Theorem iseqeq2 9920

Description: Equality theorem for the sequence builder operation.

New proofs should use seqeq2 9923 instead (together with iseqsst 9947 or iseqseq3 9963 if need be).

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

Assertion

Ref	Expression
iseqeq2	|- ( .+ = Q -> seq M ( .+ , F , S ) = seq M ( Q , F , S ) )

Proof of Theorem iseqeq2

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

Step	Hyp	Ref	Expression
1		simp1 944	. . . . . . 7 |- ( ( .+ = Q /\ x e. ( ZZ>= ` M ) /\ y e. S ) -> .+ = Q )
2	1	oveqd 5683	. . . . . 6 |- ( ( .+ = Q /\ x e. ( ZZ>= ` M ) /\ y e. S ) -> ( y .+ ( F ` ( x + 1 ) ) ) = ( y Q ( F ` ( x + 1 ) ) ) )
3	2	opeq2d 3635	. . . . 5 |- ( ( .+ = Q /\ x e. ( ZZ>= ` M ) /\ y e. S ) -> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. = <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. )
4	3	mpt2eq3dva 5727	. . . 4 |- ( .+ = Q -> ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. ) )
5		freceq1 6171	. . . 4 |- ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. ) -> 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 Q ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) )
6	4, 5	syl 14	. . 3 |- ( .+ = Q -> 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 Q ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) )
7	6	rneqd 4677	. 2 |- ( .+ = Q -> ran frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) )
8		df-iseq 9914	. 2 |- seq M ( .+ , F , S ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
9		df-iseq 9914	. 2 |- seq M ( Q , F , S ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
10	7, 8, 9	3eqtr4g 2146	1 |- ( .+ = Q -> seq M ( .+ , F , S ) = seq M ( Q , F , S ) )

Syntax hints:   -> wi 4   /\ w3a 925   = wceq 1290   e. wcel 1439   <.cop 3453   ran crn 4453   ` cfv 5028  (class class class)co 5666   |-> cmpt2 5668  freccfrec 6169   1c1 7412   + caddc 7414   ZZ>=cuz 9080   seqcseq4 9912

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-cnv 4460  df-dm 4462  df-rn 4463  df-res 4464  df-iota 4993  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-recs 6084  df-frec 6170  df-iseq 9914

This theorem is referenced by:  seqeq2  9923