Theorem iseqeq1 9846

Description: Equality theorem for the sequence builder operation.

New proofs should use seqeq1 9849 instead (together with iseqsst 9874 or iseqseq3 9890 if need be).

(Contributed by Jim Kingdon, 30-May-2020.) (New usage is discouraged.)

Assertion

Ref	Expression
iseqeq1	|- ( M = N -> seq M ( .+ , F , S ) = seq N ( .+ , F , S ) )

Proof of Theorem iseqeq1

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

Step	Hyp	Ref	Expression
1		id 19	. . . . . 6 |- ( M = N -> M = N )
2		fveq2 5299	. . . . . 6 |- ( M = N -> ( F ` M ) = ( F ` N ) )
3	1, 2	opeq12d 3628	. . . . 5 |- ( M = N -> <. M , ( F ` M ) >. = <. N , ( F ` N ) >. )
4		freceq2 6150	. . . . 5 |- ( <. M , ( F ` M ) >. = <. N , ( F ` N ) >. -> frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
5	3, 4	syl 14	. . . 4 |- ( M = N -> frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
6		fveq2 5299	. . . . . 6 |- ( M = N -> ( ZZ>= ` M ) = ( ZZ>= ` N ) )
7		eqid 2088	. . . . . 6 |- S = S
8		mpt2eq12 5701	. . . . . 6 |- ( ( ( ZZ>= ` M ) = ( ZZ>= ` N ) /\ S = S ) -> ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` N ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) )
9	6, 7, 8	sylancl 404	. . . . 5 |- ( M = N -> ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` N ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) )
10		freceq1 6149	. . . . 5 |- ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` N ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) -> frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) = frec ( ( x e. ( ZZ>= ` N ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
11	9, 10	syl 14	. . . 4 |- ( M = N -> frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) = frec ( ( x e. ( ZZ>= ` N ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
12	5, 11	eqtrd 2120	. . 3 |- ( M = N -> frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` N ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
13	12	rneqd 4660	. 2 |- ( M = N -> ran frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = ran frec ( ( x e. ( ZZ>= ` N ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
14		df-iseq 9841	. 2 |- seq M ( .+ , F , S ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
15		df-iseq 9841	. 2 |- seq N ( .+ , F , S ) = ran frec ( ( x e. ( ZZ>= ` N ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. )
16	13, 14, 15	3eqtr4g 2145	1 |- ( M = N -> seq M ( .+ , F , S ) = seq N ( .+ , F , S ) )

Syntax hints:   -> wi 4   = wceq 1289   <.cop 3447   ran crn 4437   ` cfv 5010  (class class class)co 5644   |-> cmpt2 5646  freccfrec 6147   1c1 7341   + caddc 7343   ZZ>=cuz 9009   seqcseq4 9839

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-mpt 3899  df-cnv 4444  df-dm 4446  df-rn 4447  df-res 4448  df-iota 4975  df-fv 5018  df-oprab 5648  df-mpt2 5649  df-recs 6062  df-frec 6148  df-iseq 9841

This theorem is referenced by:  seqeq1  9849  iseqid  9927  iseqz  9931  ibcval5  10159  bcn2  10160  isummolem2  10759  isummo  10760  zisum  10761