Theorem eulerpartlemsv2 29240

Description: Lemma for eulerpart 29264. 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
eulerpartlemsv2	|- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( S ` A ) = sum_ k e. ( `' A " NN ) ( ( A ` k ) x. k ) )

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

Allowed substitution hints:   S( f, k)

Proof of Theorem eulerpartlemsv2

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 29238	. 2 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( S ` A ) = sum_ k e. NN ( ( A ` k ) x. k ) )
4		cnvimass 5207	. . . 4 |- ( `' A " NN ) C_ dom A
5	1, 2	eulerpartlemelr 29239	. . . . . 6 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin ) )
6	5	simpld 465	. . . . 5 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> A : NN --> NN0 )
7		fdm 5756	. . . . 5 |- ( A : NN --> NN0 -> dom A = NN )
8	6, 7	syl 17	. . . 4 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> dom A = NN )
9	4, 8	syl5sseq 3492	. . 3 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( `' A " NN ) C_ NN )
10	6	adantr 471	. . . . . 6 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> A : NN --> NN0 )
11	9	sselda 3444	. . . . . 6 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> k e. NN )
12	10, 11	ffvelrnd 6046	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> ( A ` k ) e. NN0 )
13	11	nnnn0d 10954	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> k e. NN0 )
14	12, 13	nn0mulcld 10959	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> ( ( A ` k ) x. k ) e. NN0 )
15	14	nn0cnd 10956	. . 3 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> ( ( A ` k ) x. k ) e. CC )
16		simpr 467	. . . . . . . 8 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> k e. ( NN \ ( `' A " NN ) ) )
17	16	eldifad 3428	. . . . . . 7 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> k e. NN )
18	16	eldifbd 3429	. . . . . . . . 9 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> -. k e. ( `' A " NN ) )
19	6	adantr 471	. . . . . . . . . 10 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> A : NN --> NN0 )
20		ffn 5751	. . . . . . . . . 10 |- ( A : NN --> NN0 -> A Fn NN )
21		elpreima 6025	. . . . . . . . . 10 |- ( A Fn NN -> ( k e. ( `' A " NN ) <-> ( k e. NN /\ ( A ` k ) e. NN ) ) )
22	19, 20, 21	3syl 18	. . . . . . . . 9 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( k e. ( `' A " NN ) <-> ( k e. NN /\ ( A ` k ) e. NN ) ) )
23	18, 22	mtbid 306	. . . . . . . 8 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> -. ( k e. NN /\ ( A ` k ) e. NN ) )
24		imnan 428	. . . . . . . 8 |- ( ( k e. NN -> -. ( A ` k ) e. NN ) <-> -. ( k e. NN /\ ( A ` k ) e. NN ) )
25	23, 24	sylibr 217	. . . . . . 7 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( k e. NN -> -. ( A ` k ) e. NN ) )
26	17, 25	mpd 15	. . . . . 6 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> -. ( A ` k ) e. NN )
27	19, 17	ffvelrnd 6046	. . . . . . 7 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( A ` k ) e. NN0 )
28		elnn0 10900	. . . . . . 7 |- ( ( A ` k ) e. NN0 <-> ( ( A ` k ) e. NN \/ ( A ` k ) = 0 ) )
29	27, 28	sylib 201	. . . . . 6 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( ( A ` k ) e. NN \/ ( A ` k ) = 0 ) )
30		orel1 388	. . . . . 6 |- ( -. ( A ` k ) e. NN -> ( ( ( A ` k ) e. NN \/ ( A ` k ) = 0 ) -> ( A ` k ) = 0 ) )
31	26, 29, 30	sylc 62	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( A ` k ) = 0 )
32	31	oveq1d 6330	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( ( A ` k ) x. k ) = ( 0 x. k ) )
33	17	nncnd 10653	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> k e. CC )
34	33	mul02d 9857	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( 0 x. k ) = 0 )
35	32, 34	eqtrd 2496	. . 3 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( ( A ` k ) x. k ) = 0 )
36		nnuz 11223	. . . . 5 |- NN = ( ZZ>= ` 1 )
37	36	eqimssi 3498	. . . 4 |- NN C_ ( ZZ>= ` 1 )
38	37	a1i 11	. . 3 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> NN C_ ( ZZ>= ` 1 ) )
39	9, 15, 35, 38	sumss 13839	. 2 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> sum_ k e. ( `' A " NN ) ( ( A ` k ) x. k ) = sum_ k e. NN ( ( A ` k ) x. k ) )
40	3, 39	eqtr4d 2499	1 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( S ` A ) = sum_ k e. ( `' A " NN ) ( ( A ` k ) x. k ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   = wceq 1455   e. wcel 1898   {cab 2448   \ cdif 3413   i^i cin 3415   C_ wss 3416   |-> cmpt 4475   `'ccnv 4852   dom cdm 4853   "cima 4856   Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6315   ^m cmap 7498   Fincfn 7595   0cc0 9565   1c1 9566   x. cmul 9570   NNcn 10637   NN0cn0 10898   ZZ>=cuz 11188   sum_csu 13801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-map 7500  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-oi 8051  df-card 8399  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-n0 10899  df-z 10967  df-uz 11189  df-rp 11332  df-fz 11814  df-fzo 11947  df-seq 12246  df-exp 12305  df-hash 12548  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-clim 13601  df-sum 13802

This theorem is referenced by:  eulerpartlemsf  29241  eulerpartlemgs2  29262