| Step |
Hyp |
Ref |
Expression |
| 1 |
|
prodmo.1 |
|- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) ) |
| 2 |
|
prodmo.2 |
|- ( ( ph /\ k e. A ) -> B e. CC ) |
| 3 |
|
prodrb.4 |
|- ( ph -> M e. ZZ ) |
| 4 |
|
prodrb.5 |
|- ( ph -> N e. ZZ ) |
| 5 |
|
prodrb.6 |
|- ( ph -> A C_ ( ZZ>= ` M ) ) |
| 6 |
|
prodrb.7 |
|- ( ph -> A C_ ( ZZ>= ` N ) ) |
| 7 |
4
|
adantr |
|- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> N e. ZZ ) |
| 8 |
|
seqex |
|- seq M ( x. , F ) e. _V |
| 9 |
|
climres |
|- ( ( N e. ZZ /\ seq M ( x. , F ) e. _V ) -> ( ( seq M ( x. , F ) |` ( ZZ>= ` N ) ) ~~> C <-> seq M ( x. , F ) ~~> C ) ) |
| 10 |
7 8 9
|
sylancl |
|- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( ( seq M ( x. , F ) |` ( ZZ>= ` N ) ) ~~> C <-> seq M ( x. , F ) ~~> C ) ) |
| 11 |
2
|
adantlr |
|- ( ( ( ph /\ N e. ( ZZ>= ` M ) ) /\ k e. A ) -> B e. CC ) |
| 12 |
|
simpr |
|- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> N e. ( ZZ>= ` M ) ) |
| 13 |
1 11 12
|
prodrblem |
|- ( ( ( ph /\ N e. ( ZZ>= ` M ) ) /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( x. , F ) |` ( ZZ>= ` N ) ) = seq N ( x. , F ) ) |
| 14 |
6 13
|
mpidan |
|- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( seq M ( x. , F ) |` ( ZZ>= ` N ) ) = seq N ( x. , F ) ) |
| 15 |
14
|
breq1d |
|- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( ( seq M ( x. , F ) |` ( ZZ>= ` N ) ) ~~> C <-> seq N ( x. , F ) ~~> C ) ) |
| 16 |
10 15
|
bitr3d |
|- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( seq M ( x. , F ) ~~> C <-> seq N ( x. , F ) ~~> C ) ) |