Metamath Proof Explorer


Theorem itcovalsuc

Description: The value of the function that returns the n-th iterate of a function with regard to composition at a successor. (Contributed by AV, 4-May-2024)

Ref Expression
Assertion itcovalsuc Could not format assertion : No typesetting found for |- ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( ( IterComp ` F ) ` ( Y + 1 ) ) = ( G ( g e. _V , j e. _V |-> ( F o. g ) ) F ) ) with typecode |-

Proof

Step Hyp Ref Expression
1 simp1 Could not format ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> F e. V ) : No typesetting found for |- ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> F e. V ) with typecode |-
2 itcoval Could not format ( F e. V -> ( IterComp ` F ) = seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ) : No typesetting found for |- ( F e. V -> ( IterComp ` F ) = seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ) with typecode |-
3 2 fveq1d Could not format ( F e. V -> ( ( IterComp ` F ) ` ( Y + 1 ) ) = ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` ( Y + 1 ) ) ) : No typesetting found for |- ( F e. V -> ( ( IterComp ` F ) ` ( Y + 1 ) ) = ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` ( Y + 1 ) ) ) with typecode |-
4 1 3 syl Could not format ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( ( IterComp ` F ) ` ( Y + 1 ) ) = ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` ( Y + 1 ) ) ) : No typesetting found for |- ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( ( IterComp ` F ) ` ( Y + 1 ) ) = ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` ( Y + 1 ) ) ) with typecode |-
5 nn0uz 0 = 0
6 simp2 Could not format ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> Y e. NN0 ) : No typesetting found for |- ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> Y e. NN0 ) with typecode |-
7 eqid Y + 1 = Y + 1
8 2 adantr Could not format ( ( F e. V /\ Y e. NN0 ) -> ( IterComp ` F ) = seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ) : No typesetting found for |- ( ( F e. V /\ Y e. NN0 ) -> ( IterComp ` F ) = seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ) with typecode |-
9 8 fveq1d Could not format ( ( F e. V /\ Y e. NN0 ) -> ( ( IterComp ` F ) ` Y ) = ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` Y ) ) : No typesetting found for |- ( ( F e. V /\ Y e. NN0 ) -> ( ( IterComp ` F ) ` Y ) = ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` Y ) ) with typecode |-
10 9 eqeq1d Could not format ( ( F e. V /\ Y e. NN0 ) -> ( ( ( IterComp ` F ) ` Y ) = G <-> ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` Y ) = G ) ) : No typesetting found for |- ( ( F e. V /\ Y e. NN0 ) -> ( ( ( IterComp ` F ) ` Y ) = G <-> ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` Y ) = G ) ) with typecode |-
11 10 biimp3a Could not format ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` Y ) = G ) : No typesetting found for |- ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` Y ) = G ) with typecode |-
12 eqidd Could not format ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) = ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) : No typesetting found for |- ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) = ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) with typecode |-
13 nn0p1gt0 Y 0 0 < Y + 1
14 13 3ad2ant2 Could not format ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> 0 < ( Y + 1 ) ) : No typesetting found for |- ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> 0 < ( Y + 1 ) ) with typecode |-
15 14 gt0ne0d Could not format ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( Y + 1 ) =/= 0 ) : No typesetting found for |- ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( Y + 1 ) =/= 0 ) with typecode |-
16 15 adantr Could not format ( ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) /\ i = ( Y + 1 ) ) -> ( Y + 1 ) =/= 0 ) : No typesetting found for |- ( ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) /\ i = ( Y + 1 ) ) -> ( Y + 1 ) =/= 0 ) with typecode |-
17 neeq1 i = Y + 1 i 0 Y + 1 0
18 17 adantl Could not format ( ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) /\ i = ( Y + 1 ) ) -> ( i =/= 0 <-> ( Y + 1 ) =/= 0 ) ) : No typesetting found for |- ( ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) /\ i = ( Y + 1 ) ) -> ( i =/= 0 <-> ( Y + 1 ) =/= 0 ) ) with typecode |-
19 16 18 mpbird Could not format ( ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) /\ i = ( Y + 1 ) ) -> i =/= 0 ) : No typesetting found for |- ( ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) /\ i = ( Y + 1 ) ) -> i =/= 0 ) with typecode |-
20 19 neneqd Could not format ( ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) /\ i = ( Y + 1 ) ) -> -. i = 0 ) : No typesetting found for |- ( ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) /\ i = ( Y + 1 ) ) -> -. i = 0 ) with typecode |-
21 20 iffalsed Could not format ( ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) /\ i = ( Y + 1 ) ) -> if ( i = 0 , ( _I |` dom F ) , F ) = F ) : No typesetting found for |- ( ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) /\ i = ( Y + 1 ) ) -> if ( i = 0 , ( _I |` dom F ) , F ) = F ) with typecode |-
22 peano2nn0 Y 0 Y + 1 0
23 22 3ad2ant2 Could not format ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( Y + 1 ) e. NN0 ) : No typesetting found for |- ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( Y + 1 ) e. NN0 ) with typecode |-
24 12 21 23 1 fvmptd Could not format ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ` ( Y + 1 ) ) = F ) : No typesetting found for |- ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ` ( Y + 1 ) ) = F ) with typecode |-
25 5 6 7 11 24 seqp1d Could not format ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` ( Y + 1 ) ) = ( G ( g e. _V , j e. _V |-> ( F o. g ) ) F ) ) : No typesetting found for |- ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` ( Y + 1 ) ) = ( G ( g e. _V , j e. _V |-> ( F o. g ) ) F ) ) with typecode |-
26 4 25 eqtrd Could not format ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( ( IterComp ` F ) ` ( Y + 1 ) ) = ( G ( g e. _V , j e. _V |-> ( F o. g ) ) F ) ) : No typesetting found for |- ( ( F e. V /\ Y e. NN0 /\ ( ( IterComp ` F ) ` Y ) = G ) -> ( ( IterComp ` F ) ` ( Y + 1 ) ) = ( G ( g e. _V , j e. _V |-> ( F o. g ) ) F ) ) with typecode |-