Theorem iseqss 9541

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). New proofs should use iseqsst 9542 which is the same without those two hypotheses. (Contributed by Jim Kingdon, 18-Aug-2021.) (New usage is discouraged.)

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		eqid 2082	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		iseqss.m	. . . 4 |- ( ph -> M e. ZZ )
3		iseqss.f	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
4		iseqss.pl	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
5	1, 2, 3, 4	iseqfcl 9535	. . 3 |- ( ph -> seq M ( .+ , F , S ) : ( ZZ>= ` M ) --> S )
6		ffn 5077	. . 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		iseqss.ss	. . . . . . 7 |- ( ph -> S C_ T )
9	8	sseld 2999	. . . . . 6 |- ( ph -> ( ( F ` x ) e. S -> ( F ` x ) e. T ) )
10	9	adantr 270	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( F ` x ) e. S -> ( F ` x ) e. T ) )
11	3, 10	mpd 13	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. T )
12		iseqss.plt	. . . 4 |- ( ( ph /\ ( x e. T /\ y e. T ) ) -> ( x .+ y ) e. T )
13	1, 2, 11, 12	iseqfcl 9535	. . 3 |- ( ph -> seq M ( .+ , F , T ) : ( ZZ>= ` M ) --> T )
14		ffn 5077	. . 3 |- ( seq M ( .+ , F , T ) : ( ZZ>= ` M ) --> T -> seq M ( .+ , F , T ) Fn ( ZZ>= ` M ) )
15	13, 14	syl 14	. 2 |- ( ph -> seq M ( .+ , F , T ) Fn ( ZZ>= ` M ) )
16		fveq2 5209	. . . . . 6 |- ( w = M -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` M ) )
17		fveq2 5209	. . . . . 6 |- ( w = M -> ( seq M ( .+ , F , T ) ` w ) = ( seq M ( .+ , F , T ) ` M ) )
18	16, 17	eqeq12d 2096	. . . . 5 |- ( w = M -> ( ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , T ) ` w ) <-> ( seq M ( .+ , F , S ) ` M ) = ( seq M ( .+ , F , T ) ` M ) ) )
19	18	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 ) ) ) )
20		fveq2 5209	. . . . . 6 |- ( w = k -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` k ) )
21		fveq2 5209	. . . . . 6 |- ( w = k -> ( seq M ( .+ , F , T ) ` w ) = ( seq M ( .+ , F , T ) ` k ) )
22	20, 21	eqeq12d 2096	. . . . 5 |- ( w = k -> ( ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , T ) ` w ) <-> ( seq M ( .+ , F , S ) ` k ) = ( seq M ( .+ , F , T ) ` k ) ) )
23	22	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 ) ) ) )
24		fveq2 5209	. . . . . 6 |- ( w = ( k + 1 ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` ( k + 1 ) ) )
25		fveq2 5209	. . . . . 6 |- ( w = ( k + 1 ) -> ( seq M ( .+ , F , T ) ` w ) = ( seq M ( .+ , F , T ) ` ( k + 1 ) ) )
26	24, 25	eqeq12d 2096	. . . . 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 ) ) ) )
27	26	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 ) ) ) ) )
28		fveq2 5209	. . . . . 6 |- ( w = n -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` n ) )
29		fveq2 5209	. . . . . 6 |- ( w = n -> ( seq M ( .+ , F , T ) ` w ) = ( seq M ( .+ , F , T ) ` n ) )
30	28, 29	eqeq12d 2096	. . . . 5 |- ( w = n -> ( ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , T ) ` w ) <-> ( seq M ( .+ , F , S ) ` n ) = ( seq M ( .+ , F , T ) ` n ) ) )
31	30	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 ) ) ) )
32	2, 3, 4	iseq1 9533	. . . . . 6 |- ( ph -> ( seq M ( .+ , F , S ) ` M ) = ( F ` M ) )
33	2, 11, 12	iseq1 9533	. . . . . 6 |- ( ph -> ( seq M ( .+ , F , T ) ` M ) = ( F ` M ) )
34	32, 33	eqtr4d 2117	. . . . 5 |- ( ph -> ( seq M ( .+ , F , S ) ` M ) = ( seq M ( .+ , F , T ) ` M ) )
35	34	a1i 9	. . . 4 |- ( M e. ZZ -> ( ph -> ( seq M ( .+ , F , S ) ` M ) = ( seq M ( .+ , F , T ) ` M ) ) )
36		oveq1 5550	. . . . . . 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 ) ) ) )
37		simpr 108	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
38	3	adantlr 461	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
39	4	adantlr 461	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
40	37, 38, 39	iseqp1 9538	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
41	11	adantlr 461	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. T )
42	12	adantlr 461	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( x e. T /\ y e. T ) ) -> ( x .+ y ) e. T )
43	37, 41, 42	iseqp1 9538	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , T ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F , T ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
44	40, 43	eqeq12d 2096	. . . . . . 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 ) ) ) ) )
45	36, 44	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 ) ) ) )
46	45	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 ) ) ) ) )
47	46	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 ) ) ) ) )
48	19, 23, 27, 31, 35, 47	uzind4 8757	. . 3 |- ( n e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( .+ , F , S ) ` n ) = ( seq M ( .+ , F , T ) ` n ) ) )
49	48	impcom 123	. 2 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , S ) ` n ) = ( seq M ( .+ , F , T ) ` n ) )
50	7, 15, 49	eqfnfvd 5300	1 |- ( ph -> seq M ( .+ , F , S ) = seq M ( .+ , F , T ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   e. wcel 1434   C_ wss 2974   Fn wfn 4927   -->wf 4928   ` cfv 4932  (class class class)co 5543   1c1 7044   + caddc 7046   ZZcz 8432   ZZ>=cuz 8700   seqcseq 9521

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154

This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-iseq 9522

This theorem is referenced by:  serige0  9570  serile  9571  iserile  10318  climserile  10321