Theorem seqfeq2 12824

Description: Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Hypotheses

Ref	Expression
seqfveq2.1	|- ( ph -> K e. ( ZZ>= ` M ) )
seqfveq2.2	|- ( ph -> ( seq M ( .+ , F ) ` K ) = ( G ` K ) )
seqfeq2.4	|- ( ( ph /\ k e. ( ZZ>= ` ( K + 1 ) ) ) -> ( F ` k ) = ( G ` k ) )

Assertion

Ref	Expression
seqfeq2	|- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) = seq K ( .+ , G ) )

Distinct variable groups:   k, F   k, G   k, K   ph, k

Allowed substitution hints:   .+ ( k)   M( k)

Proof of Theorem seqfeq2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqfveq2.1	. . . 4 |- ( ph -> K e. ( ZZ>= ` M ) )
2		eluzel2 11692	. . . 4 |- ( K e. ( ZZ>= ` M ) -> M e. ZZ )
3		seqfn 12813	. . . 4 |- ( M e. ZZ -> seq M ( .+ , F ) Fn ( ZZ>= ` M ) )
4	1, 2, 3	3syl 18	. . 3 |- ( ph -> seq M ( .+ , F ) Fn ( ZZ>= ` M ) )
5		uzss 11708	. . . 4 |- ( K e. ( ZZ>= ` M ) -> ( ZZ>= ` K ) C_ ( ZZ>= ` M ) )
6	1, 5	syl 17	. . 3 |- ( ph -> ( ZZ>= ` K ) C_ ( ZZ>= ` M ) )
7		fnssres 6004	. . 3 |- ( ( seq M ( .+ , F ) Fn ( ZZ>= ` M ) /\ ( ZZ>= ` K ) C_ ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) Fn ( ZZ>= ` K ) )
8	4, 6, 7	syl2anc 693	. 2 |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) Fn ( ZZ>= ` K ) )
9		eluzelz 11697	. . 3 |- ( K e. ( ZZ>= ` M ) -> K e. ZZ )
10		seqfn 12813	. . 3 |- ( K e. ZZ -> seq K ( .+ , G ) Fn ( ZZ>= ` K ) )
11	1, 9, 10	3syl 18	. 2 |- ( ph -> seq K ( .+ , G ) Fn ( ZZ>= ` K ) )
12		fvres 6207	. . . 4 |- ( x e. ( ZZ>= ` K ) -> ( ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) ` x ) = ( seq M ( .+ , F ) ` x ) )
13	12	adantl 482	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) ` x ) = ( seq M ( .+ , F ) ` x ) )
14	1	adantr 481	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> K e. ( ZZ>= ` M ) )
15		seqfveq2.2	. . . . 5 |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( G ` K ) )
16	15	adantr 481	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( seq M ( .+ , F ) ` K ) = ( G ` K ) )
17		simpr 477	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> x e. ( ZZ>= ` K ) )
18		elfzuz 12338	. . . . . 6 |- ( k e. ( ( K + 1 ) ... x ) -> k e. ( ZZ>= ` ( K + 1 ) ) )
19		seqfeq2.4	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` ( K + 1 ) ) ) -> ( F ` k ) = ( G ` k ) )
20	18, 19	sylan2 491	. . . . 5 |- ( ( ph /\ k e. ( ( K + 1 ) ... x ) ) -> ( F ` k ) = ( G ` k ) )
21	20	adantlr 751	. . . 4 |- ( ( ( ph /\ x e. ( ZZ>= ` K ) ) /\ k e. ( ( K + 1 ) ... x ) ) -> ( F ` k ) = ( G ` k ) )
22	14, 16, 17, 21	seqfveq2 12823	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( seq M ( .+ , F ) ` x ) = ( seq K ( .+ , G ) ` x ) )
23	13, 22	eqtrd 2656	. 2 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) ` x ) = ( seq K ( .+ , G ) ` x ) )
24	8, 11, 23	eqfnfvd 6314	1 |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) = seq K ( .+ , G ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   |` cres 5116   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802

This theorem is referenced by:  seqid  12846