Theorem eulerpartlemmf 29031

Description: Lemma for eulerpart 29038 (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 14382	. . . . 5 |- (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin )
2		f1of 5822	. . . . 5 |- ( (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	. . . 4 |- (bits |` NN0 ) : NN0 --> ( ~P NN0 i^i Fin )
4		eulerpart.p	. . . . . . . . 9 |- P = { f e. ( NN0 ^m NN ) | ( ( `' f " NN ) e. Fin /\ sum_ k e. NN ( ( f ` k ) x. k ) = N ) }
5		eulerpart.o	. . . . . . . . 9 |- O = { g e. P | A. n e. ( `' g " NN ) -. 2 || n }
6		eulerpart.d	. . . . . . . . 9 |- D = { g e. P | A. n e. NN ( g ` n ) <_ 1 }
7		eulerpart.j	. . . . . . . . 9 |- J = { z e. NN | -. 2 || z }
8		eulerpart.f	. . . . . . . . 9 |- F = ( x e. J , y e. NN0 |-> ( ( 2 ^ y ) x. x ) )
9		eulerpart.h	. . . . . . . . 9 |- H = { r e. ( ( ~P NN0 i^i Fin ) ^m J ) | ( r supp (/) ) e. Fin }
10		eulerpart.m	. . . . . . . . 9 |- M = ( r e. H |-> { <. x , y >. | ( x e. J /\ y e. ( r ` x ) ) } )
11		eulerpart.r	. . . . . . . . 9 |- R = { f | ( `' f " NN ) e. Fin }
12		eulerpart.t	. . . . . . . . 9 |- T = { f e. ( NN0 ^m NN ) | ( `' f " NN ) C_ J }
13	4, 5, 6, 7, 8, 9, 10, 11, 12	eulerpartlemt0 29025	. . . . . . . 8 |- ( A e. ( T i^i R ) <-> ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) )
14	13	biimpi 197	. . . . . . 7 |- ( A e. ( T i^i R ) -> ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) )
15	14	simp1d 1017	. . . . . 6 |- ( A e. ( T i^i R ) -> A e. ( NN0 ^m NN ) )
16		nn0ex 10864	. . . . . . 7 |- NN0 e. _V
17		nnex 10604	. . . . . . 7 |- NN e. _V
18	16, 17	elmap 7499	. . . . . 6 |- ( A e. ( NN0 ^m NN ) <-> A : NN --> NN0 )
19	15, 18	sylib 199	. . . . 5 |- ( A e. ( T i^i R ) -> A : NN --> NN0 )
20		ssrab2 3543	. . . . . 6 |- { z e. NN | -. 2 || z } C_ NN
21	7, 20	eqsstri 3491	. . . . 5 |- J C_ NN
22		fssres 5757	. . . . 5 |- ( ( A : NN --> NN0 /\ J C_ NN ) -> ( A |` J ) : J --> NN0 )
23	19, 21, 22	sylancl 666	. . . 4 |- ( A e. ( T i^i R ) -> ( A |` J ) : J --> NN0 )
24		fco2 5748	. . . 4 |- ( ( (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 667	. . 3 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) : J --> ( ~P NN0 i^i Fin ) )
26	16	pwex 4599	. . . . 5 |- ~P NN0 e. _V
27	26	inex1 4557	. . . 4 |- ( ~P NN0 i^i Fin ) e. _V
28	17, 21	ssexi 4561	. . . 4 |- J e. _V
29	27, 28	elmap 7499	. . 3 |- ( (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 215	. 2 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) )
31	14	simp2d 1018	. . . . 5 |- ( A e. ( T i^i R ) -> ( `' A " NN ) e. Fin )
32		0nn0 10873	. . . . . . . . 9 |- 0 e. NN0
33		suppimacnv 6927	. . . . . . . . 9 |- ( ( A e. ( T i^i R ) /\ 0 e. NN0 ) -> ( A supp 0 ) = ( `' A " ( _V \ { 0 } ) ) )
34	32, 33	mpan2 675	. . . . . . . 8 |- ( A e. ( T i^i R ) -> ( A supp 0 ) = ( `' A " ( _V \ { 0 } ) ) )
35		frnsuppeq 6928	. . . . . . . . . 10 |- ( ( NN e. _V /\ 0 e. NN0 ) -> ( A : NN --> NN0 -> ( A supp 0 ) = ( `' A " ( NN0 \ { 0 } ) ) ) )
36	17, 32, 35	mp2an 676	. . . . . . . . 9 |- ( A : NN --> NN0 -> ( A supp 0 ) = ( `' A " ( NN0 \ { 0 } ) ) )
37	19, 36	syl 17	. . . . . . . 8 |- ( A e. ( T i^i R ) -> ( A supp 0 ) = ( `' A " ( NN0 \ { 0 } ) ) )
38	34, 37	eqtr3d 2463	. . . . . . 7 |- ( A e. ( T i^i R ) -> ( `' A " ( _V \ { 0 } ) ) = ( `' A " ( NN0 \ { 0 } ) ) )
39	38	eleq1d 2489	. . . . . 6 |- ( A e. ( T i^i R ) -> ( ( `' A " ( _V \ { 0 } ) ) e. Fin <-> ( `' A " ( NN0 \ { 0 } ) ) e. Fin ) )
40		dfn2 10871	. . . . . . . 8 |- NN = ( NN0 \ { 0 } )
41	40	imaeq2i 5177	. . . . . . 7 |- ( `' A " NN ) = ( `' A " ( NN0 \ { 0 } ) )
42	41	eleq1i 2497	. . . . . 6 |- ( ( `' A " NN ) e. Fin <-> ( `' A " ( NN0 \ { 0 } ) ) e. Fin )
43	39, 42	syl6bbr 266	. . . . 5 |- ( A e. ( T i^i R ) -> ( ( `' A " ( _V \ { 0 } ) ) e. Fin <-> ( `' A " NN ) e. Fin ) )
44	31, 43	mpbird 235	. . . 4 |- ( A e. ( T i^i R ) -> ( `' A " ( _V \ { 0 } ) ) e. Fin )
45		resss 5139	. . . . 5 |- ( A |` J ) C_ A
46		cnvss 5018	. . . . 5 |- ( ( A |` J ) C_ A -> `' ( A |` J ) C_ `' A )
47		imass1 5214	. . . . 5 |- ( `' ( A |` J ) C_ `' A -> ( `' ( A |` J ) " ( _V \ { 0 } ) ) C_ ( `' A " ( _V \ { 0 } ) ) )
48	45, 46, 47	mp2b 10	. . . 4 |- ( `' ( A |` J ) " ( _V \ { 0 } ) ) C_ ( `' A " ( _V \ { 0 } ) )
49		ssfi 7789	. . . 4 |- ( ( ( `' A " ( _V \ { 0 } ) ) e. Fin /\ ( `' ( A |` J ) " ( _V \ { 0 } ) ) C_ ( `' A " ( _V \ { 0 } ) ) ) -> ( `' ( A |` J ) " ( _V \ { 0 } ) ) e. Fin )
50	44, 48, 49	sylancl 666	. . 3 |- ( A e. ( T i^i R ) -> ( `' ( A |` J ) " ( _V \ { 0 } ) ) e. Fin )
51		cnvco 5031	. . . . . 6 |- `' (bits o. ( A |` J ) ) = ( `' ( A |` J ) o. `'bits )
52	51	imaeq1i 5176	. . . . 5 |- ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) = ( ( `' ( A |` J ) o. `'bits ) " ( _V \ { (/) } ) )
53		imaco 5351	. . . . 5 |- ( ( `' ( A |` J ) o. `'bits ) " ( _V \ { (/) } ) ) = ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) )
54	52, 53	eqtri 2449	. . . 4 |- ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) = ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) )
55		ffun 5739	. . . . . 6 |- ( A : NN --> NN0 -> Fun A )
56		funres 5631	. . . . . 6 |- ( Fun A -> Fun ( A |` J ) )
57	19, 55, 56	3syl 18	. . . . 5 |- ( A e. ( T i^i R ) -> Fun ( A |` J ) )
58		ssv 3481	. . . . . . 7 |- ( `'bits " _V ) C_ _V
59		ssdif 3597	. . . . . . 7 |- ( ( `'bits " _V ) C_ _V -> ( ( `'bits " _V ) \ ( `'bits " { (/) } ) ) C_ ( _V \ ( `'bits " { (/) } ) ) )
60	58, 59	ax-mp 5	. . . . . 6 |- ( ( `'bits " _V ) \ ( `'bits " { (/) } ) ) C_ ( _V \ ( `'bits " { (/) } ) )
61		bitsf 14364	. . . . . . 7 |- bits : ZZ --> ~P NN0
62		ffun 5739	. . . . . . 7 |- (bits : ZZ --> ~P NN0 -> Fun bits )
63		difpreima 6014	. . . . . . 7 |- ( Fun bits -> ( `'bits " ( _V \ { (/) } ) ) = ( ( `'bits " _V ) \ ( `'bits " { (/) } ) ) )
64	61, 62, 63	mp2b 10	. . . . . 6 |- ( `'bits " ( _V \ { (/) } ) ) = ( ( `'bits " _V ) \ ( `'bits " { (/) } ) )
65		bitsf1 14383	. . . . . . . . 9 |- bits : ZZ -1-1-> ~P NN0
66		0z 10937	. . . . . . . . . 10 |- 0 e. ZZ
67		snssi 4138	. . . . . . . . . 10 |- ( 0 e. ZZ -> { 0 } C_ ZZ )
68	66, 67	ax-mp 5	. . . . . . . . 9 |- { 0 } C_ ZZ
69		f1imacnv 5838	. . . . . . . . 9 |- ( (bits : ZZ -1-1-> ~P NN0 /\ { 0 } C_ ZZ ) -> ( `'bits " (bits " { 0 } ) ) = { 0 } )
70	65, 68, 69	mp2an 676	. . . . . . . 8 |- ( `'bits " (bits " { 0 } ) ) = { 0 }
71		ffn 5737	. . . . . . . . . . . 12 |- (bits : ZZ --> ~P NN0 -> bits Fn ZZ )
72	61, 71	ax-mp 5	. . . . . . . . . . 11 |- bits Fn ZZ
73		fnsnfv 5932	. . . . . . . . . . 11 |- ( (bits Fn ZZ /\ 0 e. ZZ ) -> { (bits ` 0 ) } = (bits " { 0 } ) )
74	72, 66, 73	mp2an 676	. . . . . . . . . 10 |- { (bits ` 0 ) } = (bits " { 0 } )
75		0bits 14376	. . . . . . . . . . 11 |- (bits ` 0 ) = (/)
76	75	sneqi 4004	. . . . . . . . . 10 |- { (bits ` 0 ) } = { (/) }
77	74, 76	eqtr3i 2451	. . . . . . . . 9 |- (bits " { 0 } ) = { (/) }
78	77	imaeq2i 5177	. . . . . . . 8 |- ( `'bits " (bits " { 0 } ) ) = ( `'bits " { (/) } )
79	70, 78	eqtr3i 2451	. . . . . . 7 |- { 0 } = ( `'bits " { (/) } )
80	79	difeq2i 3577	. . . . . 6 |- ( _V \ { 0 } ) = ( _V \ ( `'bits " { (/) } ) )
81	60, 64, 80	3sstr4i 3500	. . . . 5 |- ( `'bits " ( _V \ { (/) } ) ) C_ ( _V \ { 0 } )
82		sspreima 28083	. . . . 5 |- ( ( Fun ( A |` J ) /\ ( `'bits " ( _V \ { (/) } ) ) C_ ( _V \ { 0 } ) ) -> ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) )
83	57, 81, 82	sylancl 666	. . . 4 |- ( A e. ( T i^i R ) -> ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) )
84	54, 83	syl5eqss 3505	. . 3 |- ( A e. ( T i^i R ) -> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) )
85		ssfi 7789	. . 3 |- ( ( ( `' ( A |` J ) " ( _V \ { 0 } ) ) e. Fin /\ ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) ) -> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin )
86	50, 84, 85	syl2anc 665	. 2 |- ( A e. ( T i^i R ) -> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin )
87		oveq1 6303	. . . . 5 |- ( r = (bits o. ( A |` J ) ) -> ( r supp (/) ) = ( (bits o. ( A |` J ) ) supp (/) ) )
88	87	eleq1d 2489	. . . 4 |- ( r = (bits o. ( A |` J ) ) -> ( ( r supp (/) ) e. Fin <-> ( (bits o. ( A |` J ) ) supp (/) ) e. Fin ) )
89	88, 9	elrab2 3228	. . 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 ) )
90		zex 10935	. . . . . 6 |- ZZ e. _V
91		fex 6144	. . . . . 6 |- ( (bits : ZZ --> ~P NN0 /\ ZZ e. _V ) -> bits e. _V )
92	61, 90, 91	mp2an 676	. . . . 5 |- bits e. _V
93		resexg 5158	. . . . 5 |- ( A e. ( T i^i R ) -> ( A |` J ) e. _V )
94		coexg 6749	. . . . 5 |- ( (bits e. _V /\ ( A |` J ) e. _V ) -> (bits o. ( A |` J ) ) e. _V )
95	92, 93, 94	sylancr 667	. . . 4 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. _V )
96		0ex 4548	. . . . . . 7 |- (/) e. _V
97		suppimacnv 6927	. . . . . . 7 |- ( ( (bits o. ( A |` J ) ) e. _V /\ (/) e. _V ) -> ( (bits o. ( A |` J ) ) supp (/) ) = ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) )
98	96, 97	mpan2 675	. . . . . 6 |- ( (bits o. ( A |` J ) ) e. _V -> ( (bits o. ( A |` J ) ) supp (/) ) = ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) )
99	98	eleq1d 2489	. . . . 5 |- ( (bits o. ( A |` J ) ) e. _V -> ( ( (bits o. ( A |` J ) ) supp (/) ) e. Fin <-> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin ) )
100	99	anbi2d 708	. . . 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 ) ) )
101	95, 100	syl 17	. . 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 ) ) )
102	89, 101	syl5bb 260	. 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 ) ) )
103	30, 86, 102	mpbir2and 930	1 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. H )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1867   {cab 2405   A.wral 2773   {crab 2777   _Vcvv 3078   \ cdif 3430   i^i cin 3432   C_ wss 3433   (/)c0 3758   ~Pcpw 3976   {csn 3993   class class class wbr 4417   {copab 4474   |-> cmpt 4475   `'ccnv 4844   |` cres 4847   "cima 4848   o. ccom 4849   Fun wfun 5586   Fn wfn 5587   -->wf 5588   -1-1->wf1 5589   -1-1-onto->wf1o 5591   ` cfv 5592  (class class class)co 6296   |-> cmpt2 6298   supp csupp 6916   ^m cmap 7471   Fincfn 7568   0cc0 9528   1c1 9529   x. cmul 9533   <_ cle 9665   NNcn 10598   2c2 10648   NN0cn0 10858   ZZcz 10926   ^cexp 12258   sum_csu 13719   || cdvds 14272  bitscbits 14356  𝟭cind 28668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-disj 4389  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6917  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-oi 8016  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-fz 11772  df-fzo 11903  df-fl 12014  df-mod 12083  df-seq 12200  df-exp 12259  df-hash 12502  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-clim 13519  df-sum 13720  df-dvds 14273  df-bits 14359

This theorem is referenced by:  eulerpartlemgvv  29032  eulerpartlemgf  29035