Step |
Hyp |
Ref |
Expression |
1 |
|
ulmcl |
|- ( F ( ~~>u ` S ) G -> G : S --> CC ) |
2 |
1
|
adantr |
|- ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) -> G : S --> CC ) |
3 |
2
|
ffnd |
|- ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) -> G Fn S ) |
4 |
|
ulmcl |
|- ( F ( ~~>u ` S ) H -> H : S --> CC ) |
5 |
4
|
adantl |
|- ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) -> H : S --> CC ) |
6 |
5
|
ffnd |
|- ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) -> H Fn S ) |
7 |
|
eqid |
|- ( ZZ>= ` n ) = ( ZZ>= ` n ) |
8 |
|
simplr |
|- ( ( ( ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> n e. ZZ ) |
9 |
|
simpr |
|- ( ( ( ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> F : ( ZZ>= ` n ) --> ( CC ^m S ) ) |
10 |
|
simpllr |
|- ( ( ( ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> x e. S ) |
11 |
|
fvex |
|- ( ZZ>= ` n ) e. _V |
12 |
11
|
mptex |
|- ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) e. _V |
13 |
12
|
a1i |
|- ( ( ( ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) e. _V ) |
14 |
|
fveq2 |
|- ( i = k -> ( F ` i ) = ( F ` k ) ) |
15 |
14
|
fveq1d |
|- ( i = k -> ( ( F ` i ) ` x ) = ( ( F ` k ) ` x ) ) |
16 |
|
eqid |
|- ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) = ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) |
17 |
|
fvex |
|- ( ( F ` k ) ` x ) e. _V |
18 |
15 16 17
|
fvmpt |
|- ( k e. ( ZZ>= ` n ) -> ( ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) ` k ) = ( ( F ` k ) ` x ) ) |
19 |
18
|
eqcomd |
|- ( k e. ( ZZ>= ` n ) -> ( ( F ` k ) ` x ) = ( ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) ` k ) ) |
20 |
19
|
adantl |
|- ( ( ( ( ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) /\ k e. ( ZZ>= ` n ) ) -> ( ( F ` k ) ` x ) = ( ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) ` k ) ) |
21 |
|
simp-4l |
|- ( ( ( ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> F ( ~~>u ` S ) G ) |
22 |
7 8 9 10 13 20 21
|
ulmclm |
|- ( ( ( ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) ~~> ( G ` x ) ) |
23 |
|
simp-4r |
|- ( ( ( ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> F ( ~~>u ` S ) H ) |
24 |
7 8 9 10 13 20 23
|
ulmclm |
|- ( ( ( ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) ~~> ( H ` x ) ) |
25 |
|
climuni |
|- ( ( ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) ~~> ( G ` x ) /\ ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) ~~> ( H ` x ) ) -> ( G ` x ) = ( H ` x ) ) |
26 |
22 24 25
|
syl2anc |
|- ( ( ( ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> ( G ` x ) = ( H ` x ) ) |
27 |
|
ulmf |
|- ( F ( ~~>u ` S ) G -> E. n e. ZZ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) |
28 |
27
|
ad2antrr |
|- ( ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) /\ x e. S ) -> E. n e. ZZ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) |
29 |
26 28
|
r19.29a |
|- ( ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) /\ x e. S ) -> ( G ` x ) = ( H ` x ) ) |
30 |
3 6 29
|
eqfnfvd |
|- ( ( F ( ~~>u ` S ) G /\ F ( ~~>u ` S ) H ) -> G = H ) |