Theorem iseqovex 10719

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 2232	. . 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 6032	. . . . 5 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> ( z + 1 ) = ( x + 1 ) )
5	4	fveq2d 5643	. . . 4 |- ( ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) /\ ( z = x /\ w = y ) ) -> ( F ` ( z + 1 ) ) = ( F ` ( x + 1 ) ) )
6	2, 5	oveq12d 6035	. . 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 6174	. . . . 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 5639	. . . . . 6 |- ( z = ( x + 1 ) -> ( F ` z ) = ( F ` ( x + 1 ) ) )
13	12	eleq1d 2300	. . . . 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 2605	. . . . . . 7 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
16		fveq2 5639	. . . . . . . . 9 |- ( x = z -> ( F ` x ) = ( F ` z ) )
17	16	eleq1d 2300	. . . . . . . 8 |- ( x = z -> ( ( F ` x ) e. S <-> ( F ` z ) e. S ) )
18	17	cbvralv 2767	. . . . . . 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 9816	. . . . . 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 2915	. . . 4 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( F ` ( x + 1 ) ) e. S )
24	11, 8, 23	caovcld 6175	. . 3 |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( y .+ ( F ` ( x + 1 ) ) ) e. S )
25	1, 6, 7, 8, 24	ovmpod 6148	. 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 2308	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 1397   e. wcel 2202   A.wral 2510   ` cfv 5326  (class class class)co 6017   e. cmpo 6019   1c1 8032   + caddc 8034   ZZ>=cuz 9754

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755

This theorem is referenced by:  seq3val  10721  seq3-1  10723  seq3p1  10726