Theorem seqeq2 10560

Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Assertion

Ref	Expression
seqeq2	|- ( .+ = Q -> seq M ( .+ , F ) = seq M ( Q , F ) )

Proof of Theorem seqeq2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 999	. . . . . . 7 |- ( ( .+ = Q /\ x e. ( ZZ>= ` M ) /\ y e. _V ) -> .+ = Q )
2	1	oveqd 5942	. . . . . 6 |- ( ( .+ = Q /\ x e. ( ZZ>= ` M ) /\ y e. _V ) -> ( y .+ ( F ` ( x + 1 ) ) ) = ( y Q ( F ` ( x + 1 ) ) ) )
3	2	opeq2d 3816	. . . . 5 |- ( ( .+ = Q /\ x e. ( ZZ>= ` M ) /\ y e. _V ) -> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. = <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. )
4	3	mpoeq3dva 5990	. . . 4 |- ( .+ = Q -> ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. ) )
5		freceq1 6459	. . . 4 |- ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. ) -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) )
6	4, 5	syl 14	. . 3 |- ( .+ = Q -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) )
7	6	rneqd 4896	. 2 |- ( .+ = Q -> ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) )
8		df-seqfrec 10557	. 2 |- seq M ( .+ , F ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
9		df-seqfrec 10557	. 2 |- seq M ( Q , F ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
10	7, 8, 9	3eqtr4g 2254	1 |- ( .+ = Q -> seq M ( .+ , F ) = seq M ( Q , F ) )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2167   _Vcvv 2763   <.cop 3626   ran crn 4665   ` cfv 5259  (class class class)co 5925   e. cmpo 5927  freccfrec 6457   1c1 7897   + caddc 7899   ZZ>=cuz 9618   seqcseq 10556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-cnv 4672  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-recs 6372  df-frec 6458  df-seqfrec 10557

This theorem is referenced by:  seqeq2d  10563  resqrex  11208  nninfdc  12695