Theorem iseqovex 10820

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 2233	. . 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 533	. . . 4 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> w = y )
3		simprl 531	. . . . . 6 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> z = x )
4	3	oveq1d 6065	. . . . 5 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> ( z + 1 ) = ( x + 1 ) )
5	4	fveq2d 5674	. . . 4 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> ( F ` ( z + 1 ) ) = ( F ` ( x + 1 ) ) )
6	2, 5	oveq12d 6068	. . 3 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> ( w .+ ( F ` ( z + 1 ) ) ) = ( y .+ ( F ` ( x + 1 ) ) ) )
7		simprl 531	. . 3 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> x e. ( ZZ>= ` M ) )
8		simprr 533	. . 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 6207	. . . . 5 |- ( ( ph /\ ( z e. S /\ w e. S ) ) -> ( z .+ w ) e. S )
11	10	adantlr 477	. . . 4 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z e. S /\ w e. S ) ) -> ( z .+ w ) e. S )
12		fveq2 5670	. . . . . 6 |- ( z = ( x + 1 ) -> ( F ` z ) = ( F ` ( x + 1 ) ) )
13	12	eleq1d 2301	. . . . 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 2615	. . . . . . 7 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
16		fveq2 5670	. . . . . . . . 9 |- ( x = z -> ( F ` x ) = ( F ` z ) )
17	16	eleq1d 2301	. . . . . . . 8 |- ( x = z -> ( ( F ` x ) e. S <-> ( F ` z ) e. S ) )
18	17	cbvralv 2778	. . . . . . 7 |- ( A. x e. ( ZZ>= ` M ) ( F ` x ) e. S <-> A. z e. ( ZZ>= ` M ) ( F ` z ) e. S )
19	15, 18	sylib 122	. . . . . 6 |- ( ph -> A. z e. ( ZZ>= ` M ) ( F ` z ) e. S )
20	19	adantr 276	. . . . 5 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> A. z e. ( ZZ>= ` M ) ( F ` z ) e. S )
21		peano2uz 9915	. . . . . 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 2926	. . . 4 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( F ` ( x + 1 ) ) e. S )
24	11, 8, 23	caovcld 6208	. . 3 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( y .+ ( F ` ( x + 1 ) ) ) e. S )
25	1, 6, 7, 8, 24	ovmpod 6181	. 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 2309	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 104   = wceq 1398   e. wcel 2203   A.wral 2520   ` cfv 5352  (class class class)co 6050   e. cmpo 6052   1c1 8128   + caddc 8130   ZZ>=cuz 9853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854

This theorem is referenced by:  seq3val  10822  seq3-1  10824  seq3p1  10827