Theorem seqfeq3 10472

Description: Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 25-Apr-2016.)

Hypotheses

Ref	Expression
seqfeq3.m	|- ( ph -> M e. ZZ )
seqfeq3.f	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
seqfeq3.cl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
seqfeq3.id	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( x Q y ) )

Assertion

Ref	Expression
seqfeq3	|- ( ph -> seq M ( .+ , F ) = seq M ( Q , F ) )

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

Proof of Theorem seqfeq3

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2171	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		seqfeq3.m	. . . 4 |- ( ph -> M e. ZZ )
3		seqfeq3.f	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
4		seqfeq3.cl	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
5	1, 2, 3, 4	seqf 10421	. . 3 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
6	5	ffnd 5350	. 2 |- ( ph -> seq M ( .+ , F ) Fn ( ZZ>= ` M ) )
7		seqfeq3.id	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( x Q y ) )
8	7, 4	eqeltrrd 2249	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )
9	1, 2, 3, 8	seqf 10421	. . 3 |- ( ph -> seq M ( Q , F ) : ( ZZ>= ` M ) --> S )
10	9	ffnd 5350	. 2 |- ( ph -> seq M ( Q , F ) Fn ( ZZ>= ` M ) )
11	5	ffvelcdmda 5635	. . . 4 |- ( ( ph /\ a e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) ` a ) e. S )
12		fvi 5557	. . . 4 |- ( ( seq M ( .+ , F ) ` a ) e. S -> ( _I ` ( seq M ( .+ , F ) ` a ) ) = ( seq M ( .+ , F ) ` a ) )
13	11, 12	syl 14	. . 3 |- ( ( ph /\ a e. ( ZZ>= ` M ) ) -> ( _I ` ( seq M ( .+ , F ) ` a ) ) = ( seq M ( .+ , F ) ` a ) )
14	4	adantlr 475	. . . 4 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
15	3	adantlr 475	. . . 4 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
16		simpr 109	. . . 4 |- ( ( ph /\ a e. ( ZZ>= ` M ) ) -> a e. ( ZZ>= ` M ) )
17	7	adantlr 475	. . . . 5 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( x Q y ) )
18		fvi 5557	. . . . . 6 |- ( ( x .+ y ) e. S -> ( _I ` ( x .+ y ) ) = ( x .+ y ) )
19	14, 18	syl 14	. . . . 5 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( _I ` ( x .+ y ) ) = ( x .+ y ) )
20		fvi 5557	. . . . . . 7 |- ( x e. S -> ( _I ` x ) = x )
21	20	ad2antrl 488	. . . . . 6 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( _I ` x ) = x )
22		fvi 5557	. . . . . . 7 |- ( y e. S -> ( _I ` y ) = y )
23	22	ad2antll 489	. . . . . 6 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( _I ` y ) = y )
24	21, 23	oveq12d 5875	. . . . 5 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( ( _I ` x ) Q ( _I ` y ) ) = ( x Q y ) )
25	17, 19, 24	3eqtr4d 2214	. . . 4 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( _I ` ( x .+ y ) ) = ( ( _I ` x ) Q ( _I ` y ) ) )
26		fvi 5557	. . . . 5 |- ( ( F ` x ) e. S -> ( _I ` ( F ` x ) ) = ( F ` x ) )
27	15, 26	syl 14	. . . 4 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( _I ` ( F ` x ) ) = ( F ` x ) )
28	8	adantlr 475	. . . 4 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )
29	14, 15, 16, 25, 27, 15, 28	seq3homo 10470	. . 3 |- ( ( ph /\ a e. ( ZZ>= ` M ) ) -> ( _I ` ( seq M ( .+ , F ) ` a ) ) = ( seq M ( Q , F ) ` a ) )
30	13, 29	eqtr3d 2206	. 2 |- ( ( ph /\ a e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) ` a ) = ( seq M ( Q , F ) ` a ) )
31	6, 10, 30	eqfnfvd 5600	1 |- ( ph -> seq M ( .+ , F ) = seq M ( Q , F ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1349   e. wcel 2142   _I cid 4274   ` cfv 5200  (class class class)co 5857   ZZcz 9216   ZZ>=cuz 9491   seqcseq 10405

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 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-nul 4116  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522  ax-iinf 4573  ax-cnex 7869  ax-resscn 7870  ax-1cn 7871  ax-1re 7872  ax-icn 7873  ax-addcl 7874  ax-addrcl 7875  ax-mulcl 7876  ax-addcom 7878  ax-addass 7880  ax-distr 7882  ax-i2m1 7883  ax-0lt1 7884  ax-0id 7886  ax-rnegex 7887  ax-cnre 7889  ax-pre-ltirr 7890  ax-pre-ltwlin 7891  ax-pre-lttrn 7892  ax-pre-ltadd 7894

This theorem depends on definitions:  df-bi 116  df-3or 975  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-nel 2437  df-ral 2454  df-rex 2455  df-reu 2456  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-nul 3416  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-int 3833  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-tr 4089  df-id 4279  df-iord 4352  df-on 4354  df-ilim 4355  df-suc 4357  df-iom 4576  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-riota 5813  df-ov 5860  df-oprab 5861  df-mpo 5862  df-1st 6123  df-2nd 6124  df-recs 6288  df-frec 6374  df-pnf 7960  df-mnf 7961  df-xr 7962  df-ltxr 7963  df-le 7964  df-sub 8096  df-neg 8097  df-inn 8883  df-n0 9140  df-z 9217  df-uz 9492  df-seqfrec 10406

This theorem is referenced by: (None)