Theorem iseqovex 10412

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 2171	. . 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 527	. . . 4 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> w = y )
3		simprl 526	. . . . . 6 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> z = x )
4	3	oveq1d 5868	. . . . 5 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> ( z + 1 ) = ( x + 1 ) )
5	4	fveq2d 5500	. . . 4 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> ( F ` ( z + 1 ) ) = ( F ` ( x + 1 ) ) )
6	2, 5	oveq12d 5871	. . 3 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> ( w .+ ( F ` ( z + 1 ) ) ) = ( y .+ ( F ` ( x + 1 ) ) ) )
7		simprl 526	. . 3 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> x e. ( ZZ>= ` M ) )
8		simprr 527	. . 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 6005	. . . . 5 |- ( ( ph /\ ( z e. S /\ w e. S ) ) -> ( z .+ w ) e. S )
11	10	adantlr 474	. . . 4 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z e. S /\ w e. S ) ) -> ( z .+ w ) e. S )
12		fveq2 5496	. . . . . 6 |- ( z = ( x + 1 ) -> ( F ` z ) = ( F ` ( x + 1 ) ) )
13	12	eleq1d 2239	. . . . 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 2543	. . . . . . 7 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
16		fveq2 5496	. . . . . . . . 9 |- ( x = z -> ( F ` x ) = ( F ` z ) )
17	16	eleq1d 2239	. . . . . . . 8 |- ( x = z -> ( ( F ` x ) e. S <-> ( F ` z ) e. S ) )
18	17	cbvralv 2696	. . . . . . 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 9542	. . . . . 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 2839	. . . 4 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( F ` ( x + 1 ) ) e. S )
24	11, 8, 23	caovcld 6006	. . 3 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( y .+ ( F ` ( x + 1 ) ) ) e. S )
25	1, 6, 7, 8, 24	ovmpod 5980	. 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 2247	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 1348   e. wcel 2141   A.wral 2448   ` cfv 5198  (class class class)co 5853   e. cmpo 5855   1c1 7775   + caddc 7777   ZZ>=cuz 9487

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488

This theorem is referenced by:  seq3val  10414  seq3-1  10416  seq3p1  10418