Theorem seqeq3 10223

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

Assertion

Ref	Expression
seqeq3	|- ( F = G -> seq M ( .+ , F ) = seq M ( .+ , G ) )

Proof of Theorem seqeq3

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

Step	Hyp	Ref	Expression
1		simp1 981	. . . . . . . 8 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. _V ) -> F = G )
2	1	fveq1d 5423	. . . . . . 7 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. _V ) -> ( F ` ( x + 1 ) ) = ( G ` ( x + 1 ) ) )
3	2	oveq2d 5790	. . . . . 6 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. _V ) -> ( y .+ ( F ` ( x + 1 ) ) ) = ( y .+ ( G ` ( x + 1 ) ) ) )
4	3	opeq2d 3712	. . . . 5 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. _V ) -> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. = <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. )
5	4	mpoeq3dva 5835	. . . 4 |- ( F = G -> ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) )
6		fveq1 5420	. . . . 5 |- ( F = G -> ( F ` M ) = ( G ` M ) )
7	6	opeq2d 3712	. . . 4 |- ( F = G -> <. M , ( F ` M ) >. = <. M , ( G ` M ) >. )
8		freceq1 6289	. . . . 5 |- ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( G ` ( 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 .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) )
9		freceq2 6290	. . . . 5 |- ( <. M , ( F ` M ) >. = <. M , ( G ` M ) >. -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( G ` M ) >. ) )
10	8, 9	sylan9eq 2192	. . . 4 |- ( ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) /\ <. M , ( F ` M ) >. = <. M , ( G ` M ) >. ) -> 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 .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( G ` M ) >. ) )
11	5, 7, 10	syl2anc 408	. . 3 |- ( F = G -> 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 .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( G ` M ) >. ) )
12	11	rneqd 4768	. 2 |- ( F = G -> 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 .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( G ` M ) >. ) )
13		df-seqfrec 10219	. 2 |- seq M ( .+ , F ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
14		df-seqfrec 10219	. 2 |- seq M ( .+ , G ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( G ` M ) >. )
15	12, 13, 14	3eqtr4g 2197	1 |- ( F = G -> seq M ( .+ , F ) = seq M ( .+ , G ) )

Syntax hints:   -> wi 4   /\ w3a 962   = wceq 1331   e. wcel 1480   _Vcvv 2686   <.cop 3530   ran crn 4540   ` cfv 5123  (class class class)co 5774   e. cmpo 5776  freccfrec 6287   1c1 7621   + caddc 7623   ZZ>=cuz 9326   seqcseq 10218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-recs 6202  df-frec 6288  df-seqfrec 10219

This theorem is referenced by:  seqeq3d  10226  cbvsum  11129  fsumadd  11175  cbvprod  11327