Theorem eulerpartlemv 30426

Description: Lemma for eulerpart 30444. (Contributed by Thierry Arnoux, 19-Aug-2018.)

Hypothesis

Ref	Expression
eulerpart.p	|- P = { f e. ( NN0 ^m NN ) | ( ( `' f " NN ) e. Fin /\ sum_ k e. NN ( ( f ` k ) x. k ) = N ) }

Assertion

Ref	Expression
eulerpartlemv	|- ( A e. P <-> ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin /\ sum_ k e. ( `' A " NN ) ( ( A ` k ) x. k ) = N ) )

Distinct variable groups:   f, k, A   f, N, k   P, k

Allowed substitution hint:   P( f)

Proof of Theorem eulerpartlemv

Step	Hyp	Ref	Expression
1		eulerpart.p	. . 3 |- P = { f e. ( NN0 ^m NN ) | ( ( `' f " NN ) e. Fin /\ sum_ k e. NN ( ( f ` k ) x. k ) = N ) }
2	1	eulerpartleme 30425	. 2 |- ( A e. P <-> ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin /\ sum_ k e. NN ( ( A ` k ) x. k ) = N ) )
3		cnvimass 5485	. . . . . . . . 9 |- ( `' A " NN ) C_ dom A
4		fdm 6051	. . . . . . . . 9 |- ( A : NN --> NN0 -> dom A = NN )
5	3, 4	syl5sseq 3653	. . . . . . . 8 |- ( A : NN --> NN0 -> ( `' A " NN ) C_ NN )
6		simpl 473	. . . . . . . . . . 11 |- ( ( A : NN --> NN0 /\ k e. ( `' A " NN ) ) -> A : NN --> NN0 )
7	5	sselda 3603	. . . . . . . . . . 11 |- ( ( A : NN --> NN0 /\ k e. ( `' A " NN ) ) -> k e. NN )
8	6, 7	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( A : NN --> NN0 /\ k e. ( `' A " NN ) ) -> ( A ` k ) e. NN0 )
9	7	nnnn0d 11351	. . . . . . . . . 10 |- ( ( A : NN --> NN0 /\ k e. ( `' A " NN ) ) -> k e. NN0 )
10	8, 9	nn0mulcld 11356	. . . . . . . . 9 |- ( ( A : NN --> NN0 /\ k e. ( `' A " NN ) ) -> ( ( A ` k ) x. k ) e. NN0 )
11	10	nn0cnd 11353	. . . . . . . 8 |- ( ( A : NN --> NN0 /\ k e. ( `' A " NN ) ) -> ( ( A ` k ) x. k ) e. CC )
12		simpr 477	. . . . . . . . . . . . 13 |- ( ( A : NN --> NN0 /\ k e. ( NN \ ( `' A " NN ) ) ) -> k e. ( NN \ ( `' A " NN ) ) )
13	12	eldifad 3586	. . . . . . . . . . . 12 |- ( ( A : NN --> NN0 /\ k e. ( NN \ ( `' A " NN ) ) ) -> k e. NN )
14	12	eldifbd 3587	. . . . . . . . . . . . . 14 |- ( ( A : NN --> NN0 /\ k e. ( NN \ ( `' A " NN ) ) ) -> -. k e. ( `' A " NN ) )
15		simpl 473	. . . . . . . . . . . . . . 15 |- ( ( A : NN --> NN0 /\ k e. ( NN \ ( `' A " NN ) ) ) -> A : NN --> NN0 )
16		ffn 6045	. . . . . . . . . . . . . . 15 |- ( A : NN --> NN0 -> A Fn NN )
17		elpreima 6337	. . . . . . . . . . . . . . 15 |- ( A Fn NN -> ( k e. ( `' A " NN ) <-> ( k e. NN /\ ( A ` k ) e. NN ) ) )
18	15, 16, 17	3syl 18	. . . . . . . . . . . . . 14 |- ( ( A : NN --> NN0 /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( k e. ( `' A " NN ) <-> ( k e. NN /\ ( A ` k ) e. NN ) ) )
19	14, 18	mtbid 314	. . . . . . . . . . . . 13 |- ( ( A : NN --> NN0 /\ k e. ( NN \ ( `' A " NN ) ) ) -> -. ( k e. NN /\ ( A ` k ) e. NN ) )
20		imnan 438	. . . . . . . . . . . . 13 |- ( ( k e. NN -> -. ( A ` k ) e. NN ) <-> -. ( k e. NN /\ ( A ` k ) e. NN ) )
21	19, 20	sylibr 224	. . . . . . . . . . . 12 |- ( ( A : NN --> NN0 /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( k e. NN -> -. ( A ` k ) e. NN ) )
22	13, 21	mpd 15	. . . . . . . . . . 11 |- ( ( A : NN --> NN0 /\ k e. ( NN \ ( `' A " NN ) ) ) -> -. ( A ` k ) e. NN )
23	15, 13	ffvelrnd 6360	. . . . . . . . . . . 12 |- ( ( A : NN --> NN0 /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( A ` k ) e. NN0 )
24		elnn0 11294	. . . . . . . . . . . 12 |- ( ( A ` k ) e. NN0 <-> ( ( A ` k ) e. NN \/ ( A ` k ) = 0 ) )
25	23, 24	sylib 208	. . . . . . . . . . 11 |- ( ( A : NN --> NN0 /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( ( A ` k ) e. NN \/ ( A ` k ) = 0 ) )
26		orel1 397	. . . . . . . . . . 11 |- ( -. ( A ` k ) e. NN -> ( ( ( A ` k ) e. NN \/ ( A ` k ) = 0 ) -> ( A ` k ) = 0 ) )
27	22, 25, 26	sylc 65	. . . . . . . . . 10 |- ( ( A : NN --> NN0 /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( A ` k ) = 0 )
28	27	oveq1d 6665	. . . . . . . . 9 |- ( ( A : NN --> NN0 /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( ( A ` k ) x. k ) = ( 0 x. k ) )
29	13	nncnd 11036	. . . . . . . . . 10 |- ( ( A : NN --> NN0 /\ k e. ( NN \ ( `' A " NN ) ) ) -> k e. CC )
30	29	mul02d 10234	. . . . . . . . 9 |- ( ( A : NN --> NN0 /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( 0 x. k ) = 0 )
31	28, 30	eqtrd 2656	. . . . . . . 8 |- ( ( A : NN --> NN0 /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( ( A ` k ) x. k ) = 0 )
32		nnuz 11723	. . . . . . . . . 10 |- NN = ( ZZ>= ` 1 )
33	32	eqimssi 3659	. . . . . . . . 9 |- NN C_ ( ZZ>= ` 1 )
34	33	a1i 11	. . . . . . . 8 |- ( A : NN --> NN0 -> NN C_ ( ZZ>= ` 1 ) )
35	5, 11, 31, 34	sumss 14455	. . . . . . 7 |- ( A : NN --> NN0 -> sum_ k e. ( `' A " NN ) ( ( A ` k ) x. k ) = sum_ k e. NN ( ( A ` k ) x. k ) )
36	35	eqcomd 2628	. . . . . 6 |- ( A : NN --> NN0 -> sum_ k e. NN ( ( A ` k ) x. k ) = sum_ k e. ( `' A " NN ) ( ( A ` k ) x. k ) )
37	36	adantr 481	. . . . 5 |- ( ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin ) -> sum_ k e. NN ( ( A ` k ) x. k ) = sum_ k e. ( `' A " NN ) ( ( A ` k ) x. k ) )
38	37	eqeq1d 2624	. . . 4 |- ( ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin ) -> ( sum_ k e. NN ( ( A ` k ) x. k ) = N <-> sum_ k e. ( `' A " NN ) ( ( A ` k ) x. k ) = N ) )
39	38	pm5.32i 669	. . 3 |- ( ( ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin ) /\ sum_ k e. NN ( ( A ` k ) x. k ) = N ) <-> ( ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin ) /\ sum_ k e. ( `' A " NN ) ( ( A ` k ) x. k ) = N ) )
40		df-3an 1039	. . 3 |- ( ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin /\ sum_ k e. NN ( ( A ` k ) x. k ) = N ) <-> ( ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin ) /\ sum_ k e. NN ( ( A ` k ) x. k ) = N ) )
41		df-3an 1039	. . 3 |- ( ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin /\ sum_ k e. ( `' A " NN ) ( ( A ` k ) x. k ) = N ) <-> ( ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin ) /\ sum_ k e. ( `' A " NN ) ( ( A ` k ) x. k ) = N ) )
42	39, 40, 41	3bitr4i 292	. 2 |- ( ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin /\ sum_ k e. NN ( ( A ` k ) x. k ) = N ) <-> ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin /\ sum_ k e. ( `' A " NN ) ( ( A ` k ) x. k ) = N ) )
43	2, 42	bitri 264	1 |- ( A e. P <-> ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin /\ sum_ k e. ( `' A " NN ) ( ( A ` k ) x. k ) = N ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571   C_ wss 3574   `'ccnv 5113   dom cdm 5114   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   0cc0 9936   1c1 9937   x. cmul 9941   NNcn 11020   NN0cn0 11292   ZZ>=cuz 11687   sum_csu 14416

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-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

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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  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

This theorem is referenced by: (None)