Theorem seqeq3 10713

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 1023	. . . . . . . 8 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. _V ) -> F = G )
2	1	fveq1d 5641	. . . . . . 7 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. _V ) -> ( F ` ( x + 1 ) ) = ( G ` ( x + 1 ) ) )
3	2	oveq2d 6033	. . . . . 6 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. _V ) -> ( y .+ ( F ` ( x + 1 ) ) ) = ( y .+ ( G ` ( x + 1 ) ) ) )
4	3	opeq2d 3869	. . . . 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 6084	. . . 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 5638	. . . . 5 |- ( F = G -> ( F ` M ) = ( G ` M ) )
7	6	opeq2d 3869	. . . 4 |- ( F = G -> <. M , ( F ` M ) >. = <. M , ( G ` M ) >. )
8		freceq1 6557	. . . . 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 6558	. . . . 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 2284	. . . 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 4961	. 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 10709	. 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 10709	. 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 2289	1 |- ( F = G -> seq M ( .+ , F ) = seq M ( .+ , G ) )

Syntax hints:   -> wi 4   /\ w3a 1004   = wceq 1397   e. wcel 2202   _Vcvv 2802   <.cop 3672   ran crn 4726   ` cfv 5326  (class class class)co 6017   e. cmpo 6019  freccfrec 6555   1c1 8032   + caddc 8034   ZZ>=cuz 9754   seqcseq 10708

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-recs 6470  df-frec 6556  df-seqfrec 10709

This theorem is referenced by:  seqeq3d  10716  cbvsum  11920  fsumadd  11966  cbvprod  12118