Theorem eulerpartlemt0 30431

Description: Lemma for eulerpart 30444. (Contributed by Thierry Arnoux, 19-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
eulerpartlemt0	|- ( A e. ( T i^i R ) <-> ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) )

Distinct variable groups:   A, f   f, J

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

Proof of Theorem eulerpartlemt0

Step	Hyp	Ref	Expression
1		cnveq 5296	. . . . . 6 |- ( f = A -> `' f = `' A )
2	1	imaeq1d 5465	. . . . 5 |- ( f = A -> ( `' f " NN ) = ( `' A " NN ) )
3	2	sseq1d 3632	. . . 4 |- ( f = A -> ( ( `' f " NN ) C_ J <-> ( `' A " NN ) C_ J ) )
4		eulerpart.t	. . . 4 |- T = { f e. ( NN0 ^m NN ) | ( `' f " NN ) C_ J }
5	3, 4	elrab2 3366	. . 3 |- ( A e. T <-> ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) C_ J ) )
6	2	eleq1d 2686	. . . 4 |- ( f = A -> ( ( `' f " NN ) e. Fin <-> ( `' A " NN ) e. Fin ) )
7		eulerpart.r	. . . 4 |- R = { f | ( `' f " NN ) e. Fin }
8	6, 7	elab4g 3355	. . 3 |- ( A e. R <-> ( A e. _V /\ ( `' A " NN ) e. Fin ) )
9	5, 8	anbi12i 733	. 2 |- ( ( A e. T /\ A e. R ) <-> ( ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) C_ J ) /\ ( A e. _V /\ ( `' A " NN ) e. Fin ) ) )
10		elin 3796	. 2 |- ( A e. ( T i^i R ) <-> ( A e. T /\ A e. R ) )
11		elex 3212	. . . . 5 |- ( A e. ( NN0 ^m NN ) -> A e. _V )
12	11	pm4.71i 664	. . . 4 |- ( A e. ( NN0 ^m NN ) <-> ( A e. ( NN0 ^m NN ) /\ A e. _V ) )
13	12	anbi1i 731	. . 3 |- ( ( A e. ( NN0 ^m NN ) /\ ( ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) ) <-> ( ( A e. ( NN0 ^m NN ) /\ A e. _V ) /\ ( ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) ) )
14		3anass 1042	. . 3 |- ( ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) <-> ( A e. ( NN0 ^m NN ) /\ ( ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) ) )
15		an42 866	. . 3 |- ( ( ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) C_ J ) /\ ( A e. _V /\ ( `' A " NN ) e. Fin ) ) <-> ( ( A e. ( NN0 ^m NN ) /\ A e. _V ) /\ ( ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) ) )
16	13, 14, 15	3bitr4i 292	. 2 |- ( ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) <-> ( ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) C_ J ) /\ ( A e. _V /\ ( `' A " NN ) e. Fin ) ) )
17	9, 10, 16	3bitr4i 292	1 |- ( A e. ( T i^i R ) <-> ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   {copab 4712   |-> cmpt 4729   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   supp csupp 7295   ^m cmap 7857   Fincfn 7955   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  eulerpartlemf  30432  eulerpartlemt  30433  eulerpartlemmf  30437  eulerpartlemgvv  30438  eulerpartlemgu  30439  eulerpartlemgh  30440  eulerpartlemgs2  30442  eulerpartlemn  30443