Theorem iseqss 9226

Description: Specifying a larger universe for seq. As long as F and .+ are closed over S, then any set which contains S can be used as the last argument to seq. This theorem does not allow T to be a proper class, however. It also currently requires that .+ be closed over T (as well as S). (Contributed by Jim Kingdon, 18-Aug-2021.)

Hypotheses

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

Assertion

Ref	Expression
iseqss	|- ( 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

Allowed substitution hints:   V( x, y)

Proof of Theorem iseqss

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

Step	Hyp	Ref	Expression
1		iseqss.m	. . 3 |- ( ph -> M e. ZZ )
2		iseqss.t	. . . 4 |- ( ph -> T e. V )
3		iseqss.ss	. . . 4 |- ( ph -> S C_ T )
4	2, 3	ssexd 3897	. . 3 |- ( ph -> S e. _V )
5		iseqss.f	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
6		iseqss.pl	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
7	1, 4, 5, 6	iseqfn 9221	. 2 |- ( ph -> seq M ( .+ , F , S ) Fn ( ZZ>= ` M ) )
8	3	sseld 2944	. . . . 5 |- ( ph -> ( ( F ` x ) e. S -> ( F ` x ) e. T ) )
9	8	adantr 261	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( F ` x ) e. S -> ( F ` x ) e. T ) )
10	5, 9	mpd 13	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. T )
11		iseqss.plt	. . 3 |- ( ( ph /\ ( x e. T /\ y e. T ) ) -> ( x .+ y ) e. T )
12	1, 2, 10, 11	iseqfn 9221	. 2 |- ( ph -> seq M ( .+ , F , T ) Fn ( ZZ>= ` M ) )
13		fveq2 5178	. . . . . 6 |- ( w = M -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` M ) )
14		fveq2 5178	. . . . . 6 |- ( w = M -> ( seq M ( .+ , F , T ) ` w ) = ( seq M ( .+ , F , T ) ` M ) )
15	13, 14	eqeq12d 2054	. . . . 5 |- ( w = M -> ( ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , T ) ` w ) <-> ( seq M ( .+ , F , S ) ` M ) = ( seq M ( .+ , F , T ) ` M ) ) )
16	15	imbi2d 219	. . . 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 ) ) ) )
17		fveq2 5178	. . . . . 6 |- ( w = k -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` k ) )
18		fveq2 5178	. . . . . 6 |- ( w = k -> ( seq M ( .+ , F , T ) ` w ) = ( seq M ( .+ , F , T ) ` k ) )
19	17, 18	eqeq12d 2054	. . . . 5 |- ( w = k -> ( ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , T ) ` w ) <-> ( seq M ( .+ , F , S ) ` k ) = ( seq M ( .+ , F , T ) ` k ) ) )
20	19	imbi2d 219	. . . 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 ) ) ) )
21		fveq2 5178	. . . . . 6 |- ( w = ( k + 1 ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` ( k + 1 ) ) )
22		fveq2 5178	. . . . . 6 |- ( w = ( k + 1 ) -> ( seq M ( .+ , F , T ) ` w ) = ( seq M ( .+ , F , T ) ` ( k + 1 ) ) )
23	21, 22	eqeq12d 2054	. . . . 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 ) ) ) )
24	23	imbi2d 219	. . . 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 ) ) ) ) )
25		fveq2 5178	. . . . . 6 |- ( w = n -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` n ) )
26		fveq2 5178	. . . . . 6 |- ( w = n -> ( seq M ( .+ , F , T ) ` w ) = ( seq M ( .+ , F , T ) ` n ) )
27	25, 26	eqeq12d 2054	. . . . 5 |- ( w = n -> ( ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , T ) ` w ) <-> ( seq M ( .+ , F , S ) ` n ) = ( seq M ( .+ , F , T ) ` n ) ) )
28	27	imbi2d 219	. . . 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 ) ) ) )
29	1, 4, 5, 6	iseq1 9222	. . . . . 6 |- ( ph -> ( seq M ( .+ , F , S ) ` M ) = ( F ` M ) )
30	1, 2, 10, 11	iseq1 9222	. . . . . 6 |- ( ph -> ( seq M ( .+ , F , T ) ` M ) = ( F ` M ) )
31	29, 30	eqtr4d 2075	. . . . 5 |- ( ph -> ( seq M ( .+ , F , S ) ` M ) = ( seq M ( .+ , F , T ) ` M ) )
32	31	a1i 9	. . . 4 |- ( M e. ZZ -> ( ph -> ( seq M ( .+ , F , S ) ` M ) = ( seq M ( .+ , F , T ) ` M ) ) )
33		oveq1 5519	. . . . . . 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 ) ) ) )
34		simpr 103	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
35	4	adantr 261	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> S e. _V )
36	5	adantlr 446	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
37	6	adantlr 446	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
38	34, 35, 36, 37	iseqp1 9225	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
39	2	adantr 261	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> T e. V )
40	10	adantlr 446	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. T )
41	11	adantlr 446	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( x e. T /\ y e. T ) ) -> ( x .+ y ) e. T )
42	34, 39, 40, 41	iseqp1 9225	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , T ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F , T ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
43	38, 42	eqeq12d 2054	. . . . . . 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 ) ) ) ) )
44	33, 43	syl5ibr 145	. . . . . 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 ) ) ) )
45	44	expcom 109	. . . . 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 ) ) ) ) )
46	45	a2d 23	. . . 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 ) ) ) ) )
47	16, 20, 24, 28, 32, 46	uzind4 8531	. . 3 |- ( n e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( .+ , F , S ) ` n ) = ( seq M ( .+ , F , T ) ` n ) ) )
48	47	impcom 116	. 2 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , S ) ` n ) = ( seq M ( .+ , F , T ) ` n ) )
49	7, 12, 48	eqfnfvd 5268	1 |- ( ph -> seq M ( .+ , F , S ) = seq M ( .+ , F , T ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   _Vcvv 2557   C_ wss 2917   ` cfv 4902  (class class class)co 5512   1c1 6890   + caddc 6892   ZZcz 8245   ZZ>=cuz 8473   seqcseq 9211

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474  df-iseq 9212

This theorem is referenced by:  serige0  9252  serile  9253  iserile  9862  climserile  9865