Theorem seqeq2 10673

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 1021	. . . . . . 7 |- ( ( .+ = Q /\ x e. ( ZZ>= ` M ) /\ y e. _V ) -> .+ = Q )
2	1	oveqd 6018	. . . . . 6 |- ( ( .+ = Q /\ x e. ( ZZ>= ` M ) /\ y e. _V ) -> ( y .+ ( F ` ( x + 1 ) ) ) = ( y Q ( F ` ( x + 1 ) ) ) )
3	2	opeq2d 3864	. . . . 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 6068	. . . 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 6538	. . . 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 4953	. 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 10670	. 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 10670	. 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 2287	1 |- ( .+ = Q -> seq M ( .+ , F ) = seq M ( Q , F ) )

Syntax hints:   -> wi 4   /\ w3a 1002   = wceq 1395   e. wcel 2200   _Vcvv 2799   <.cop 3669   ran crn 4720   ` cfv 5318  (class class class)co 6001   e. cmpo 6003  freccfrec 6536   1c1 8000   + caddc 8002   ZZ>=cuz 9722   seqcseq 10669

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-recs 6451  df-frec 6537  df-seqfrec 10670

This theorem is referenced by:  seqeq2d  10676  resqrex  11537  nninfdc  13024