Theorem mertenslemub 11716

Description: Lemma for mertensabs 11719. An upper bound for 𝑇. (Contributed by Jim Kingdon, 3-Dec-2022.)

Hypotheses

Ref	Expression
mertenslemub.gb	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
mertenslemub.b	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
mertenslemub.cvg	⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )
mertenslemub.t	⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑆 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))}
mertenslemub.elt	⊢ (𝜑 → 𝑋 ∈ 𝑇)
mertenslemub.s	⊢ (𝜑 → 𝑆 ∈ ℕ)

Assertion

Ref	Expression
mertenslemub	⊢ (𝜑 → 𝑋 ≤ Σ𝑛 ∈ (0...(𝑆 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))

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

Allowed substitution hints:   𝜑(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 ⊢ (𝜑 → 𝑋 ∈ 𝑇)
2		eqeq1 2203	. . . . . . 7 ⊢ (𝑧 = 𝑋 → (𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
3	2	rexbidv 2498	. . . . . 6 ⊢ (𝑧 = 𝑋 → (∃𝑛 ∈ (0...(𝑆 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ ∃𝑛 ∈ (0...(𝑆 − 1))𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
4		mertenslemub.t	. . . . . 6 ⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑆 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))}
5	3, 4	elab2g 2911	. . . . 5 ⊢ (𝑋 ∈ 𝑇 → (𝑋 ∈ 𝑇 ↔ ∃𝑛 ∈ (0...(𝑆 − 1))𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
6	1, 5	syl 14	. . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝑇 ↔ ∃𝑛 ∈ (0...(𝑆 − 1))𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
7	1, 6	mpbid 147	. . 3 ⊢ (𝜑 → ∃𝑛 ∈ (0...(𝑆 − 1))𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
8		fvoveq1 5948	. . . . . . 7 ⊢ (𝑛 = 𝑎 → (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑎 + 1)))
9	8	sumeq1d 11548	. . . . . 6 ⊢ (𝑛 = 𝑎 → Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘))
10	9	fveq2d 5565	. . . . 5 ⊢ (𝑛 = 𝑎 → (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))
11	10	eqeq2d 2208	. . . 4 ⊢ (𝑛 = 𝑎 → (𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘))))
12	11	cbvrexv 2730	. . 3 ⊢ (∃𝑛 ∈ (0...(𝑆 − 1))𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ ∃𝑎 ∈ (0...(𝑆 − 1))𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))
13	7, 12	sylib 122	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (0...(𝑆 − 1))𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))
14		simprr 531	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))
15		0zd 9355	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 0 ∈ ℤ)
16		mertenslemub.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℕ)
17	16	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 𝑆 ∈ ℕ)
18	17	nnzd 9464	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 𝑆 ∈ ℤ)
19		1zzd 9370	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 1 ∈ ℤ)
20	18, 19	zsubcld 9470	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → (𝑆 − 1) ∈ ℤ)
21	15, 20	fzfigd 10540	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → (0...(𝑆 − 1)) ∈ Fin)
22		eqid 2196	. . . . . . 7 ⊢ (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑛 + 1))
23		elfzelz 10117	. . . . . . . . 9 ⊢ (𝑛 ∈ (0...(𝑆 − 1)) → 𝑛 ∈ ℤ)
24	23	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → 𝑛 ∈ ℤ)
25	24	peano2zd 9468	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → (𝑛 + 1) ∈ ℤ)
26		eqidd 2197	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → (𝐺‘𝑘) = (𝐺‘𝑘))
27		simpll 527	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → 𝜑)
28		elfznn0 10206	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (0...(𝑆 − 1)) → 𝑛 ∈ ℕ0)
29	28	ad2antlr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑛 ∈ ℕ0)
30		peano2nn0 9306	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
31	29, 30	syl 14	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ0)
32		eluznn0 9690	. . . . . . . . 9 ⊢ (((𝑛 + 1) ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑘 ∈ ℕ0)
33	31, 32	sylancom 420	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑘 ∈ ℕ0)
34		mertenslemub.gb	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
35		mertenslemub.b	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
36	34, 35	eqeltrd 2273	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) ∈ ℂ)
37	27, 33, 36	syl2anc 411	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → (𝐺‘𝑘) ∈ ℂ)
38		mertenslemub.cvg	. . . . . . . . 9 ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )
39	38	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → seq0( + , 𝐺) ∈ dom ⇝ )
40		nn0uz 9653	. . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0)
41	28	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → 𝑛 ∈ ℕ0)
42	41, 30	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → (𝑛 + 1) ∈ ℕ0)
43	36	adantlr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) ∈ ℂ)
44	40, 42, 43	iserex 11521	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq(𝑛 + 1)( + , 𝐺) ∈ dom ⇝ ))
45	39, 44	mpbid 147	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → seq(𝑛 + 1)( + , 𝐺) ∈ dom ⇝ )
46	22, 25, 26, 37, 45	isumcl 11607	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘) ∈ ℂ)
47	46	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) ∧ 𝑛 ∈ (0...(𝑆 − 1))) → Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘) ∈ ℂ)
48	47	abscld 11363	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) ∧ 𝑛 ∈ (0...(𝑆 − 1))) → (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ∈ ℝ)
49	47	absge0d 11366	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) ∧ 𝑛 ∈ (0...(𝑆 − 1))) → 0 ≤ (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
50		simprl 529	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 𝑎 ∈ (0...(𝑆 − 1)))
51	21, 48, 49, 10, 50	fsumge1 11643	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)) ≤ Σ𝑛 ∈ (0...(𝑆 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
52	14, 51	eqbrtrd 4056	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 𝑋 ≤ Σ𝑛 ∈ (0...(𝑆 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
53	13, 52	rexlimddv 2619	1 ⊢ (𝜑 → 𝑋 ≤ Σ𝑛 ∈ (0...(𝑆 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  {cab 2182  ∃wrex 2476   class class class wbr 4034  dom cdm 4664  ‘cfv 5259  (class class class)co 5925  ℂcc 7894  0cc0 7896  1c1 7897   + caddc 7899   ≤ cle 8079   − cmin 8214  ℕcn 9007  ℕ₀cn0 9266  ℤcz 9343  ℤ_≥cuz 9618  ...cfz 10100  seqcseq 10556  abscabs 11179   ⇝ cli 11460  Σcsu 11535

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-oadd 6487  df-er 6601  df-en 6809  df-dom 6810  df-fin 6811  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-ico 9986  df-fz 10101  df-fzo 10235  df-seqfrec 10557  df-exp 10648  df-ihash 10885  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-clim 11461  df-sumdc 11536

This theorem is referenced by:  mertenslem2  11718