Theorem eulerpartlemgf 30441

Description: Lemma for eulerpart 30444: Images under G have finite support. (Contributed by Thierry Arnoux, 29-Aug-2018.)

Hypotheses

Ref	Expression
eulerpart.p	|- P = { f e. ( NN0 ^m NN ) | ( ( `' f " NN ) e. Fin /\ sum_ k e. NN ( ( f ` k ) x. k ) = N ) }
eulerpart.o	|- O = { g e. P | A. n e. ( `' g " NN ) -. 2 || n }
eulerpart.d	|- D = { g e. P | A. n e. NN ( g ` n ) <_ 1 }
eulerpart.j	|- J = { z e. NN | -. 2 || z }
eulerpart.f	|- F = ( x e. J , y e. NN0 |-> ( ( 2 ^ y ) x. x ) )
eulerpart.h	|- H = { r e. ( ( ~P NN0 i^i Fin ) ^m J ) | ( r supp (/) ) e. Fin }
eulerpart.m	|- M = ( r e. H |-> { <. x , y >. | ( x e. J /\ y e. ( r ` x ) ) } )
eulerpart.r	|- R = { f | ( `' f " NN ) e. Fin }
eulerpart.t	|- T = { f e. ( NN0 ^m NN ) | ( `' f " NN ) C_ J }
eulerpart.g	|- G = ( o e. ( T i^i R ) |-> ( (𝟭 ` NN ) ` ( F " ( M ` (bits o. ( o |` J ) ) ) ) ) )

Assertion

Ref	Expression
eulerpartlemgf	|- ( A e. ( T i^i R ) -> ( `' ( G ` A ) " NN ) e. Fin )

Distinct variable groups:   f, g, k, n, o, x, y, z   f, r, A, o   o, F   o, H, r   f, J   n, r, J, o, x, y   o, M, r   f, N, g, x   P, g, n   R, f, o   T, f, o

Allowed substitution hints:   A( x, y, z, g, k, n)   D( x, y, z, f, g, k, n, o, r)   P( x, y, z, f, k, o, r)   R( x, y, z, g, k, n, r)   T( x, y, z, g, k, n, r)   F( x, y, z, f, g, k, n, r)   G( x, y, z, f, g, k, n, o, r)   H( x, y, z, f, g, k, n)   J( z, g, k)   M( x, y, z, f, g, k, n)   N( y, z, k, n, o, r)   O( x, y, z, f, g, k, n, o, r)

Proof of Theorem eulerpartlemgf

Step	Hyp	Ref	Expression
1		eulerpart.p	. . . . . . 7 |- P = { f e. ( NN0 ^m NN ) | ( ( `' f " NN ) e. Fin /\ sum_ k e. NN ( ( f ` k ) x. k ) = N ) }
2		eulerpart.o	. . . . . . 7 |- O = { g e. P | A. n e. ( `' g " NN ) -. 2 || n }
3		eulerpart.d	. . . . . . 7 |- D = { g e. P | A. n e. NN ( g ` n ) <_ 1 }
4		eulerpart.j	. . . . . . 7 |- J = { z e. NN | -. 2 || z }
5		eulerpart.f	. . . . . . 7 |- F = ( x e. J , y e. NN0 |-> ( ( 2 ^ y ) x. x ) )
6		eulerpart.h	. . . . . . 7 |- H = { r e. ( ( ~P NN0 i^i Fin ) ^m J ) | ( r supp (/) ) e. Fin }
7		eulerpart.m	. . . . . . 7 |- M = ( r e. H |-> { <. x , y >. | ( x e. J /\ y e. ( r ` x ) ) } )
8		eulerpart.r	. . . . . . 7 |- R = { f | ( `' f " NN ) e. Fin }
9		eulerpart.t	. . . . . . 7 |- T = { f e. ( NN0 ^m NN ) | ( `' f " NN ) C_ J }
10		eulerpart.g	. . . . . . 7 |- G = ( o e. ( T i^i R ) |-> ( (𝟭 ` NN ) ` ( F " ( M ` (bits o. ( o |` J ) ) ) ) ) )
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	eulerpartlemgv 30435	. . . . . 6 |- ( A e. ( T i^i R ) -> ( G ` A ) = ( (𝟭 ` NN ) ` ( F " ( M ` (bits o. ( A |` J ) ) ) ) ) )
12	11	cnveqd 5298	. . . . 5 |- ( A e. ( T i^i R ) -> `' ( G ` A ) = `' ( (𝟭 ` NN ) ` ( F " ( M ` (bits o. ( A |` J ) ) ) ) ) )
13	12	imaeq1d 5465	. . . 4 |- ( A e. ( T i^i R ) -> ( `' ( G ` A ) " { 1 } ) = ( `' ( (𝟭 ` NN ) ` ( F " ( M ` (bits o. ( A |` J ) ) ) ) ) " { 1 } ) )
14		nnex 11026	. . . . 5 |- NN e. _V
15		imassrn 5477	. . . . . 6 |- ( F " ( M ` (bits o. ( A |` J ) ) ) ) C_ ran F
16	4, 5	oddpwdc 30416	. . . . . . 7 |- F : ( J X. NN0 ) -1-1-onto-> NN
17		f1of 6137	. . . . . . 7 |- ( F : ( J X. NN0 ) -1-1-onto-> NN -> F : ( J X. NN0 ) --> NN )
18		frn 6053	. . . . . . 7 |- ( F : ( J X. NN0 ) --> NN -> ran F C_ NN )
19	16, 17, 18	mp2b 10	. . . . . 6 |- ran F C_ NN
20	15, 19	sstri 3612	. . . . 5 |- ( F " ( M ` (bits o. ( A |` J ) ) ) ) C_ NN
21		indpi1 30082	. . . . 5 |- ( ( NN e. _V /\ ( F " ( M ` (bits o. ( A |` J ) ) ) ) C_ NN ) -> ( `' ( (𝟭 ` NN ) ` ( F " ( M ` (bits o. ( A |` J ) ) ) ) ) " { 1 } ) = ( F " ( M ` (bits o. ( A |` J ) ) ) ) )
22	14, 20, 21	mp2an 708	. . . 4 |- ( `' ( (𝟭 ` NN ) ` ( F " ( M ` (bits o. ( A |` J ) ) ) ) ) " { 1 } ) = ( F " ( M ` (bits o. ( A |` J ) ) ) )
23	13, 22	syl6eq 2672	. . 3 |- ( A e. ( T i^i R ) -> ( `' ( G ` A ) " { 1 } ) = ( F " ( M ` (bits o. ( A |` J ) ) ) ) )
24		ffun 6048	. . . . 5 |- ( F : ( J X. NN0 ) --> NN -> Fun F )
25	16, 17, 24	mp2b 10	. . . 4 |- Fun F
26		inss2 3834	. . . . 5 |- ( ~P ( J X. NN0 ) i^i Fin ) C_ Fin
27	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	eulerpartlemmf 30437	. . . . . 6 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. H )
28	1, 2, 3, 4, 5, 6, 7	eulerpartlem1 30429	. . . . . . . 8 |- M : H -1-1-onto-> ( ~P ( J X. NN0 ) i^i Fin )
29		f1of 6137	. . . . . . . 8 |- ( M : H -1-1-onto-> ( ~P ( J X. NN0 ) i^i Fin ) -> M : H --> ( ~P ( J X. NN0 ) i^i Fin ) )
30	28, 29	ax-mp 5	. . . . . . 7 |- M : H --> ( ~P ( J X. NN0 ) i^i Fin )
31	30	ffvelrni 6358	. . . . . 6 |- ( (bits o. ( A |` J ) ) e. H -> ( M ` (bits o. ( A |` J ) ) ) e. ( ~P ( J X. NN0 ) i^i Fin ) )
32	27, 31	syl 17	. . . . 5 |- ( A e. ( T i^i R ) -> ( M ` (bits o. ( A |` J ) ) ) e. ( ~P ( J X. NN0 ) i^i Fin ) )
33	26, 32	sseldi 3601	. . . 4 |- ( A e. ( T i^i R ) -> ( M ` (bits o. ( A |` J ) ) ) e. Fin )
34		imafi 8259	. . . 4 |- ( ( Fun F /\ ( M ` (bits o. ( A |` J ) ) ) e. Fin ) -> ( F " ( M ` (bits o. ( A |` J ) ) ) ) e. Fin )
35	25, 33, 34	sylancr 695	. . 3 |- ( A e. ( T i^i R ) -> ( F " ( M ` (bits o. ( A |` J ) ) ) ) e. Fin )
36	23, 35	eqeltrd 2701	. 2 |- ( A e. ( T i^i R ) -> ( `' ( G ` A ) " { 1 } ) e. Fin )
37	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	eulerpartgbij 30434	. . . . . . . 8 |- G : ( T i^i R ) -1-1-onto-> ( ( { 0 , 1 } ^m NN ) i^i R )
38		f1of 6137	. . . . . . . 8 |- ( G : ( T i^i R ) -1-1-onto-> ( ( { 0 , 1 } ^m NN ) i^i R ) -> G : ( T i^i R ) --> ( ( { 0 , 1 } ^m NN ) i^i R ) )
39	37, 38	ax-mp 5	. . . . . . 7 |- G : ( T i^i R ) --> ( ( { 0 , 1 } ^m NN ) i^i R )
40	39	ffvelrni 6358	. . . . . 6 |- ( A e. ( T i^i R ) -> ( G ` A ) e. ( ( { 0 , 1 } ^m NN ) i^i R ) )
41		elin 3796	. . . . . . 7 |- ( ( G ` A ) e. ( ( { 0 , 1 } ^m NN ) i^i R ) <-> ( ( G ` A ) e. ( { 0 , 1 } ^m NN ) /\ ( G ` A ) e. R ) )
42	41	simplbi 476	. . . . . 6 |- ( ( G ` A ) e. ( ( { 0 , 1 } ^m NN ) i^i R ) -> ( G ` A ) e. ( { 0 , 1 } ^m NN ) )
43		elmapi 7879	. . . . . 6 |- ( ( G ` A ) e. ( { 0 , 1 } ^m NN ) -> ( G ` A ) : NN --> { 0 , 1 } )
44	40, 42, 43	3syl 18	. . . . 5 |- ( A e. ( T i^i R ) -> ( G ` A ) : NN --> { 0 , 1 } )
45		ffun 6048	. . . . 5 |- ( ( G ` A ) : NN --> { 0 , 1 } -> Fun ( G ` A ) )
46	44, 45	syl 17	. . . 4 |- ( A e. ( T i^i R ) -> Fun ( G ` A ) )
47		ssv 3625	. . . . 5 |- NN0 C_ _V
48		dfn2 11305	. . . . . 6 |- NN = ( NN0 \ { 0 } )
49		ssdif 3745	. . . . . 6 |- ( NN0 C_ _V -> ( NN0 \ { 0 } ) C_ ( _V \ { 0 } ) )
50	48, 49	syl5eqss 3649	. . . . 5 |- ( NN0 C_ _V -> NN C_ ( _V \ { 0 } ) )
51	47, 50	ax-mp 5	. . . 4 |- NN C_ ( _V \ { 0 } )
52		sspreima 29447	. . . 4 |- ( ( Fun ( G ` A ) /\ NN C_ ( _V \ { 0 } ) ) -> ( `' ( G ` A ) " NN ) C_ ( `' ( G ` A ) " ( _V \ { 0 } ) ) )
53	46, 51, 52	sylancl 694	. . 3 |- ( A e. ( T i^i R ) -> ( `' ( G ` A ) " NN ) C_ ( `' ( G ` A ) " ( _V \ { 0 } ) ) )
54		fvex 6201	. . . . 5 |- ( G ` A ) e. _V
55		0nn0 11307	. . . . 5 |- 0 e. NN0
56		suppimacnv 7306	. . . . 5 |- ( ( ( G ` A ) e. _V /\ 0 e. NN0 ) -> ( ( G ` A ) supp 0 ) = ( `' ( G ` A ) " ( _V \ { 0 } ) ) )
57	54, 55, 56	mp2an 708	. . . 4 |- ( ( G ` A ) supp 0 ) = ( `' ( G ` A ) " ( _V \ { 0 } ) )
58		0ne1 11088	. . . . . . . . 9 |- 0 =/= 1
59		difprsn1 4330	. . . . . . . . 9 |- ( 0 =/= 1 -> ( { 0 , 1 } \ { 0 } ) = { 1 } )
60	58, 59	ax-mp 5	. . . . . . . 8 |- ( { 0 , 1 } \ { 0 } ) = { 1 }
61	60	eqcomi 2631	. . . . . . 7 |- { 1 } = ( { 0 , 1 } \ { 0 } )
62	61	ffs2 29503	. . . . . 6 |- ( ( NN e. _V /\ 0 e. NN0 /\ ( G ` A ) : NN --> { 0 , 1 } ) -> ( ( G ` A ) supp 0 ) = ( `' ( G ` A ) " { 1 } ) )
63	14, 55, 62	mp3an12 1414	. . . . 5 |- ( ( G ` A ) : NN --> { 0 , 1 } -> ( ( G ` A ) supp 0 ) = ( `' ( G ` A ) " { 1 } ) )
64	44, 63	syl 17	. . . 4 |- ( A e. ( T i^i R ) -> ( ( G ` A ) supp 0 ) = ( `' ( G ` A ) " { 1 } ) )
65	57, 64	syl5eqr 2670	. . 3 |- ( A e. ( T i^i R ) -> ( `' ( G ` A ) " ( _V \ { 0 } ) ) = ( `' ( G ` A ) " { 1 } ) )
66	53, 65	sseqtrd 3641	. 2 |- ( A e. ( T i^i R ) -> ( `' ( G ` A ) " NN ) C_ ( `' ( G ` A ) " { 1 } ) )
67		ssfi 8180	. 2 |- ( ( ( `' ( G ` A ) " { 1 } ) e. Fin /\ ( `' ( G ` A ) " NN ) C_ ( `' ( G ` A ) " { 1 } ) ) -> ( `' ( G ` A ) " NN ) e. Fin )
68	36, 66, 67	syl2anc 693	1 |- ( A e. ( T i^i R ) -> ( `' ( G ` A ) " NN ) e. Fin )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   {cpr 4179   class class class wbr 4653   {copab 4712   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Fun wfun 5882   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   supp csupp 7295   ^m cmap 7857   Fincfn 7955   0cc0 9936   1c1 9937   x. cmul 9941   <_ cle 10075   NNcn 11020   2c2 11070   NN0cn0 11292   ^cexp 12860   sum_csu 14416   || cdvds 14983  bitscbits 15141  𝟭cind 30072

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-inf2 8538  ax-ac2 9285  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  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-rmo 2920  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-int 4476  df-iun 4522  df-disj 4621  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-bits 15144  df-ind 30073

This theorem is referenced by:  eulerpartlemgs2  30442