Theorem eulerpartlemf 30432

Description: Lemma for eulerpart 30444: Odd partitions are zero for even numbers. (Contributed by Thierry Arnoux, 9-Sep-2017.)

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 }

Assertion

Ref	Expression
eulerpartlemf	|- ( ( A e. ( T i^i R ) /\ t e. ( NN \ J ) ) -> ( A ` t ) = 0 )

Distinct variable groups:   z, t   f, g, k, n, t, A   f, J   f, N   P, g

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

Proof of Theorem eulerpartlemf

Step	Hyp	Ref	Expression
1		eldif 3584	. . . . . 6 |- ( t e. ( NN \ J ) <-> ( t e. NN /\ -. t e. J ) )
2		breq2 4657	. . . . . . . . . . 11 |- ( z = t -> ( 2 || z <-> 2 || t ) )
3	2	notbid 308	. . . . . . . . . 10 |- ( z = t -> ( -. 2 || z <-> -. 2 || t ) )
4		eulerpart.j	. . . . . . . . . 10 |- J = { z e. NN | -. 2 || z }
5	3, 4	elrab2 3366	. . . . . . . . 9 |- ( t e. J <-> ( t e. NN /\ -. 2 || t ) )
6	5	simplbi2 655	. . . . . . . 8 |- ( t e. NN -> ( -. 2 || t -> t e. J ) )
7	6	con1d 139	. . . . . . 7 |- ( t e. NN -> ( -. t e. J -> 2 || t ) )
8	7	imp 445	. . . . . 6 |- ( ( t e. NN /\ -. t e. J ) -> 2 || t )
9	1, 8	sylbi 207	. . . . 5 |- ( t e. ( NN \ J ) -> 2 || t )
10	9	adantl 482	. . . 4 |- ( ( A e. ( T i^i R ) /\ t e. ( NN \ J ) ) -> 2 || t )
11	10	adantr 481	. . 3 |- ( ( ( A e. ( T i^i R ) /\ t e. ( NN \ J ) ) /\ ( A ` t ) e. NN ) -> 2 || t )
12		simpll 790	. . . 4 |- ( ( ( A e. ( T i^i R ) /\ t e. ( NN \ J ) ) /\ ( A ` t ) e. NN ) -> A e. ( T i^i R ) )
13		eldifi 3732	. . . . . 6 |- ( t e. ( NN \ J ) -> t e. NN )
14		eulerpart.p	. . . . . . . . . . 11 |- P = { f e. ( NN0 ^m NN ) | ( ( `' f " NN ) e. Fin /\ sum_ k e. NN ( ( f ` k ) x. k ) = N ) }
15		eulerpart.o	. . . . . . . . . . 11 |- O = { g e. P | A. n e. ( `' g " NN ) -. 2 || n }
16		eulerpart.d	. . . . . . . . . . 11 |- D = { g e. P | A. n e. NN ( g ` n ) <_ 1 }
17		eulerpart.f	. . . . . . . . . . 11 |- F = ( x e. J , y e. NN0 |-> ( ( 2 ^ y ) x. x ) )
18		eulerpart.h	. . . . . . . . . . 11 |- H = { r e. ( ( ~P NN0 i^i Fin ) ^m J ) | ( r supp (/) ) e. Fin }
19		eulerpart.m	. . . . . . . . . . 11 |- M = ( r e. H |-> { <. x , y >. | ( x e. J /\ y e. ( r ` x ) ) } )
20		eulerpart.r	. . . . . . . . . . 11 |- R = { f | ( `' f " NN ) e. Fin }
21		eulerpart.t	. . . . . . . . . . 11 |- T = { f e. ( NN0 ^m NN ) | ( `' f " NN ) C_ J }
22	14, 15, 16, 4, 17, 18, 19, 20, 21	eulerpartlemt0 30431	. . . . . . . . . 10 |- ( A e. ( T i^i R ) <-> ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) )
23	22	simp1bi 1076	. . . . . . . . 9 |- ( A e. ( T i^i R ) -> A e. ( NN0 ^m NN ) )
24		elmapi 7879	. . . . . . . . 9 |- ( A e. ( NN0 ^m NN ) -> A : NN --> NN0 )
25	23, 24	syl 17	. . . . . . . 8 |- ( A e. ( T i^i R ) -> A : NN --> NN0 )
26		ffn 6045	. . . . . . . 8 |- ( A : NN --> NN0 -> A Fn NN )
27		elpreima 6337	. . . . . . . 8 |- ( A Fn NN -> ( t e. ( `' A " NN ) <-> ( t e. NN /\ ( A ` t ) e. NN ) ) )
28	25, 26, 27	3syl 18	. . . . . . 7 |- ( A e. ( T i^i R ) -> ( t e. ( `' A " NN ) <-> ( t e. NN /\ ( A ` t ) e. NN ) ) )
29	28	baibd 948	. . . . . 6 |- ( ( A e. ( T i^i R ) /\ t e. NN ) -> ( t e. ( `' A " NN ) <-> ( A ` t ) e. NN ) )
30	13, 29	sylan2 491	. . . . 5 |- ( ( A e. ( T i^i R ) /\ t e. ( NN \ J ) ) -> ( t e. ( `' A " NN ) <-> ( A ` t ) e. NN ) )
31	30	biimpar 502	. . . 4 |- ( ( ( A e. ( T i^i R ) /\ t e. ( NN \ J ) ) /\ ( A ` t ) e. NN ) -> t e. ( `' A " NN ) )
32	22	simp3bi 1078	. . . . . 6 |- ( A e. ( T i^i R ) -> ( `' A " NN ) C_ J )
33	32	sselda 3603	. . . . 5 |- ( ( A e. ( T i^i R ) /\ t e. ( `' A " NN ) ) -> t e. J )
34	5	simprbi 480	. . . . 5 |- ( t e. J -> -. 2 || t )
35	33, 34	syl 17	. . . 4 |- ( ( A e. ( T i^i R ) /\ t e. ( `' A " NN ) ) -> -. 2 || t )
36	12, 31, 35	syl2anc 693	. . 3 |- ( ( ( A e. ( T i^i R ) /\ t e. ( NN \ J ) ) /\ ( A ` t ) e. NN ) -> -. 2 || t )
37	11, 36	pm2.65da 600	. 2 |- ( ( A e. ( T i^i R ) /\ t e. ( NN \ J ) ) -> -. ( A ` t ) e. NN )
38	25	adantr 481	. . . 4 |- ( ( A e. ( T i^i R ) /\ t e. ( NN \ J ) ) -> A : NN --> NN0 )
39	13	adantl 482	. . . 4 |- ( ( A e. ( T i^i R ) /\ t e. ( NN \ J ) ) -> t e. NN )
40	38, 39	ffvelrnd 6360	. . 3 |- ( ( A e. ( T i^i R ) /\ t e. ( NN \ J ) ) -> ( A ` t ) e. NN0 )
41		elnn0 11294	. . 3 |- ( ( A ` t ) e. NN0 <-> ( ( A ` t ) e. NN \/ ( A ` t ) = 0 ) )
42	40, 41	sylib 208	. 2 |- ( ( A e. ( T i^i R ) /\ t e. ( NN \ J ) ) -> ( ( A ` t ) e. NN \/ ( A ` t ) = 0 ) )
43		orel1 397	. 2 |- ( -. ( A ` t ) e. NN -> ( ( ( A ` t ) e. NN \/ ( A ` t ) = 0 ) -> ( A ` t ) = 0 ) )
44	37, 42, 43	sylc 65	1 |- ( ( A e. ( T i^i R ) /\ t e. ( NN \ J ) ) -> ( A ` t ) = 0 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   {copab 4712   |-> cmpt 4729   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` 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

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-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-n0 11293

This theorem is referenced by:  eulerpartlemgh  30440