Theorem seqfeq3 10854

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 2231	. . . 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 10789	. . 3 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
6	5	ffnd 5490	. 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 2309	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )
9	1, 2, 3, 8	seqf 10789	. . 3 |- ( ph -> seq M ( Q , F ) : ( ZZ>= ` M ) --> S )
10	9	ffnd 5490	. 2 |- ( ph -> seq M ( Q , F ) Fn ( ZZ>= ` M ) )
11	5	ffvelcdmda 5790	. . . 4 |- ( ( ph /\ a e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) ` a ) e. S )
12		fvi 5712	. . . 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 477	. . . 4 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
15	3	adantlr 477	. . . 4 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
16		simpr 110	. . . 4 |- ( ( ph /\ a e. ( ZZ>= ` M ) ) -> a e. ( ZZ>= ` M ) )
17	7	adantlr 477	. . . . 5 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( x Q y ) )
18		fvi 5712	. . . . . 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 5712	. . . . . . 7 |- ( x e. S -> ( _I ` x ) = x )
21	20	ad2antrl 490	. . . . . 6 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( _I ` x ) = x )
22		fvi 5712	. . . . . . 7 |- ( y e. S -> ( _I ` y ) = y )
23	22	ad2antll 491	. . . . . 6 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( _I ` y ) = y )
24	21, 23	oveq12d 6046	. . . . 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 2274	. . . 4 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( _I ` ( x .+ y ) ) = ( ( _I ` x ) Q ( _I ` y ) ) )
26		fvi 5712	. . . . 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 477	. . . 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 10852	. . 3 |- ( ( ph /\ a e. ( ZZ>= ` M ) ) -> ( _I ` ( seq M ( .+ , F ) ` a ) ) = ( seq M ( Q , F ) ` a ) )
30	13, 29	eqtr3d 2266	. 2 |- ( ( ph /\ a e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) ` a ) = ( seq M ( Q , F ) ` a ) )
31	6, 10, 30	eqfnfvd 5756	1 |- ( ph -> seq M ( .+ , F ) = seq M ( Q , F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _I cid 4391   ` cfv 5333  (class class class)co 6028   ZZcz 9540   ZZ>=cuz 9816   seqcseq 10772

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817  df-seqfrec 10773

This theorem is referenced by:  mulgpropdg  13831