Theorem seqeq3 10634

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 1000	. . . . . . . 8 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. _V ) -> F = G )
2	1	fveq1d 5601	. . . . . . 7 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. _V ) -> ( F ` ( x + 1 ) ) = ( G ` ( x + 1 ) ) )
3	2	oveq2d 5983	. . . . . 6 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. _V ) -> ( y .+ ( F ` ( x + 1 ) ) ) = ( y .+ ( G ` ( x + 1 ) ) ) )
4	3	opeq2d 3840	. . . . 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 6032	. . . 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 5598	. . . . 5 |- ( F = G -> ( F ` M ) = ( G ` M ) )
7	6	opeq2d 3840	. . . 4 |- ( F = G -> <. M , ( F ` M ) >. = <. M , ( G ` M ) >. )
8		freceq1 6501	. . . . 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 6502	. . . . 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 2260	. . . 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 411	. . 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 4926	. 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 10630	. 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 10630	. 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 2265	1 |- ( F = G -> seq M ( .+ , F ) = seq M ( .+ , G ) )

Syntax hints:   -> wi 4   /\ w3a 981   = wceq 1373   e. wcel 2178   _Vcvv 2776   <.cop 3646   ran crn 4694   ` cfv 5290  (class class class)co 5967   e. cmpo 5969  freccfrec 6499   1c1 7961   + caddc 7963   ZZ>=cuz 9683   seqcseq 10629

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-recs 6414  df-frec 6500  df-seqfrec 10630

This theorem is referenced by:  seqeq3d  10637  cbvsum  11786  fsumadd  11832  cbvprod  11984