Theorem mertenslemub 12301

Description: Lemma for mertensabs 12304. 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 2245	. . . . . . 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 2551	. . . . . 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 2973	. . . . 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 6108	. . . . . . 7 |- ( n = a -> ( ZZ>= ` ( n + 1 ) ) = ( ZZ>= ` ( a + 1 ) ) )
9	8	sumeq1d 12132	. . . . . 6 |- ( n = a -> sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) = sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) )
10	9	fveq2d 5699	. . . . 5 |- ( n = a -> ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) )
11	10	eqeq2d 2250	. . . 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 2787	. . 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 537	. . 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 9656	. . . . 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 9767	. . . . . 6 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> S e. ZZ )
19		1zzd 9671	. . . . . 6 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> 1 e. ZZ )
20	18, 19	zsubcld 9773	. . . . 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 10868	. . . 4 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> ( 0 ... ( S - 1 ) ) e. Fin )
22		eqid 2238	. . . . . . 7 |- ( ZZ>= ` ( n + 1 ) ) = ( ZZ>= ` ( n + 1 ) )
23		elfzelz 10428	. . . . . . . . 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 9771	. . . . . . 7 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> ( n + 1 ) e. ZZ )
26		eqidd 2239	. . . . . . 7 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> ( G ` k ) = ( G ` k ) )
27		simpll 531	. . . . . . . 8 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> ph )
28		elfznn0 10521	. . . . . . . . . . 11 |- ( n e. ( 0 ... ( S - 1 ) ) -> n e. NN0 )
29	28	ad2antlr 493	. . . . . . . . . 10 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> n e. NN0 )
30		peano2nn0 9603	. . . . . . . . . 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 9999	. . . . . . . . 9 |- ( ( ( n + 1 ) e. NN0 /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> k e. NN0 )
33	31, 32	sylancom 424	. . . . . . . 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 2315	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> ( G ` k ) e. CC )
37	27, 33, 36	syl2anc 415	. . . . . . 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 9957	. . . . . . . . 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 481	. . . . . . . . 9 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. NN0 ) -> ( G ` k ) e. CC )
44	40, 42, 43	iserex 12105	. . . . . . . 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 12192	. . . . . 6 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) e. CC )
47	46	adantlr 481	. . . . 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 11947	. . . 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 11950	. . . 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 535	. . . 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 12228	. . 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 4152	. 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 2673	1 |- ( ph -> X <_ sum_ n e. ( 0 ... ( S - 1 ) ) ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   E.wrex 2529   class class class wbr 4130   dom cdm 4774   ` cfv 5377  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   <_ cle 8361   - cmin 8497   NNcn 9304   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411   seqcseq 10884   abscabs 11763   ~~> cli 12044   sum_csu 12119

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-ico 10296  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-ihash 11215  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-sumdc 12120

This theorem is used by:  mertenslem2  12303