Theorem eulerpartlemsv3 30423

Description: Lemma for eulerpart 30444. Value of the sum of a finite partition A (Contributed by Thierry Arnoux, 19-Aug-2018.)

Hypotheses

Ref	Expression
eulerpartlems.r	|- R = { f | ( `' f " NN ) e. Fin }
eulerpartlems.s	|- S = ( f e. ( ( NN0 ^m NN ) i^i R ) |-> sum_ k e. NN ( ( f ` k ) x. k ) )

Assertion

Ref	Expression
eulerpartlemsv3	|- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( S ` A ) = sum_ k e. ( 1 ... ( S ` A ) ) ( ( A ` k ) x. k ) )

Distinct variable groups:   f, k, A   R, f, k   S, k

Allowed substitution hint:   S( f)

Proof of Theorem eulerpartlemsv3

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eulerpartlems.r	. . 3 |- R = { f | ( `' f " NN ) e. Fin }
2		eulerpartlems.s	. . 3 |- S = ( f e. ( ( NN0 ^m NN ) i^i R ) |-> sum_ k e. NN ( ( f ` k ) x. k ) )
3	1, 2	eulerpartlemsv1 30418	. 2 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( S ` A ) = sum_ k e. NN ( ( A ` k ) x. k ) )
4		fzssuz 12382	. . . . 5 |- ( 1 ... ( S ` A ) ) C_ ( ZZ>= ` 1 )
5		nnuz 11723	. . . . 5 |- NN = ( ZZ>= ` 1 )
6	4, 5	sseqtr4i 3638	. . . 4 |- ( 1 ... ( S ` A ) ) C_ NN
7	6	a1i 11	. . 3 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( 1 ... ( S ` A ) ) C_ NN )
8	1, 2	eulerpartlemelr 30419	. . . . . . . 8 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin ) )
9	8	simpld 475	. . . . . . 7 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> A : NN --> NN0 )
10	9	adantr 481	. . . . . 6 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( 1 ... ( S ` A ) ) ) -> A : NN --> NN0 )
11	7	sselda 3603	. . . . . 6 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( 1 ... ( S ` A ) ) ) -> k e. NN )
12	10, 11	ffvelrnd 6360	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( 1 ... ( S ` A ) ) ) -> ( A ` k ) e. NN0 )
13	12	nn0cnd 11353	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( 1 ... ( S ` A ) ) ) -> ( A ` k ) e. CC )
14	11	nncnd 11036	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( 1 ... ( S ` A ) ) ) -> k e. CC )
15	13, 14	mulcld 10060	. . 3 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( 1 ... ( S ` A ) ) ) -> ( ( A ` k ) x. k ) e. CC )
16	1, 2	eulerpartlems 30422	. . . . . . . . 9 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ t e. ( ZZ>= ` ( ( S ` A ) + 1 ) ) ) -> ( A ` t ) = 0 )
17	16	ralrimiva 2966	. . . . . . . 8 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> A. t e. ( ZZ>= ` ( ( S ` A ) + 1 ) ) ( A ` t ) = 0 )
18		fveq2 6191	. . . . . . . . . 10 |- ( k = t -> ( A ` k ) = ( A ` t ) )
19	18	eqeq1d 2624	. . . . . . . . 9 |- ( k = t -> ( ( A ` k ) = 0 <-> ( A ` t ) = 0 ) )
20	19	cbvralv 3171	. . . . . . . 8 |- ( A. k e. ( ZZ>= ` ( ( S ` A ) + 1 ) ) ( A ` k ) = 0 <-> A. t e. ( ZZ>= ` ( ( S ` A ) + 1 ) ) ( A ` t ) = 0 )
21	17, 20	sylibr 224	. . . . . . 7 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> A. k e. ( ZZ>= ` ( ( S ` A ) + 1 ) ) ( A ` k ) = 0 )
22	1, 2	eulerpartlemsf 30421	. . . . . . . . . 10 |- S : ( ( NN0 ^m NN ) i^i R ) --> NN0
23	22	ffvelrni 6358	. . . . . . . . 9 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( S ` A ) e. NN0 )
24		nndiffz1 29548	. . . . . . . . 9 |- ( ( S ` A ) e. NN0 -> ( NN \ ( 1 ... ( S ` A ) ) ) = ( ZZ>= ` ( ( S ` A ) + 1 ) ) )
25	23, 24	syl 17	. . . . . . . 8 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( NN \ ( 1 ... ( S ` A ) ) ) = ( ZZ>= ` ( ( S ` A ) + 1 ) ) )
26	25	raleqdv 3144	. . . . . . 7 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( A. k e. ( NN \ ( 1 ... ( S ` A ) ) ) ( A ` k ) = 0 <-> A. k e. ( ZZ>= ` ( ( S ` A ) + 1 ) ) ( A ` k ) = 0 ) )
27	21, 26	mpbird 247	. . . . . 6 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> A. k e. ( NN \ ( 1 ... ( S ` A ) ) ) ( A ` k ) = 0 )
28	27	r19.21bi 2932	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( 1 ... ( S ` A ) ) ) ) -> ( A ` k ) = 0 )
29	28	oveq1d 6665	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( 1 ... ( S ` A ) ) ) ) -> ( ( A ` k ) x. k ) = ( 0 x. k ) )
30		simpr 477	. . . . . . 7 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( 1 ... ( S ` A ) ) ) ) -> k e. ( NN \ ( 1 ... ( S ` A ) ) ) )
31	30	eldifad 3586	. . . . . 6 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( 1 ... ( S ` A ) ) ) ) -> k e. NN )
32	31	nncnd 11036	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( 1 ... ( S ` A ) ) ) ) -> k e. CC )
33	32	mul02d 10234	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( 1 ... ( S ` A ) ) ) ) -> ( 0 x. k ) = 0 )
34	29, 33	eqtrd 2656	. . 3 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( 1 ... ( S ` A ) ) ) ) -> ( ( A ` k ) x. k ) = 0 )
35	5	eqimssi 3659	. . . 4 |- NN C_ ( ZZ>= ` 1 )
36	35	a1i 11	. . 3 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> NN C_ ( ZZ>= ` 1 ) )
37	7, 15, 34, 36	sumss 14455	. 2 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> sum_ k e. ( 1 ... ( S ` A ) ) ( ( A ` k ) x. k ) = sum_ k e. NN ( ( A ` k ) x. k ) )
38	3, 37	eqtr4d 2659	1 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( S ` A ) = sum_ k e. ( 1 ... ( S ` A ) ) ( ( A ` k ) x. k ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   \ cdif 3571   i^i cin 3573   C_ wss 3574   |-> cmpt 4729   `'ccnv 5113   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NNcn 11020   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326   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  ax-pre-sup 10014

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-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  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-rlim 14220  df-sum 14417

This theorem is referenced by:  eulerpartlemgc  30424