Theorem eulerpartlemmf 27982

Description: Lemma for eulerpart 27989 (Contributed by Thierry Arnoux, 30-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.)

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
eulerpartlemmf	|- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. H )

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

Proof of Theorem eulerpartlemmf

Step	Hyp	Ref	Expression
1		bitsf1o 13954	. . . . . 6 |- (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin )
2		f1of 5816	. . . . . 6 |- ( (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin ) -> (bits |` NN0 ) : NN0 --> ( ~P NN0 i^i Fin ) )
3	1, 2	ax-mp 5	. . . . 5 |- (bits |` NN0 ) : NN0 --> ( ~P NN0 i^i Fin )
4		eulerpart.p	. . . . . . . . . 10 |- P = { f e. ( NN0 ^m NN ) | ( ( `' f " NN ) e. Fin /\ sum_ k e. NN ( ( f ` k ) x. k ) = N ) }
5		eulerpart.o	. . . . . . . . . 10 |- O = { g e. P | A. n e. ( `' g " NN ) -. 2 || n }
6		eulerpart.d	. . . . . . . . . 10 |- D = { g e. P | A. n e. NN ( g ` n ) <_ 1 }
7		eulerpart.j	. . . . . . . . . 10 |- J = { z e. NN | -. 2 || z }
8		eulerpart.f	. . . . . . . . . 10 |- F = ( x e. J , y e. NN0 |-> ( ( 2 ^ y ) x. x ) )
9		eulerpart.h	. . . . . . . . . 10 |- H = { r e. ( ( ~P NN0 i^i Fin ) ^m J ) | ( r supp (/) ) e. Fin }
10		eulerpart.m	. . . . . . . . . 10 |- M = ( r e. H |-> { <. x , y >. | ( x e. J /\ y e. ( r ` x ) ) } )
11		eulerpart.r	. . . . . . . . . 10 |- R = { f | ( `' f " NN ) e. Fin }
12		eulerpart.t	. . . . . . . . . 10 |- T = { f e. ( NN0 ^m NN ) | ( `' f " NN ) C_ J }
13	4, 5, 6, 7, 8, 9, 10, 11, 12	eulerpartlemt0 27976	. . . . . . . . 9 |- ( A e. ( T i^i R ) <-> ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) )
14	13	biimpi 194	. . . . . . . 8 |- ( A e. ( T i^i R ) -> ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) )
15	14	simp1d 1008	. . . . . . 7 |- ( A e. ( T i^i R ) -> A e. ( NN0 ^m NN ) )
16		nn0ex 10801	. . . . . . . 8 |- NN0 e. _V
17		nnex 10542	. . . . . . . 8 |- NN e. _V
18	16, 17	elmap 7447	. . . . . . 7 |- ( A e. ( NN0 ^m NN ) <-> A : NN --> NN0 )
19	15, 18	sylib 196	. . . . . 6 |- ( A e. ( T i^i R ) -> A : NN --> NN0 )
20		ssrab2 3585	. . . . . . 7 |- { z e. NN | -. 2 || z } C_ NN
21	7, 20	eqsstri 3534	. . . . . 6 |- J C_ NN
22		fssres 5751	. . . . . 6 |- ( ( A : NN --> NN0 /\ J C_ NN ) -> ( A |` J ) : J --> NN0 )
23	19, 21, 22	sylancl 662	. . . . 5 |- ( A e. ( T i^i R ) -> ( A |` J ) : J --> NN0 )
24		fco2 5742	. . . . 5 |- ( ( (bits |` NN0 ) : NN0 --> ( ~P NN0 i^i Fin ) /\ ( A |` J ) : J --> NN0 ) -> (bits o. ( A |` J ) ) : J --> ( ~P NN0 i^i Fin ) )
25	3, 23, 24	sylancr 663	. . . 4 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) : J --> ( ~P NN0 i^i Fin ) )
26	16	pwex 4630	. . . . . 6 |- ~P NN0 e. _V
27	26	inex1 4588	. . . . 5 |- ( ~P NN0 i^i Fin ) e. _V
28	17, 21	ssexi 4592	. . . . 5 |- J e. _V
29	27, 28	elmap 7447	. . . 4 |- ( (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) <-> (bits o. ( A |` J ) ) : J --> ( ~P NN0 i^i Fin ) )
30	25, 29	sylibr 212	. . 3 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) )
31	14	simp2d 1009	. . . . . 6 |- ( A e. ( T i^i R ) -> ( `' A " NN ) e. Fin )
32		0nn0 10810	. . . . . . . . . 10 |- 0 e. NN0
33		suppimacnv 6912	. . . . . . . . . 10 |- ( ( A e. ( T i^i R ) /\ 0 e. NN0 ) -> ( A supp 0 ) = ( `' A " ( _V \ { 0 } ) ) )
34	32, 33	mpan2 671	. . . . . . . . 9 |- ( A e. ( T i^i R ) -> ( A supp 0 ) = ( `' A " ( _V \ { 0 } ) ) )
35		frnsuppeq 6913	. . . . . . . . . . 11 |- ( ( NN e. _V /\ 0 e. NN0 ) -> ( A : NN --> NN0 -> ( A supp 0 ) = ( `' A " ( NN0 \ { 0 } ) ) ) )
36	17, 32, 35	mp2an 672	. . . . . . . . . 10 |- ( A : NN --> NN0 -> ( A supp 0 ) = ( `' A " ( NN0 \ { 0 } ) ) )
37	19, 36	syl 16	. . . . . . . . 9 |- ( A e. ( T i^i R ) -> ( A supp 0 ) = ( `' A " ( NN0 \ { 0 } ) ) )
38	34, 37	eqtr3d 2510	. . . . . . . 8 |- ( A e. ( T i^i R ) -> ( `' A " ( _V \ { 0 } ) ) = ( `' A " ( NN0 \ { 0 } ) ) )
39	38	eleq1d 2536	. . . . . . 7 |- ( A e. ( T i^i R ) -> ( ( `' A " ( _V \ { 0 } ) ) e. Fin <-> ( `' A " ( NN0 \ { 0 } ) ) e. Fin ) )
40		dfn2 10808	. . . . . . . . 9 |- NN = ( NN0 \ { 0 } )
41	40	imaeq2i 5335	. . . . . . . 8 |- ( `' A " NN ) = ( `' A " ( NN0 \ { 0 } ) )
42	41	eleq1i 2544	. . . . . . 7 |- ( ( `' A " NN ) e. Fin <-> ( `' A " ( NN0 \ { 0 } ) ) e. Fin )
43	39, 42	syl6bbr 263	. . . . . 6 |- ( A e. ( T i^i R ) -> ( ( `' A " ( _V \ { 0 } ) ) e. Fin <-> ( `' A " NN ) e. Fin ) )
44	31, 43	mpbird 232	. . . . 5 |- ( A e. ( T i^i R ) -> ( `' A " ( _V \ { 0 } ) ) e. Fin )
45		resss 5297	. . . . . . 7 |- ( A |` J ) C_ A
46		cnvss 5175	. . . . . . 7 |- ( ( A |` J ) C_ A -> `' ( A |` J ) C_ `' A )
47		imass1 5371	. . . . . . 7 |- ( `' ( A |` J ) C_ `' A -> ( `' ( A |` J ) " ( _V \ { 0 } ) ) C_ ( `' A " ( _V \ { 0 } ) ) )
48	45, 46, 47	mp2b 10	. . . . . 6 |- ( `' ( A |` J ) " ( _V \ { 0 } ) ) C_ ( `' A " ( _V \ { 0 } ) )
49		ssfi 7740	. . . . . 6 |- ( ( ( `' A " ( _V \ { 0 } ) ) e. Fin /\ ( `' ( A |` J ) " ( _V \ { 0 } ) ) C_ ( `' A " ( _V \ { 0 } ) ) ) -> ( `' ( A |` J ) " ( _V \ { 0 } ) ) e. Fin )
50	48, 49	mpan2 671	. . . . 5 |- ( ( `' A " ( _V \ { 0 } ) ) e. Fin -> ( `' ( A |` J ) " ( _V \ { 0 } ) ) e. Fin )
51	44, 50	syl 16	. . . 4 |- ( A e. ( T i^i R ) -> ( `' ( A |` J ) " ( _V \ { 0 } ) ) e. Fin )
52		cnvco 5188	. . . . . . 7 |- `' (bits o. ( A |` J ) ) = ( `' ( A |` J ) o. `'bits )
53	52	imaeq1i 5334	. . . . . 6 |- ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) = ( ( `' ( A |` J ) o. `'bits ) " ( _V \ { (/) } ) )
54		imaco 5512	. . . . . 6 |- ( ( `' ( A |` J ) o. `'bits ) " ( _V \ { (/) } ) ) = ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) )
55	53, 54	eqtri 2496	. . . . 5 |- ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) = ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) )
56		ffun 5733	. . . . . . 7 |- ( A : NN --> NN0 -> Fun A )
57		funres 5627	. . . . . . 7 |- ( Fun A -> Fun ( A |` J ) )
58	19, 56, 57	3syl 20	. . . . . 6 |- ( A e. ( T i^i R ) -> Fun ( A |` J ) )
59		ssv 3524	. . . . . . . 8 |- ( `'bits " _V ) C_ _V
60		ssdif 3639	. . . . . . . 8 |- ( ( `'bits " _V ) C_ _V -> ( ( `'bits " _V ) \ ( `'bits " { (/) } ) ) C_ ( _V \ ( `'bits " { (/) } ) ) )
61	59, 60	ax-mp 5	. . . . . . 7 |- ( ( `'bits " _V ) \ ( `'bits " { (/) } ) ) C_ ( _V \ ( `'bits " { (/) } ) )
62		bitsf 13936	. . . . . . . . 9 |- bits : ZZ --> ~P NN0
63		ffun 5733	. . . . . . . . 9 |- (bits : ZZ --> ~P NN0 -> Fun bits )
64	62, 63	ax-mp 5	. . . . . . . 8 |- Fun bits
65		difpreima 6009	. . . . . . . 8 |- ( Fun bits -> ( `'bits " ( _V \ { (/) } ) ) = ( ( `'bits " _V ) \ ( `'bits " { (/) } ) ) )
66	64, 65	ax-mp 5	. . . . . . 7 |- ( `'bits " ( _V \ { (/) } ) ) = ( ( `'bits " _V ) \ ( `'bits " { (/) } ) )
67		bitsf1 13955	. . . . . . . . . 10 |- bits : ZZ -1-1-> ~P NN0
68		0z 10875	. . . . . . . . . . 11 |- 0 e. ZZ
69		snssi 4171	. . . . . . . . . . 11 |- ( 0 e. ZZ -> { 0 } C_ ZZ )
70	68, 69	ax-mp 5	. . . . . . . . . 10 |- { 0 } C_ ZZ
71		f1imacnv 5832	. . . . . . . . . 10 |- ( (bits : ZZ -1-1-> ~P NN0 /\ { 0 } C_ ZZ ) -> ( `'bits " (bits " { 0 } ) ) = { 0 } )
72	67, 70, 71	mp2an 672	. . . . . . . . 9 |- ( `'bits " (bits " { 0 } ) ) = { 0 }
73		ffn 5731	. . . . . . . . . . . . 13 |- (bits : ZZ --> ~P NN0 -> bits Fn ZZ )
74	62, 73	ax-mp 5	. . . . . . . . . . . 12 |- bits Fn ZZ
75		fnsnfv 5927	. . . . . . . . . . . 12 |- ( (bits Fn ZZ /\ 0 e. ZZ ) -> { (bits ` 0 ) } = (bits " { 0 } ) )
76	74, 68, 75	mp2an 672	. . . . . . . . . . 11 |- { (bits ` 0 ) } = (bits " { 0 } )
77		0bits 13948	. . . . . . . . . . . 12 |- (bits ` 0 ) = (/)
78	77	sneqi 4038	. . . . . . . . . . 11 |- { (bits ` 0 ) } = { (/) }
79	76, 78	eqtr3i 2498	. . . . . . . . . 10 |- (bits " { 0 } ) = { (/) }
80	79	imaeq2i 5335	. . . . . . . . 9 |- ( `'bits " (bits " { 0 } ) ) = ( `'bits " { (/) } )
81	72, 80	eqtr3i 2498	. . . . . . . 8 |- { 0 } = ( `'bits " { (/) } )
82	81	difeq2i 3619	. . . . . . 7 |- ( _V \ { 0 } ) = ( _V \ ( `'bits " { (/) } ) )
83	61, 66, 82	3sstr4i 3543	. . . . . 6 |- ( `'bits " ( _V \ { (/) } ) ) C_ ( _V \ { 0 } )
84		sspreima 27185	. . . . . 6 |- ( ( Fun ( A |` J ) /\ ( `'bits " ( _V \ { (/) } ) ) C_ ( _V \ { 0 } ) ) -> ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) )
85	58, 83, 84	sylancl 662	. . . . 5 |- ( A e. ( T i^i R ) -> ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) )
86	55, 85	syl5eqss 3548	. . . 4 |- ( A e. ( T i^i R ) -> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) )
87		ssfi 7740	. . . 4 |- ( ( ( `' ( A |` J ) " ( _V \ { 0 } ) ) e. Fin /\ ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) ) -> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin )
88	51, 86, 87	syl2anc 661	. . 3 |- ( A e. ( T i^i R ) -> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin )
89	30, 88	jca 532	. 2 |- ( A e. ( T i^i R ) -> ( (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) /\ ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin ) )
90		oveq1 6291	. . . . 5 |- ( r = (bits o. ( A |` J ) ) -> ( r supp (/) ) = ( (bits o. ( A |` J ) ) supp (/) ) )
91	90	eleq1d 2536	. . . 4 |- ( r = (bits o. ( A |` J ) ) -> ( ( r supp (/) ) e. Fin <-> ( (bits o. ( A |` J ) ) supp (/) ) e. Fin ) )
92	91, 9	elrab2 3263	. . 3 |- ( (bits o. ( A |` J ) ) e. H <-> ( (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) /\ ( (bits o. ( A |` J ) ) supp (/) ) e. Fin ) )
93		zex 10873	. . . . . 6 |- ZZ e. _V
94		fex 6133	. . . . . 6 |- ( (bits : ZZ --> ~P NN0 /\ ZZ e. _V ) -> bits e. _V )
95	62, 93, 94	mp2an 672	. . . . 5 |- bits e. _V
96		resexg 5316	. . . . 5 |- ( A e. ( T i^i R ) -> ( A |` J ) e. _V )
97		coexg 6735	. . . . 5 |- ( (bits e. _V /\ ( A |` J ) e. _V ) -> (bits o. ( A |` J ) ) e. _V )
98	95, 96, 97	sylancr 663	. . . 4 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. _V )
99		0ex 4577	. . . . . . 7 |- (/) e. _V
100		suppimacnv 6912	. . . . . . 7 |- ( ( (bits o. ( A |` J ) ) e. _V /\ (/) e. _V ) -> ( (bits o. ( A |` J ) ) supp (/) ) = ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) )
101	99, 100	mpan2 671	. . . . . 6 |- ( (bits o. ( A |` J ) ) e. _V -> ( (bits o. ( A |` J ) ) supp (/) ) = ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) )
102	101	eleq1d 2536	. . . . 5 |- ( (bits o. ( A |` J ) ) e. _V -> ( ( (bits o. ( A |` J ) ) supp (/) ) e. Fin <-> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin ) )
103	102	anbi2d 703	. . . 4 |- ( (bits o. ( A |` J ) ) e. _V -> ( ( (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) /\ ( (bits o. ( A |` J ) ) supp (/) ) e. Fin ) <-> ( (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) /\ ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin ) ) )
104	98, 103	syl 16	. . 3 |- ( A e. ( T i^i R ) -> ( ( (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) /\ ( (bits o. ( A |` J ) ) supp (/) ) e. Fin ) <-> ( (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) /\ ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin ) ) )
105	92, 104	syl5bb 257	. 2 |- ( A e. ( T i^i R ) -> ( (bits o. ( A |` J ) ) e. H <-> ( (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) /\ ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin ) ) )
106	89, 105	mpbird 232	1 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. H )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   {cab 2452   A.wral 2814   {crab 2818   _Vcvv 3113   \ cdif 3473   i^i cin 3475   C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027   class class class wbr 4447   {copab 4504   |-> cmpt 4505   `'ccnv 4998   |` cres 5001   "cima 5002   o. ccom 5003   Fun wfun 5582   Fn wfn 5583   -->wf 5584   -1-1->wf1 5585   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284   |-> cmpt2 6286   supp csupp 6901   ^m cmap 7420   Fincfn 7516   0cc0 9492   1c1 9493   x. cmul 9497   <_ cle 9629   NNcn 10536   2c2 10585   NN0cn0 10795   ZZcz 10864   ^cexp 12134   sum_csu 13471   || cdivides 13847  bitscbits 13928  𝟭cind 27692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472  df-dvds 13848  df-bits 13931

This theorem is referenced by:  eulerpartlemgvv  27983  eulerpartlemgf  27986