Theorem iseqovex 10391

Description: Closure of a function used in proving sequence builder theorems. This can be thought of as a lemma for the small number of sequence builder theorems which need it. (Contributed by Jim Kingdon, 31-May-2020.)

Hypotheses

Ref	Expression
iseqovex.f	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
iseqovex.pl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )

Assertion

Ref	Expression
iseqovex	|- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) e. S )

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

Allowed substitution hint:   M( y)

Proof of Theorem iseqovex

Step	Hyp	Ref	Expression
1		eqidd 2166	. . 3 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) = ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) )
2		simprr 522	. . . 4 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> w = y )
3		simprl 521	. . . . . 6 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> z = x )
4	3	oveq1d 5857	. . . . 5 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> ( z + 1 ) = ( x + 1 ) )
5	4	fveq2d 5490	. . . 4 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> ( F ` ( z + 1 ) ) = ( F ` ( x + 1 ) ) )
6	2, 5	oveq12d 5860	. . 3 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> ( w .+ ( F ` ( z + 1 ) ) ) = ( y .+ ( F ` ( x + 1 ) ) ) )
7		simprl 521	. . 3 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> x e. ( ZZ>= ` M ) )
8		simprr 522	. . 3 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> y e. S )
9		iseqovex.pl	. . . . . 6 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
10	9	caovclg 5994	. . . . 5 |- ( ( ph /\ ( z e. S /\ w e. S ) ) -> ( z .+ w ) e. S )
11	10	adantlr 469	. . . 4 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z e. S /\ w e. S ) ) -> ( z .+ w ) e. S )
12		fveq2 5486	. . . . . 6 |- ( z = ( x + 1 ) -> ( F ` z ) = ( F ` ( x + 1 ) ) )
13	12	eleq1d 2235	. . . . 5 |- ( z = ( x + 1 ) -> ( ( F ` z ) e. S <-> ( F ` ( x + 1 ) ) e. S ) )
14		iseqovex.f	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
15	14	ralrimiva 2539	. . . . . . 7 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
16		fveq2 5486	. . . . . . . . 9 |- ( x = z -> ( F ` x ) = ( F ` z ) )
17	16	eleq1d 2235	. . . . . . . 8 |- ( x = z -> ( ( F ` x ) e. S <-> ( F ` z ) e. S ) )
18	17	cbvralv 2692	. . . . . . 7 |- ( A. x e. ( ZZ>= ` M ) ( F ` x ) e. S <-> A. z e. ( ZZ>= ` M ) ( F ` z ) e. S )
19	15, 18	sylib 121	. . . . . 6 |- ( ph -> A. z e. ( ZZ>= ` M ) ( F ` z ) e. S )
20	19	adantr 274	. . . . 5 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> A. z e. ( ZZ>= ` M ) ( F ` z ) e. S )
21		peano2uz 9521	. . . . . 6 |- ( x e. ( ZZ>= ` M ) -> ( x + 1 ) e. ( ZZ>= ` M ) )
22	7, 21	syl 14	. . . . 5 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( x + 1 ) e. ( ZZ>= ` M ) )
23	13, 20, 22	rspcdva 2835	. . . 4 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( F ` ( x + 1 ) ) e. S )
24	11, 8, 23	caovcld 5995	. . 3 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( y .+ ( F ` ( x + 1 ) ) ) e. S )
25	1, 6, 7, 8, 24	ovmpod 5969	. 2 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) = ( y .+ ( F ` ( x + 1 ) ) ) )
26	25, 24	eqeltrd 2243	1 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) e. S )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   A.wral 2444   ` cfv 5188  (class class class)co 5842   e. cmpo 5844   1c1 7754   + caddc 7756   ZZ>=cuz 9466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467

This theorem is referenced by:  seq3val  10393  seq3-1  10395  seq3p1  10397