Theorem mertenslemub 12061

Description: Lemma for mertensabs 12064. An upper bound for T. (Contributed by Jim Kingdon, 3-Dec-2022.)

Hypotheses

Ref	Expression
mertenslemub.gb	|- ( ( ph /\ k e. NN0 ) -> ( G ` k ) = B )
mertenslemub.b	|- ( ( ph /\ k e. NN0 ) -> B e. CC )
mertenslemub.cvg	|- ( ph -> seq 0 ( + , G ) e. dom ~~> )
mertenslemub.t	|- T = { z | E. n e. ( 0 ... ( S - 1 ) ) z = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) }
mertenslemub.elt	|- ( ph -> X e. T )
mertenslemub.s	|- ( ph -> S e. NN )

Assertion

Ref	Expression
mertenslemub	|- ( ph -> X <_ sum_ n e. ( 0 ... ( S - 1 ) ) ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) )

Distinct variable groups:   k, G, n, z   S, k, n, z   n, X, z   ph, k, n

Allowed substitution hints:   ph( z)   B( z, k, n)   T( z, k, n)   X( k)

Proof of Theorem mertenslemub

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mertenslemub.elt	. . . 4 |- ( ph -> X e. T )
2		eqeq1 2236	. . . . . . 7 |- ( z = X -> ( z = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) <-> X = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) ) )
3	2	rexbidv 2531	. . . . . 6 |- ( z = X -> ( E. n e. ( 0 ... ( S - 1 ) ) z = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) <-> E. n e. ( 0 ... ( S - 1 ) ) X = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) ) )
4		mertenslemub.t	. . . . . 6 |- T = { z | E. n e. ( 0 ... ( S - 1 ) ) z = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) }
5	3, 4	elab2g 2950	. . . . 5 |- ( X e. T -> ( X e. T <-> E. n e. ( 0 ... ( S - 1 ) ) X = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) ) )
6	1, 5	syl 14	. . . 4 |- ( ph -> ( X e. T <-> E. n e. ( 0 ... ( S - 1 ) ) X = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) ) )
7	1, 6	mpbid 147	. . 3 |- ( ph -> E. n e. ( 0 ... ( S - 1 ) ) X = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) )
8		fvoveq1 6030	. . . . . . 7 |- ( n = a -> ( ZZ>= ` ( n + 1 ) ) = ( ZZ>= ` ( a + 1 ) ) )
9	8	sumeq1d 11893	. . . . . 6 |- ( n = a -> sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) = sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) )
10	9	fveq2d 5633	. . . . 5 |- ( n = a -> ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) )
11	10	eqeq2d 2241	. . . 4 |- ( n = a -> ( X = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) <-> X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) )
12	11	cbvrexv 2766	. . 3 |- ( E. n e. ( 0 ... ( S - 1 ) ) X = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) <-> E. a e. ( 0 ... ( S - 1 ) ) X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) )
13	7, 12	sylib 122	. 2 |- ( ph -> E. a e. ( 0 ... ( S - 1 ) ) X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) )
14		simprr 531	. . 3 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) )
15		0zd 9469	. . . . 5 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> 0 e. ZZ )
16		mertenslemub.s	. . . . . . . 8 |- ( ph -> S e. NN )
17	16	adantr 276	. . . . . . 7 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> S e. NN )
18	17	nnzd 9579	. . . . . 6 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> S e. ZZ )
19		1zzd 9484	. . . . . 6 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> 1 e. ZZ )
20	18, 19	zsubcld 9585	. . . . 5 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> ( S - 1 ) e. ZZ )
21	15, 20	fzfigd 10665	. . . 4 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> ( 0 ... ( S - 1 ) ) e. Fin )
22		eqid 2229	. . . . . . 7 |- ( ZZ>= ` ( n + 1 ) ) = ( ZZ>= ` ( n + 1 ) )
23		elfzelz 10233	. . . . . . . . 9 |- ( n e. ( 0 ... ( S - 1 ) ) -> n e. ZZ )
24	23	adantl 277	. . . . . . . 8 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> n e. ZZ )
25	24	peano2zd 9583	. . . . . . 7 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> ( n + 1 ) e. ZZ )
26		eqidd 2230	. . . . . . 7 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> ( G ` k ) = ( G ` k ) )
27		simpll 527	. . . . . . . 8 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> ph )
28		elfznn0 10322	. . . . . . . . . . 11 |- ( n e. ( 0 ... ( S - 1 ) ) -> n e. NN0 )
29	28	ad2antlr 489	. . . . . . . . . 10 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> n e. NN0 )
30		peano2nn0 9420	. . . . . . . . . 10 |- ( n e. NN0 -> ( n + 1 ) e. NN0 )
31	29, 30	syl 14	. . . . . . . . 9 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> ( n + 1 ) e. NN0 )
32		eluznn0 9806	. . . . . . . . 9 |- ( ( ( n + 1 ) e. NN0 /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> k e. NN0 )
33	31, 32	sylancom 420	. . . . . . . 8 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> k e. NN0 )
34		mertenslemub.gb	. . . . . . . . 9 |- ( ( ph /\ k e. NN0 ) -> ( G ` k ) = B )
35		mertenslemub.b	. . . . . . . . 9 |- ( ( ph /\ k e. NN0 ) -> B e. CC )
36	34, 35	eqeltrd 2306	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> ( G ` k ) e. CC )
37	27, 33, 36	syl2anc 411	. . . . . . 7 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> ( G ` k ) e. CC )
38		mertenslemub.cvg	. . . . . . . . 9 |- ( ph -> seq 0 ( + , G ) e. dom ~~> )
39	38	adantr 276	. . . . . . . 8 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> seq 0 ( + , G ) e. dom ~~> )
40		nn0uz 9769	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
41	28	adantl 277	. . . . . . . . . 10 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> n e. NN0 )
42	41, 30	syl 14	. . . . . . . . 9 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> ( n + 1 ) e. NN0 )
43	36	adantlr 477	. . . . . . . . 9 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. NN0 ) -> ( G ` k ) e. CC )
44	40, 42, 43	iserex 11866	. . . . . . . 8 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> ( seq 0 ( + , G ) e. dom ~~> <-> seq ( n + 1 ) ( + , G ) e. dom ~~> ) )
45	39, 44	mpbid 147	. . . . . . 7 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> seq ( n + 1 ) ( + , G ) e. dom ~~> )
46	22, 25, 26, 37, 45	isumcl 11952	. . . . . 6 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) e. CC )
47	46	adantlr 477	. . . . 5 |- ( ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) /\ n e. ( 0 ... ( S - 1 ) ) ) -> sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) e. CC )
48	47	abscld 11708	. . . 4 |- ( ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) /\ n e. ( 0 ... ( S - 1 ) ) ) -> ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) e. RR )
49	47	absge0d 11711	. . . 4 |- ( ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) /\ n e. ( 0 ... ( S - 1 ) ) ) -> 0 <_ ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) )
50		simprl 529	. . . 4 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> a e. ( 0 ... ( S - 1 ) ) )
51	21, 48, 49, 10, 50	fsumge1 11988	. . 3 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) <_ sum_ n e. ( 0 ... ( S - 1 ) ) ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) )
52	14, 51	eqbrtrd 4105	. 2 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> X <_ sum_ n e. ( 0 ... ( S - 1 ) ) ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) )
53	13, 52	rexlimddv 2653	1 |- ( ph -> X <_ sum_ n e. ( 0 ... ( S - 1 ) ) ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {cab 2215   E.wrex 2509   class class class wbr 4083   dom cdm 4719   ` cfv 5318  (class class class)co 6007   CCcc 8008   0cc0 8010   1c1 8011   + caddc 8013   <_ cle 8193   - cmin 8328   NNcn 9121   NN0cn0 9380   ZZcz 9457   ZZ>=cuz 9733   ...cfz 10216   seqcseq 10681   abscabs 11524   ~~> cli 11805   sum_csu 11880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-frec 6543  df-1o 6568  df-oadd 6572  df-er 6688  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-ico 10102  df-fz 10217  df-fzo 10351  df-seqfrec 10682  df-exp 10773  df-ihash 11010  df-cj 11369  df-re 11370  df-im 11371  df-rsqrt 11525  df-abs 11526  df-clim 11806  df-sumdc 11881

This theorem is referenced by:  mertenslem2  12063