Step |
Hyp |
Ref |
Expression |
1 |
|
smuval.a |
|- ( ph -> A C_ NN0 ) |
2 |
|
smuval.b |
|- ( ph -> B C_ NN0 ) |
3 |
|
smuval.p |
|- P = seq 0 ( ( p e. ~P NN0 , m e. NN0 |-> ( p sadd { n e. NN0 | ( m e. A /\ ( n - m ) e. B ) } ) ) , ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ) |
4 |
|
0z |
|- 0 e. ZZ |
5 |
3
|
fveq1i |
|- ( P ` 0 ) = ( seq 0 ( ( p e. ~P NN0 , m e. NN0 |-> ( p sadd { n e. NN0 | ( m e. A /\ ( n - m ) e. B ) } ) ) , ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ) ` 0 ) |
6 |
|
seq1 |
|- ( 0 e. ZZ -> ( seq 0 ( ( p e. ~P NN0 , m e. NN0 |-> ( p sadd { n e. NN0 | ( m e. A /\ ( n - m ) e. B ) } ) ) , ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ) ` 0 ) = ( ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ` 0 ) ) |
7 |
5 6
|
syl5eq |
|- ( 0 e. ZZ -> ( P ` 0 ) = ( ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ` 0 ) ) |
8 |
4 7
|
mp1i |
|- ( ph -> ( P ` 0 ) = ( ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ` 0 ) ) |
9 |
|
0nn0 |
|- 0 e. NN0 |
10 |
|
iftrue |
|- ( n = 0 -> if ( n = 0 , (/) , ( n - 1 ) ) = (/) ) |
11 |
|
eqid |
|- ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) = ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) |
12 |
|
0ex |
|- (/) e. _V |
13 |
10 11 12
|
fvmpt |
|- ( 0 e. NN0 -> ( ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ` 0 ) = (/) ) |
14 |
9 13
|
mp1i |
|- ( ph -> ( ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ` 0 ) = (/) ) |
15 |
8 14
|
eqtrd |
|- ( ph -> ( P ` 0 ) = (/) ) |