| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dvivth.1 |
|- ( ph -> M e. ( A (,) B ) ) |
| 2 |
|
dvivth.2 |
|- ( ph -> N e. ( A (,) B ) ) |
| 3 |
|
dvivth.3 |
|- ( ph -> F e. ( ( A (,) B ) -cn-> RR ) ) |
| 4 |
|
dvivth.4 |
|- ( ph -> dom ( RR _D F ) = ( A (,) B ) ) |
| 5 |
|
dvivth.5 |
|- ( ph -> M < N ) |
| 6 |
|
dvivth.6 |
|- ( ph -> C e. ( ( ( RR _D F ) ` N ) [,] ( ( RR _D F ) ` M ) ) ) |
| 7 |
|
dvivth.7 |
|- G = ( y e. ( A (,) B ) |-> ( ( F ` y ) - ( C x. y ) ) ) |
| 8 |
1 2 3 4 5 6 7
|
dvivthlem1 |
|- ( ph -> E. x e. ( M [,] N ) ( ( RR _D F ) ` x ) = C ) |
| 9 |
|
dvf |
|- ( RR _D F ) : dom ( RR _D F ) --> CC |
| 10 |
4
|
feq2d |
|- ( ph -> ( ( RR _D F ) : dom ( RR _D F ) --> CC <-> ( RR _D F ) : ( A (,) B ) --> CC ) ) |
| 11 |
9 10
|
mpbii |
|- ( ph -> ( RR _D F ) : ( A (,) B ) --> CC ) |
| 12 |
11
|
ffnd |
|- ( ph -> ( RR _D F ) Fn ( A (,) B ) ) |
| 13 |
|
iccssioo2 |
|- ( ( M e. ( A (,) B ) /\ N e. ( A (,) B ) ) -> ( M [,] N ) C_ ( A (,) B ) ) |
| 14 |
1 2 13
|
syl2anc |
|- ( ph -> ( M [,] N ) C_ ( A (,) B ) ) |
| 15 |
14
|
sselda |
|- ( ( ph /\ x e. ( M [,] N ) ) -> x e. ( A (,) B ) ) |
| 16 |
|
fnfvelrn |
|- ( ( ( RR _D F ) Fn ( A (,) B ) /\ x e. ( A (,) B ) ) -> ( ( RR _D F ) ` x ) e. ran ( RR _D F ) ) |
| 17 |
12 15 16
|
syl2an2r |
|- ( ( ph /\ x e. ( M [,] N ) ) -> ( ( RR _D F ) ` x ) e. ran ( RR _D F ) ) |
| 18 |
|
eleq1 |
|- ( ( ( RR _D F ) ` x ) = C -> ( ( ( RR _D F ) ` x ) e. ran ( RR _D F ) <-> C e. ran ( RR _D F ) ) ) |
| 19 |
17 18
|
syl5ibcom |
|- ( ( ph /\ x e. ( M [,] N ) ) -> ( ( ( RR _D F ) ` x ) = C -> C e. ran ( RR _D F ) ) ) |
| 20 |
19
|
rexlimdva |
|- ( ph -> ( E. x e. ( M [,] N ) ( ( RR _D F ) ` x ) = C -> C e. ran ( RR _D F ) ) ) |
| 21 |
8 20
|
mpd |
|- ( ph -> C e. ran ( RR _D F ) ) |