Theorem iseqss 9446

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 3918	. . 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 9441	. 2 |- ( ph -> seq M ( .+ , F , S ) Fn ( ZZ>= ` M ) )
8	3	sseld 2998	. . . . 5 |- ( ph -> ( ( F ` x ) e. S -> ( F ` x ) e. T ) )
9	8	adantr 270	. . . 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 9441	. 2 |- ( ph -> seq M ( .+ , F , T ) Fn ( ZZ>= ` M ) )
13		fveq2 5198	. . . . . 6 |- ( w = M -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` M ) )
14		fveq2 5198	. . . . . 6 |- ( w = M -> ( seq M ( .+ , F , T ) ` w ) = ( seq M ( .+ , F , T ) ` M ) )
15	13, 14	eqeq12d 2095	. . . . 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 228	. . . 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 5198	. . . . . 6 |- ( w = k -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` k ) )
18		fveq2 5198	. . . . . 6 |- ( w = k -> ( seq M ( .+ , F , T ) ` w ) = ( seq M ( .+ , F , T ) ` k ) )
19	17, 18	eqeq12d 2095	. . . . 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 228	. . . 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 5198	. . . . . 6 |- ( w = ( k + 1 ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` ( k + 1 ) ) )
22		fveq2 5198	. . . . . 6 |- ( w = ( k + 1 ) -> ( seq M ( .+ , F , T ) ` w ) = ( seq M ( .+ , F , T ) ` ( k + 1 ) ) )
23	21, 22	eqeq12d 2095	. . . . 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 228	. . . 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 5198	. . . . . 6 |- ( w = n -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` n ) )
26		fveq2 5198	. . . . . 6 |- ( w = n -> ( seq M ( .+ , F , T ) ` w ) = ( seq M ( .+ , F , T ) ` n ) )
27	25, 26	eqeq12d 2095	. . . . 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 228	. . . 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 9442	. . . . . 6 |- ( ph -> ( seq M ( .+ , F , S ) ` M ) = ( F ` M ) )
30	1, 2, 10, 11	iseq1 9442	. . . . . 6 |- ( ph -> ( seq M ( .+ , F , T ) ` M ) = ( F ` M ) )
31	29, 30	eqtr4d 2116	. . . . 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 5539	. . . . . . 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 108	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
35	4	adantr 270	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> S e. _V )
36	5	adantlr 460	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
37	6	adantlr 460	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
38	34, 35, 36, 37	iseqp1 9445	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
39	2	adantr 270	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> T e. V )
40	10	adantlr 460	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. T )
41	11	adantlr 460	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( x e. T /\ y e. T ) ) -> ( x .+ y ) e. T )
42	34, 39, 40, 41	iseqp1 9445	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , T ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F , T ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
43	38, 42	eqeq12d 2095	. . . . . . 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 154	. . . . . 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 114	. . . . 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 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 ) ) ) ) )
47	16, 20, 24, 28, 32, 46	uzind4 8676	. . 3 |- ( n e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( .+ , F , S ) ` n ) = ( seq M ( .+ , F , T ) ` n ) ) )
48	47	impcom 123	. 2 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , S ) ` n ) = ( seq M ( .+ , F , T ) ` n ) )
49	7, 12, 48	eqfnfvd 5289	1 |- ( ph -> seq M ( .+ , F , S ) = seq M ( .+ , F , T ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   _Vcvv 2601   C_ wss 2973   ` cfv 4922  (class class class)co 5532   1c1 6982   + caddc 6984   ZZcz 8351   ZZ>=cuz 8619   seqcseq 9431

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-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432

This theorem is referenced by:  serige0  9473  serile  9474  iserile  10180  climserile  10183