/= 0 ) |
40 |
28
|
adantr |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> seq m ( x. , G ) ~~> z ) |
41 |
26
|
adantr |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> seq n ( x. , F ) ~~> y ) |
42 |
|
simpl1 |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> ph ) |
43 |
42 5
|
sylan |
|- ( ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) /\ k e. Z ) -> ( G ` k ) e. CC ) |
44 |
42 3
|
sylan |
|- ( ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) /\ k e. Z ) -> ( F ` k ) e. CC ) |
45 |
|
simpr |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> m <_ n ) |
46 |
3 5
|
mulcomd |
|- ( ( ph /\ k e. Z ) -> ( ( F ` k ) x. ( G ` k ) ) = ( ( G ` k ) x. ( F ` k ) ) ) |
47 |
6 46
|
eqtrd |
|- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( ( G ` k ) x. ( F ` k ) ) ) |
48 |
42 47
|
sylan |
|- ( ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) /\ k e. Z ) -> ( H ` k ) = ( ( G ` k ) x. ( F ` k ) ) ) |
49 |
1 36 37 38 39 40 41 43 44 45 48
|
ntrivcvgmullem |
|- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , H ) ~~> w ) ) |
50 |
16 19 35 49
|
lecasei |
|- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , H ) ~~> w ) ) |
51 |
50
|
3expia |
|- ( ( ph /\ ( n e. Z /\ m e. Z ) ) -> ( ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , H ) ~~> w ) ) ) |
52 |
51
|
exlimdvv |
|- ( ( ph /\ ( n e. Z /\ m e. Z ) ) -> ( E. y E. z ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , H ) ~~> w ) ) ) |
53 |
52
|
rexlimdvva |
|- ( ph -> ( E. n e. Z E. m e. Z E. y E. z ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , H ) ~~> w ) ) ) |
54 |
11 53
|
mpd |
|- ( ph -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , H ) ~~> w ) ) |