Theorem seqof 12858

Description: Distribute function operation through a sequence. Note that G ( z ) is an implicit function on z. (Contributed by Mario Carneiro, 3-Mar-2015.)

Hypotheses

Ref	Expression
seqof.1	|- ( ph -> A e. V )
seqof.2	|- ( ph -> N e. ( ZZ>= ` M ) )
seqof.3	|- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = ( z e. A |-> ( G ` x ) ) )

Assertion

Ref	Expression
seqof	|- ( ph -> ( seq M ( oF .+ , F ) ` N ) = ( z e. A |-> ( seq M ( .+ , G ) ` N ) ) )

Distinct variable groups:   x, z, A   x, F, z   x, G   x, M, z   x, N, z   x, .+ , z   ph, x, z

Allowed substitution hints:   G( z)   V( x, z)

Proof of Theorem seqof

Dummy variables y w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqof.2	. . . . 5 |- ( ph -> N e. ( ZZ>= ` M ) )
2		fvex 6201	. . . . . . . . 9 |- ( G ` x ) e. _V
3	2	rgenw 2924	. . . . . . . 8 |- A. z e. A ( G ` x ) e. _V
4		eqid 2622	. . . . . . . . 9 |- ( z e. A |-> ( G ` x ) ) = ( z e. A |-> ( G ` x ) )
5	4	fnmpt 6020	. . . . . . . 8 |- ( A. z e. A ( G ` x ) e. _V -> ( z e. A |-> ( G ` x ) ) Fn A )
6	3, 5	mp1i 13	. . . . . . 7 |- ( ( ph /\ x e. ( M ... N ) ) -> ( z e. A |-> ( G ` x ) ) Fn A )
7		seqof.3	. . . . . . . 8 |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = ( z e. A |-> ( G ` x ) ) )
8	7	fneq1d 5981	. . . . . . 7 |- ( ( ph /\ x e. ( M ... N ) ) -> ( ( F ` x ) Fn A <-> ( z e. A |-> ( G ` x ) ) Fn A ) )
9	6, 8	mpbird 247	. . . . . 6 |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) Fn A )
10		fvex 6201	. . . . . . 7 |- ( F ` x ) e. _V
11		fneq1 5979	. . . . . . 7 |- ( z = ( F ` x ) -> ( z Fn A <-> ( F ` x ) Fn A ) )
12	10, 11	elab 3350	. . . . . 6 |- ( ( F ` x ) e. { z | z Fn A } <-> ( F ` x ) Fn A )
13	9, 12	sylibr 224	. . . . 5 |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) e. { z | z Fn A } )
14		simprl 794	. . . . . . . . 9 |- ( ( ph /\ ( x Fn A /\ y Fn A ) ) -> x Fn A )
15		simprr 796	. . . . . . . . 9 |- ( ( ph /\ ( x Fn A /\ y Fn A ) ) -> y Fn A )
16		seqof.1	. . . . . . . . . 10 |- ( ph -> A e. V )
17	16	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( x Fn A /\ y Fn A ) ) -> A e. V )
18		inidm 3822	. . . . . . . . 9 |- ( A i^i A ) = A
19	14, 15, 17, 17, 18	offn 6908	. . . . . . . 8 |- ( ( ph /\ ( x Fn A /\ y Fn A ) ) -> ( x oF .+ y ) Fn A )
20	19	ex 450	. . . . . . 7 |- ( ph -> ( ( x Fn A /\ y Fn A ) -> ( x oF .+ y ) Fn A ) )
21		vex 3203	. . . . . . . . 9 |- x e. _V
22		fneq1 5979	. . . . . . . . 9 |- ( z = x -> ( z Fn A <-> x Fn A ) )
23	21, 22	elab 3350	. . . . . . . 8 |- ( x e. { z | z Fn A } <-> x Fn A )
24		vex 3203	. . . . . . . . 9 |- y e. _V
25		fneq1 5979	. . . . . . . . 9 |- ( z = y -> ( z Fn A <-> y Fn A ) )
26	24, 25	elab 3350	. . . . . . . 8 |- ( y e. { z | z Fn A } <-> y Fn A )
27	23, 26	anbi12i 733	. . . . . . 7 |- ( ( x e. { z | z Fn A } /\ y e. { z | z Fn A } ) <-> ( x Fn A /\ y Fn A ) )
28		ovex 6678	. . . . . . . 8 |- ( x oF .+ y ) e. _V
29		fneq1 5979	. . . . . . . 8 |- ( z = ( x oF .+ y ) -> ( z Fn A <-> ( x oF .+ y ) Fn A ) )
30	28, 29	elab 3350	. . . . . . 7 |- ( ( x oF .+ y ) e. { z | z Fn A } <-> ( x oF .+ y ) Fn A )
31	20, 27, 30	3imtr4g 285	. . . . . 6 |- ( ph -> ( ( x e. { z | z Fn A } /\ y e. { z | z Fn A } ) -> ( x oF .+ y ) e. { z | z Fn A } ) )
32	31	imp 445	. . . . 5 |- ( ( ph /\ ( x e. { z | z Fn A } /\ y e. { z | z Fn A } ) ) -> ( x oF .+ y ) e. { z | z Fn A } )
33	1, 13, 32	seqcl 12821	. . . 4 |- ( ph -> ( seq M ( oF .+ , F ) ` N ) e. { z | z Fn A } )
34		fvex 6201	. . . . 5 |- ( seq M ( oF .+ , F ) ` N ) e. _V
35		fneq1 5979	. . . . 5 |- ( z = ( seq M ( oF .+ , F ) ` N ) -> ( z Fn A <-> ( seq M ( oF .+ , F ) ` N ) Fn A ) )
36	34, 35	elab 3350	. . . 4 |- ( ( seq M ( oF .+ , F ) ` N ) e. { z | z Fn A } <-> ( seq M ( oF .+ , F ) ` N ) Fn A )
37	33, 36	sylib 208	. . 3 |- ( ph -> ( seq M ( oF .+ , F ) ` N ) Fn A )
38		dffn5 6241	. . 3 |- ( ( seq M ( oF .+ , F ) ` N ) Fn A <-> ( seq M ( oF .+ , F ) ` N ) = ( z e. A |-> ( ( seq M ( oF .+ , F ) ` N ) ` z ) ) )
39	37, 38	sylib 208	. 2 |- ( ph -> ( seq M ( oF .+ , F ) ` N ) = ( z e. A |-> ( ( seq M ( oF .+ , F ) ` N ) ` z ) ) )
40		fveq1 6190	. . . . . 6 |- ( w = ( seq M ( oF .+ , F ) ` N ) -> ( w ` z ) = ( ( seq M ( oF .+ , F ) ` N ) ` z ) )
41		eqid 2622	. . . . . 6 |- ( w e. _V |-> ( w ` z ) ) = ( w e. _V |-> ( w ` z ) )
42		fvex 6201	. . . . . 6 |- ( ( seq M ( oF .+ , F ) ` N ) ` z ) e. _V
43	40, 41, 42	fvmpt 6282	. . . . 5 |- ( ( seq M ( oF .+ , F ) ` N ) e. _V -> ( ( w e. _V |-> ( w ` z ) ) ` ( seq M ( oF .+ , F ) ` N ) ) = ( ( seq M ( oF .+ , F ) ` N ) ` z ) )
44	34, 43	mp1i 13	. . . 4 |- ( ( ph /\ z e. A ) -> ( ( w e. _V |-> ( w ` z ) ) ` ( seq M ( oF .+ , F ) ` N ) ) = ( ( seq M ( oF .+ , F ) ` N ) ` z ) )
45	32	adantlr 751	. . . . 5 |- ( ( ( ph /\ z e. A ) /\ ( x e. { z | z Fn A } /\ y e. { z | z Fn A } ) ) -> ( x oF .+ y ) e. { z | z Fn A } )
46	13	adantlr 751	. . . . 5 |- ( ( ( ph /\ z e. A ) /\ x e. ( M ... N ) ) -> ( F ` x ) e. { z | z Fn A } )
47	1	adantr 481	. . . . 5 |- ( ( ph /\ z e. A ) -> N e. ( ZZ>= ` M ) )
48		eqidd 2623	. . . . . . . . 9 |- ( ( ( ph /\ ( x Fn A /\ y Fn A ) ) /\ z e. A ) -> ( x ` z ) = ( x ` z ) )
49		eqidd 2623	. . . . . . . . 9 |- ( ( ( ph /\ ( x Fn A /\ y Fn A ) ) /\ z e. A ) -> ( y ` z ) = ( y ` z ) )
50	14, 15, 17, 17, 18, 48, 49	ofval 6906	. . . . . . . 8 |- ( ( ( ph /\ ( x Fn A /\ y Fn A ) ) /\ z e. A ) -> ( ( x oF .+ y ) ` z ) = ( ( x ` z ) .+ ( y ` z ) ) )
51	50	an32s 846	. . . . . . 7 |- ( ( ( ph /\ z e. A ) /\ ( x Fn A /\ y Fn A ) ) -> ( ( x oF .+ y ) ` z ) = ( ( x ` z ) .+ ( y ` z ) ) )
52		fveq1 6190	. . . . . . . . 9 |- ( w = ( x oF .+ y ) -> ( w ` z ) = ( ( x oF .+ y ) ` z ) )
53		fvex 6201	. . . . . . . . 9 |- ( ( x oF .+ y ) ` z ) e. _V
54	52, 41, 53	fvmpt 6282	. . . . . . . 8 |- ( ( x oF .+ y ) e. _V -> ( ( w e. _V |-> ( w ` z ) ) ` ( x oF .+ y ) ) = ( ( x oF .+ y ) ` z ) )
55	28, 54	ax-mp 5	. . . . . . 7 |- ( ( w e. _V |-> ( w ` z ) ) ` ( x oF .+ y ) ) = ( ( x oF .+ y ) ` z )
56		fveq1 6190	. . . . . . . . . 10 |- ( w = x -> ( w ` z ) = ( x ` z ) )
57		fvex 6201	. . . . . . . . . 10 |- ( x ` z ) e. _V
58	56, 41, 57	fvmpt 6282	. . . . . . . . 9 |- ( x e. _V -> ( ( w e. _V |-> ( w ` z ) ) ` x ) = ( x ` z ) )
59	21, 58	ax-mp 5	. . . . . . . 8 |- ( ( w e. _V |-> ( w ` z ) ) ` x ) = ( x ` z )
60		fveq1 6190	. . . . . . . . . 10 |- ( w = y -> ( w ` z ) = ( y ` z ) )
61		fvex 6201	. . . . . . . . . 10 |- ( y ` z ) e. _V
62	60, 41, 61	fvmpt 6282	. . . . . . . . 9 |- ( y e. _V -> ( ( w e. _V |-> ( w ` z ) ) ` y ) = ( y ` z ) )
63	24, 62	ax-mp 5	. . . . . . . 8 |- ( ( w e. _V |-> ( w ` z ) ) ` y ) = ( y ` z )
64	59, 63	oveq12i 6662	. . . . . . 7 |- ( ( ( w e. _V |-> ( w ` z ) ) ` x ) .+ ( ( w e. _V |-> ( w ` z ) ) ` y ) ) = ( ( x ` z ) .+ ( y ` z ) )
65	51, 55, 64	3eqtr4g 2681	. . . . . 6 |- ( ( ( ph /\ z e. A ) /\ ( x Fn A /\ y Fn A ) ) -> ( ( w e. _V |-> ( w ` z ) ) ` ( x oF .+ y ) ) = ( ( ( w e. _V |-> ( w ` z ) ) ` x ) .+ ( ( w e. _V |-> ( w ` z ) ) ` y ) ) )
66	27, 65	sylan2b 492	. . . . 5 |- ( ( ( ph /\ z e. A ) /\ ( x e. { z | z Fn A } /\ y e. { z | z Fn A } ) ) -> ( ( w e. _V |-> ( w ` z ) ) ` ( x oF .+ y ) ) = ( ( ( w e. _V |-> ( w ` z ) ) ` x ) .+ ( ( w e. _V |-> ( w ` z ) ) ` y ) ) )
67		fveq1 6190	. . . . . . . 8 |- ( w = ( F ` x ) -> ( w ` z ) = ( ( F ` x ) ` z ) )
68		fvex 6201	. . . . . . . 8 |- ( ( F ` x ) ` z ) e. _V
69	67, 41, 68	fvmpt 6282	. . . . . . 7 |- ( ( F ` x ) e. _V -> ( ( w e. _V |-> ( w ` z ) ) ` ( F ` x ) ) = ( ( F ` x ) ` z ) )
70	10, 69	ax-mp 5	. . . . . 6 |- ( ( w e. _V |-> ( w ` z ) ) ` ( F ` x ) ) = ( ( F ` x ) ` z )
71	7	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ z e. A ) /\ x e. ( M ... N ) ) -> ( F ` x ) = ( z e. A |-> ( G ` x ) ) )
72	71	fveq1d 6193	. . . . . . 7 |- ( ( ( ph /\ z e. A ) /\ x e. ( M ... N ) ) -> ( ( F ` x ) ` z ) = ( ( z e. A |-> ( G ` x ) ) ` z ) )
73		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ z e. A ) /\ x e. ( M ... N ) ) -> z e. A )
74	4	fvmpt2 6291	. . . . . . . 8 |- ( ( z e. A /\ ( G ` x ) e. _V ) -> ( ( z e. A |-> ( G ` x ) ) ` z ) = ( G ` x ) )
75	73, 2, 74	sylancl 694	. . . . . . 7 |- ( ( ( ph /\ z e. A ) /\ x e. ( M ... N ) ) -> ( ( z e. A |-> ( G ` x ) ) ` z ) = ( G ` x ) )
76	72, 75	eqtrd 2656	. . . . . 6 |- ( ( ( ph /\ z e. A ) /\ x e. ( M ... N ) ) -> ( ( F ` x ) ` z ) = ( G ` x ) )
77	70, 76	syl5eq 2668	. . . . 5 |- ( ( ( ph /\ z e. A ) /\ x e. ( M ... N ) ) -> ( ( w e. _V |-> ( w ` z ) ) ` ( F ` x ) ) = ( G ` x ) )
78	45, 46, 47, 66, 77	seqhomo 12848	. . . 4 |- ( ( ph /\ z e. A ) -> ( ( w e. _V |-> ( w ` z ) ) ` ( seq M ( oF .+ , F ) ` N ) ) = ( seq M ( .+ , G ) ` N ) )
79	44, 78	eqtr3d 2658	. . 3 |- ( ( ph /\ z e. A ) -> ( ( seq M ( oF .+ , F ) ` N ) ` z ) = ( seq M ( .+ , G ) ` N ) )
80	79	mpteq2dva 4744	. 2 |- ( ph -> ( z e. A |-> ( ( seq M ( oF .+ , F ) ` N ) ` z ) ) = ( z e. A |-> ( seq M ( .+ , G ) ` N ) ) )
81	39, 80	eqtrd 2656	1 |- ( ph -> ( seq M ( oF .+ , F ) ` N ) = ( z e. A |-> ( seq M ( .+ , G ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   |-> cmpt 4729   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   oFcof 6895   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-rep 4771  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-of 6897  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:  seqof2  12859  mtest  24158  pserulm  24176  knoppcnlem7  32489