Theorem iseqovex 10708

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 2230	. . 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 531	. . . 4 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> w = y )
3		simprl 529	. . . . . 6 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> z = x )
4	3	oveq1d 6026	. . . . 5 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> ( z + 1 ) = ( x + 1 ) )
5	4	fveq2d 5637	. . . 4 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> ( F ` ( z + 1 ) ) = ( F ` ( x + 1 ) ) )
6	2, 5	oveq12d 6029	. . 3 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> ( w .+ ( F ` ( z + 1 ) ) ) = ( y .+ ( F ` ( x + 1 ) ) ) )
7		simprl 529	. . 3 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> x e. ( ZZ>= ` M ) )
8		simprr 531	. . 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 6168	. . . . 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 5633	. . . . . 6 |- ( z = ( x + 1 ) -> ( F ` z ) = ( F ` ( x + 1 ) ) )
13	12	eleq1d 2298	. . . . 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 2603	. . . . . . 7 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
16		fveq2 5633	. . . . . . . . 9 |- ( x = z -> ( F ` x ) = ( F ` z ) )
17	16	eleq1d 2298	. . . . . . . 8 |- ( x = z -> ( ( F ` x ) e. S <-> ( F ` z ) e. S ) )
18	17	cbvralv 2765	. . . . . . 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 9805	. . . . . 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 2913	. . . 4 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( F ` ( x + 1 ) ) e. S )
24	11, 8, 23	caovcld 6169	. . 3 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( y .+ ( F ` ( x + 1 ) ) ) e. S )
25	1, 6, 7, 8, 24	ovmpod 6142	. 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 2306	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 1395   e. wcel 2200   A.wral 2508   ` cfv 5322  (class class class)co 6011   e. cmpo 6013   1c1 8021   + caddc 8023   ZZ>=cuz 9743

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4203  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-addcom 8120  ax-addass 8122  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-0id 8128  ax-rnegex 8129  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-ltadd 8136

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-br 4085  df-opab 4147  df-mpt 4148  df-id 4386  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-inn 9132  df-n0 9391  df-z 9468  df-uz 9744

This theorem is referenced by:  seq3val  10710  seq3-1  10712  seq3p1  10715