Step |
Hyp |
Ref |
Expression |
1 |
|
df-ack |
Could not format Ack = seq 0 ( ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) , ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) ) : No typesetting found for |- Ack = seq 0 ( ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) , ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) ) with typecode |- |
2 |
1
|
fveq1i |
Could not format ( Ack ` ( M + 1 ) ) = ( seq 0 ( ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) , ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) ) ` ( M + 1 ) ) : No typesetting found for |- ( Ack ` ( M + 1 ) ) = ( seq 0 ( ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) , ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) ) ` ( M + 1 ) ) with typecode |- |
3 |
|
nn0uz |
|
4 |
|
id |
|
5 |
|
eqid |
|
6 |
1
|
eqcomi |
Could not format seq 0 ( ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) , ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) ) = Ack : No typesetting found for |- seq 0 ( ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) , ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) ) = Ack with typecode |- |
7 |
6
|
fveq1i |
Could not format ( seq 0 ( ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) , ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) ) ` M ) = ( Ack ` M ) : No typesetting found for |- ( seq 0 ( ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) , ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) ) ` M ) = ( Ack ` M ) with typecode |- |
8 |
7
|
a1i |
Could not format ( M e. NN0 -> ( seq 0 ( ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) , ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) ) ` M ) = ( Ack ` M ) ) : No typesetting found for |- ( M e. NN0 -> ( seq 0 ( ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) , ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) ) ` M ) = ( Ack ` M ) ) with typecode |- |
9 |
|
eqidd |
|
10 |
|
nn0p1gt0 |
|
11 |
10
|
gt0ne0d |
|
12 |
11
|
adantr |
|
13 |
|
neeq1 |
|
14 |
13
|
adantl |
|
15 |
12 14
|
mpbird |
|
16 |
15
|
neneqd |
|
17 |
16
|
iffalsed |
|
18 |
|
simpr |
|
19 |
17 18
|
eqtrd |
|
20 |
|
peano2nn0 |
|
21 |
9 19 20 20
|
fvmptd |
|
22 |
3 4 5 8 21
|
seqp1d |
Could not format ( M e. NN0 -> ( seq 0 ( ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) , ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) ) ` ( M + 1 ) ) = ( ( Ack ` M ) ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) ( M + 1 ) ) ) : No typesetting found for |- ( M e. NN0 -> ( seq 0 ( ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) , ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) ) ` ( M + 1 ) ) = ( ( Ack ` M ) ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) ( M + 1 ) ) ) with typecode |- |
23 |
|
eqidd |
Could not format ( M e. NN0 -> ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) = ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) ) : No typesetting found for |- ( M e. NN0 -> ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) = ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) ) with typecode |- |
24 |
|
fveq2 |
Could not format ( f = ( Ack ` M ) -> ( IterComp ` f ) = ( IterComp ` ( Ack ` M ) ) ) : No typesetting found for |- ( f = ( Ack ` M ) -> ( IterComp ` f ) = ( IterComp ` ( Ack ` M ) ) ) with typecode |- |
25 |
24
|
fveq1d |
Could not format ( f = ( Ack ` M ) -> ( ( IterComp ` f ) ` ( n + 1 ) ) = ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ) : No typesetting found for |- ( f = ( Ack ` M ) -> ( ( IterComp ` f ) ` ( n + 1 ) ) = ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ) with typecode |- |
26 |
25
|
fveq1d |
Could not format ( f = ( Ack ` M ) -> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) = ( ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ` 1 ) ) : No typesetting found for |- ( f = ( Ack ` M ) -> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) = ( ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ` 1 ) ) with typecode |- |
27 |
26
|
mpteq2dv |
Could not format ( f = ( Ack ` M ) -> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) = ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ` 1 ) ) ) : No typesetting found for |- ( f = ( Ack ` M ) -> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) = ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ` 1 ) ) ) with typecode |- |
28 |
27
|
ad2antrl |
Could not format ( ( M e. NN0 /\ ( f = ( Ack ` M ) /\ j = ( M + 1 ) ) ) -> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) = ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ` 1 ) ) ) : No typesetting found for |- ( ( M e. NN0 /\ ( f = ( Ack ` M ) /\ j = ( M + 1 ) ) ) -> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) = ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ` 1 ) ) ) with typecode |- |
29 |
|
fvexd |
Could not format ( M e. NN0 -> ( Ack ` M ) e. _V ) : No typesetting found for |- ( M e. NN0 -> ( Ack ` M ) e. _V ) with typecode |- |
30 |
|
ovexd |
|
31 |
|
nn0ex |
|
32 |
31
|
mptex |
Could not format ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ` 1 ) ) e. _V : No typesetting found for |- ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ` 1 ) ) e. _V with typecode |- |
33 |
32
|
a1i |
Could not format ( M e. NN0 -> ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ` 1 ) ) e. _V ) : No typesetting found for |- ( M e. NN0 -> ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ` 1 ) ) e. _V ) with typecode |- |
34 |
23 28 29 30 33
|
ovmpod |
Could not format ( M e. NN0 -> ( ( Ack ` M ) ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) ( M + 1 ) ) = ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ` 1 ) ) ) : No typesetting found for |- ( M e. NN0 -> ( ( Ack ` M ) ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) ( M + 1 ) ) = ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ` 1 ) ) ) with typecode |- |
35 |
22 34
|
eqtrd |
Could not format ( M e. NN0 -> ( seq 0 ( ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) , ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) ) ` ( M + 1 ) ) = ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ` 1 ) ) ) : No typesetting found for |- ( M e. NN0 -> ( seq 0 ( ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) , ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) ) ` ( M + 1 ) ) = ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ` 1 ) ) ) with typecode |- |
36 |
2 35
|
syl5eq |
Could not format ( M e. NN0 -> ( Ack ` ( M + 1 ) ) = ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ` 1 ) ) ) : No typesetting found for |- ( M e. NN0 -> ( Ack ` ( M + 1 ) ) = ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` M ) ) ` ( n + 1 ) ) ` 1 ) ) ) with typecode |- |