| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ftc1.g |
|- G = ( x e. ( A [,] B ) |-> S. ( A (,) x ) ( F ` t ) _d t ) |
| 2 |
|
ftc1.a |
|- ( ph -> A e. RR ) |
| 3 |
|
ftc1.b |
|- ( ph -> B e. RR ) |
| 4 |
|
ftc1.le |
|- ( ph -> A <_ B ) |
| 5 |
|
ftc1.s |
|- ( ph -> ( A (,) B ) C_ D ) |
| 6 |
|
ftc1.d |
|- ( ph -> D C_ RR ) |
| 7 |
|
ftc1.i |
|- ( ph -> F e. L^1 ) |
| 8 |
|
ftc1.c |
|- ( ph -> C e. ( A (,) B ) ) |
| 9 |
|
ftc1.f |
|- ( ph -> F e. ( ( K CnP L ) ` C ) ) |
| 10 |
|
ftc1.j |
|- J = ( L |`t RR ) |
| 11 |
|
ftc1.k |
|- K = ( L |`t D ) |
| 12 |
|
ftc1.l |
|- L = ( TopOpen ` CCfld ) |
| 13 |
12
|
cnfldtopon |
|- L e. ( TopOn ` CC ) |
| 14 |
|
ax-resscn |
|- RR C_ CC |
| 15 |
6 14
|
sstrdi |
|- ( ph -> D C_ CC ) |
| 16 |
|
resttopon |
|- ( ( L e. ( TopOn ` CC ) /\ D C_ CC ) -> ( L |`t D ) e. ( TopOn ` D ) ) |
| 17 |
13 15 16
|
sylancr |
|- ( ph -> ( L |`t D ) e. ( TopOn ` D ) ) |
| 18 |
11 17
|
eqeltrid |
|- ( ph -> K e. ( TopOn ` D ) ) |
| 19 |
13
|
a1i |
|- ( ph -> L e. ( TopOn ` CC ) ) |
| 20 |
|
cnpf2 |
|- ( ( K e. ( TopOn ` D ) /\ L e. ( TopOn ` CC ) /\ F e. ( ( K CnP L ) ` C ) ) -> F : D --> CC ) |
| 21 |
18 19 9 20
|
syl3anc |
|- ( ph -> F : D --> CC ) |