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Theorem iserile 9862

Description: Comparison of the limits of two infinite series. (Contributed by Jim Kingdon, 22-Aug-2021.)

**Hypotheses**
Ref	Expression
clim2ser.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
iserile.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
iserile.4	⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴)
iserile.5	⊢ (𝜑 → seq𝑀( + , 𝐺, ℂ) ⇝ 𝐵)
iserile.6	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
iserile.7	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)
iserile.8	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘))

**Assertion**
Ref	Expression
iserile	⊢ (𝜑 → 𝐴 ≤ 𝐵)

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝑘,𝐹 𝑘,𝑀 𝑘,𝐺 𝜑,𝑘 𝑘,𝑍

Proof of Theorem iserile Dummy variables 𝑗 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clim2ser.1	. 2 ⊢ 𝑍 = (ℤ_≥‘𝑀)
2		iserile.2	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		iserile.4	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴)
4		iserile.5	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐺, ℂ) ⇝ 𝐵)
5		cnex 7005	. . . . . . 7 ⊢ ℂ ∈ V
6	5	a1i 9	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
7		ax-resscn 6976	. . . . . . 7 ⊢ ℝ ⊆ ℂ
8	7	a1i 9	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
9	1	eleq2i 2104	. . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ_≥‘𝑀))
10		iserile.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
11	9, 10	sylan2br 272	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
12		readdcl 7007	. . . . . . 7 ⊢ ((𝑘 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑘 + 𝑥) ∈ ℝ)
13	12	adantl 262	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ℝ ∧ 𝑥 ∈ ℝ)) → (𝑘 + 𝑥) ∈ ℝ)
14		addcl 7006	. . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 + 𝑥) ∈ ℂ)
15	14	adantl 262	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 + 𝑥) ∈ ℂ)
16	2, 6, 8, 11, 13, 15	iseqss 9226	. . . . 5 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℝ) = seq𝑀( + , 𝐹, ℂ))
17	16	adantr 261	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → seq𝑀( + , 𝐹, ℝ) = seq𝑀( + , 𝐹, ℂ))
18	17	fveq1d 5180	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹, ℝ)‘𝑗) = (seq𝑀( + , 𝐹, ℂ)‘𝑗))
19	1, 2, 10	iserfre 9234	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℝ):𝑍⟶ℝ)
20	19	ffvelrnda 5302	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹, ℝ)‘𝑗) ∈ ℝ)
21	18, 20	eqeltrrd 2115	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹, ℂ)‘𝑗) ∈ ℝ)
22		iserile.7	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)
23	9, 22	sylan2br 272	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑘) ∈ ℝ)
24	2, 6, 8, 23, 13, 15	iseqss 9226	. . . . 5 ⊢ (𝜑 → seq𝑀( + , 𝐺, ℝ) = seq𝑀( + , 𝐺, ℂ))
25	24	adantr 261	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → seq𝑀( + , 𝐺, ℝ) = seq𝑀( + , 𝐺, ℂ))
26	25	fveq1d 5180	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐺, ℝ)‘𝑗) = (seq𝑀( + , 𝐺, ℂ)‘𝑗))
27	1, 2, 22	iserfre 9234	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐺, ℝ):𝑍⟶ℝ)
28	27	ffvelrnda 5302	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐺, ℝ)‘𝑗) ∈ ℝ)
29	26, 28	eqeltrrd 2115	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐺, ℂ)‘𝑗) ∈ ℝ)
30		simpr 103	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
31	30, 1	syl6eleq 2130	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑀))
32	11	adantlr 446	. . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
33	23	adantlr 446	. . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑘) ∈ ℝ)
34		simpll 481	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝜑)
35	9	biimpri 124	. . . . 5 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑘 ∈ 𝑍)
36	35	adantl 262	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ 𝑍)
37		iserile.8	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘))
38	34, 36, 37	syl2anc 391	. . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ≤ (𝐺‘𝑘))
39	31, 32, 33, 38	serile 9253	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹, ℂ)‘𝑗) ≤ (seq𝑀( + , 𝐺, ℂ)‘𝑗))
40	1, 2, 3, 4, 21, 29, 39	climle 9854	1 ⊢ (𝜑 → 𝐴 ≤ 𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ⊆ wss 2917 class class class wbr 3764 ‘cfv 4902 (class class class)co 5512 ℂcc 6887 ℝcr 6888 + caddc 6892 ≤ cle 7061 ℤcz 8245 ℤ_≥cuz 8473 seqcseq 9211 ⇝ cli 9799

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 ax-pre-mulext 7002 ax-arch 7003 ax-caucvg 7004

This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-if 3332 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-frec 5978 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 df-div 7652 df-inn 7915 df-2 7973 df-3 7974 df-4 7975 df-n0 8182 df-z 8246 df-uz 8474 df-rp 8584 df-fz 8875 df-fzo 9000 df-iseq 9212 df-iexp 9255 df-cj 9442 df-re 9443 df-im 9444 df-rsqrt 9596 df-abs 9597 df-clim 9800

This theorem is referenced by: iserige0 9863

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