Theorem iseqsst 9947

Description: Specifying a larger universe for seq. As long as F and .+ are closed over S, then any class which contains S can be used as the last argument to seq.

Together with df-seq3 9915 it can be used to convert between the df-iseq 9914 syntax and the df-seq3 9915 syntax (in many cases iseqseq3 9963 is an even more convenient way to do this).

(Contributed by Jim Kingdon, 28-Apr-2022.)

Hypotheses

Ref	Expression
iseqsst.m	|- ( ph -> M e. ZZ )
iseqsst.ss	|- ( ph -> S C_ T )
iseqsst.f	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
iseqsst.pl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )

Assertion

Ref	Expression
iseqsst	|- ( ph -> seq M ( .+ , F , S ) = seq M ( .+ , F , T ) )

Distinct variable groups:   x, .+ , y   x, F, y   x, M, y   x, S, y   x, T, y   ph, x, y

Proof of Theorem iseqsst

Dummy variables k w n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2089	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		iseqsst.m	. . . 4 |- ( ph -> M e. ZZ )
3		iseqsst.f	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
4		iseqsst.pl	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
5	1, 2, 3, 4	iseqfcl 9939	. . 3 |- ( ph -> seq M ( .+ , F , S ) : ( ZZ>= ` M ) --> S )
6		ffn 5174	. . 3 |- ( seq M ( .+ , F , S ) : ( ZZ>= ` M ) --> S -> seq M ( .+ , F , S ) Fn ( ZZ>= ` M ) )
7	5, 6	syl 14	. 2 |- ( ph -> seq M ( .+ , F , S ) Fn ( ZZ>= ` M ) )
8		iseqsst.ss	. . . 4 |- ( ph -> S C_ T )
9	1, 2, 3, 4, 8	iseqfclt 9940	. . 3 |- ( ph -> seq M ( .+ , F , T ) : ( ZZ>= ` M ) --> S )
10		ffn 5174	. . 3 |- ( seq M ( .+ , F , T ) : ( ZZ>= ` M ) --> S -> seq M ( .+ , F , T ) Fn ( ZZ>= ` M ) )
11	9, 10	syl 14	. 2 |- ( ph -> seq M ( .+ , F , T ) Fn ( ZZ>= ` M ) )
12		fveq2 5318	. . . . . 6 |- ( w = M -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` M ) )
13		fveq2 5318	. . . . . 6 |- ( w = M -> ( seq M ( .+ , F , T ) ` w ) = ( seq M ( .+ , F , T ) ` M ) )
14	12, 13	eqeq12d 2103	. . . . 5 |- ( w = M -> ( ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , T ) ` w ) <-> ( seq M ( .+ , F , S ) ` M ) = ( seq M ( .+ , F , T ) ` M ) ) )
15	14	imbi2d 229	. . . 4 |- ( w = M -> ( ( ph -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , T ) ` w ) ) <-> ( ph -> ( seq M ( .+ , F , S ) ` M ) = ( seq M ( .+ , F , T ) ` M ) ) ) )
16		fveq2 5318	. . . . . 6 |- ( w = k -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` k ) )
17		fveq2 5318	. . . . . 6 |- ( w = k -> ( seq M ( .+ , F , T ) ` w ) = ( seq M ( .+ , F , T ) ` k ) )
18	16, 17	eqeq12d 2103	. . . . 5 |- ( w = k -> ( ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , T ) ` w ) <-> ( seq M ( .+ , F , S ) ` k ) = ( seq M ( .+ , F , T ) ` k ) ) )
19	18	imbi2d 229	. . . 4 |- ( w = k -> ( ( ph -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , T ) ` w ) ) <-> ( ph -> ( seq M ( .+ , F , S ) ` k ) = ( seq M ( .+ , F , T ) ` k ) ) ) )
20		fveq2 5318	. . . . . 6 |- ( w = ( k + 1 ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` ( k + 1 ) ) )
21		fveq2 5318	. . . . . 6 |- ( w = ( k + 1 ) -> ( seq M ( .+ , F , T ) ` w ) = ( seq M ( .+ , F , T ) ` ( k + 1 ) ) )
22	20, 21	eqeq12d 2103	. . . . 5 |- ( w = ( k + 1 ) -> ( ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , T ) ` w ) <-> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( seq M ( .+ , F , T ) ` ( k + 1 ) ) ) )
23	22	imbi2d 229	. . . 4 |- ( w = ( k + 1 ) -> ( ( ph -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , T ) ` w ) ) <-> ( ph -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( seq M ( .+ , F , T ) ` ( k + 1 ) ) ) ) )
24		fveq2 5318	. . . . . 6 |- ( w = n -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` n ) )
25		fveq2 5318	. . . . . 6 |- ( w = n -> ( seq M ( .+ , F , T ) ` w ) = ( seq M ( .+ , F , T ) ` n ) )
26	24, 25	eqeq12d 2103	. . . . 5 |- ( w = n -> ( ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , T ) ` w ) <-> ( seq M ( .+ , F , S ) ` n ) = ( seq M ( .+ , F , T ) ` n ) ) )
27	26	imbi2d 229	. . . 4 |- ( w = n -> ( ( ph -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , T ) ` w ) ) <-> ( ph -> ( seq M ( .+ , F , S ) ` n ) = ( seq M ( .+ , F , T ) ` n ) ) ) )
28	2, 3, 4	iseq1 9936	. . . . . 6 |- ( ph -> ( seq M ( .+ , F , S ) ` M ) = ( F ` M ) )
29	2, 3, 4, 8	iseq1t 9937	. . . . . 6 |- ( ph -> ( seq M ( .+ , F , T ) ` M ) = ( F ` M ) )
30	28, 29	eqtr4d 2124	. . . . 5 |- ( ph -> ( seq M ( .+ , F , S ) ` M ) = ( seq M ( .+ , F , T ) ` M ) )
31	30	a1i 9	. . . 4 |- ( M e. ZZ -> ( ph -> ( seq M ( .+ , F , S ) ` M ) = ( seq M ( .+ , F , T ) ` M ) ) )
32		oveq1 5673	. . . . . . 7 |- ( ( seq M ( .+ , F , S ) ` k ) = ( seq M ( .+ , F , T ) ` k ) -> ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) = ( ( seq M ( .+ , F , T ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
33		simpr 109	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
34	3	adantlr 462	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
35	4	adantlr 462	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
36	33, 34, 35	iseqp1 9943	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
37	8	adantr 271	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> S C_ T )
38	33, 34, 35, 37	iseqp1t 9944	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , T ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F , T ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
39	36, 38	eqeq12d 2103	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( seq M ( .+ , F , T ) ` ( k + 1 ) ) <-> ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) = ( ( seq M ( .+ , F , T ) ` k ) .+ ( F ` ( k + 1 ) ) ) ) )
40	32, 39	syl5ibr 155	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( seq M ( .+ , F , S ) ` k ) = ( seq M ( .+ , F , T ) ` k ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( seq M ( .+ , F , T ) ` ( k + 1 ) ) ) )
41	40	expcom 115	. . . . 5 |- ( k e. ( ZZ>= ` M ) -> ( ph -> ( ( seq M ( .+ , F , S ) ` k ) = ( seq M ( .+ , F , T ) ` k ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( seq M ( .+ , F , T ) ` ( k + 1 ) ) ) ) )
42	41	a2d 26	. . . 4 |- ( k e. ( ZZ>= ` M ) -> ( ( ph -> ( seq M ( .+ , F , S ) ` k ) = ( seq M ( .+ , F , T ) ` k ) ) -> ( ph -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( seq M ( .+ , F , T ) ` ( k + 1 ) ) ) ) )
43	15, 19, 23, 27, 31, 42	uzind4 9137	. . 3 |- ( n e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( .+ , F , S ) ` n ) = ( seq M ( .+ , F , T ) ` n ) ) )
44	43	impcom 124	. 2 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , S ) ` n ) = ( seq M ( .+ , F , T ) ` n ) )
45	7, 11, 44	eqfnfvd 5414	1 |- ( ph -> seq M ( .+ , F , S ) = seq M ( .+ , F , T ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439   C_ wss 3000   Fn wfn 5023   -->wf 5024   ` cfv 5028  (class class class)co 5666   1c1 7412   + caddc 7414   ZZcz 8811   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-in1 580  ax-in2 581  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-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-addcom 7506  ax-addass 7508  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-0id 7514  ax-rnegex 7515  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-ltadd 7522

This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-frec 6170  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-inn 8484  df-n0 8735  df-z 8812  df-uz 9081  df-iseq 9914

This theorem is referenced by:  seq3feq  9958  seq3shft2  9960  iseqseq3  9963