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Theorem iserile 10180

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 7097	. . . . . . 7 ⊢ ℂ ∈ V
6	5	a1i 9	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
7		ax-resscn 7068	. . . . . . 7 ⊢ ℝ ⊆ ℂ
8	7	a1i 9	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
9	1	eleq2i 2145	. . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ_≥‘𝑀))
10		iserile.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
11	9, 10	sylan2br 282	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
12		readdcl 7099	. . . . . . 7 ⊢ ((𝑘 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑘 + 𝑥) ∈ ℝ)
13	12	adantl 271	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ℝ ∧ 𝑥 ∈ ℝ)) → (𝑘 + 𝑥) ∈ ℝ)
14		addcl 7098	. . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 + 𝑥) ∈ ℂ)
15	14	adantl 271	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 + 𝑥) ∈ ℂ)
16	2, 6, 8, 11, 13, 15	iseqss 9446	. . . . 5 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℝ) = seq𝑀( + , 𝐹, ℂ))
17	16	adantr 270	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → seq𝑀( + , 𝐹, ℝ) = seq𝑀( + , 𝐹, ℂ))
18	17	fveq1d 5200	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹, ℝ)‘𝑗) = (seq𝑀( + , 𝐹, ℂ)‘𝑗))
19	1, 2, 10	iserfre 9454	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℝ):𝑍⟶ℝ)
20	19	ffvelrnda 5323	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹, ℝ)‘𝑗) ∈ ℝ)
21	18, 20	eqeltrrd 2156	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹, ℂ)‘𝑗) ∈ ℝ)
22		iserile.7	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)
23	9, 22	sylan2br 282	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑘) ∈ ℝ)
24	2, 6, 8, 23, 13, 15	iseqss 9446	. . . . 5 ⊢ (𝜑 → seq𝑀( + , 𝐺, ℝ) = seq𝑀( + , 𝐺, ℂ))
25	24	adantr 270	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → seq𝑀( + , 𝐺, ℝ) = seq𝑀( + , 𝐺, ℂ))
26	25	fveq1d 5200	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐺, ℝ)‘𝑗) = (seq𝑀( + , 𝐺, ℂ)‘𝑗))
27	1, 2, 22	iserfre 9454	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐺, ℝ):𝑍⟶ℝ)
28	27	ffvelrnda 5323	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐺, ℝ)‘𝑗) ∈ ℝ)
29	26, 28	eqeltrrd 2156	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐺, ℂ)‘𝑗) ∈ ℝ)
30		simpr 108	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
31	30, 1	syl6eleq 2171	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑀))
32	11	adantlr 460	. . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
33	23	adantlr 460	. . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑘) ∈ ℝ)
34		simpll 495	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝜑)
35	9	biimpri 131	. . . . 5 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑘 ∈ 𝑍)
36	35	adantl 271	. . . 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 403	. . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ≤ (𝐺‘𝑘))
39	31, 32, 33, 38	serile 9474	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹, ℂ)‘𝑗) ≤ (seq𝑀( + , 𝐺, ℂ)‘𝑗))
40	1, 2, 3, 4, 21, 29, 39	climle 10172	1 ⊢ (𝜑 → 𝐴 ≤ 𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 Vcvv 2601 ⊆ wss 2973 class class class wbr 3785 ‘cfv 4922 (class class class)co 5532 ℂcc 6979 ℝcr 6980 + caddc 6984 ≤ cle 7154 ℤcz 8351 ℤ_≥cuz 8619 seqcseq 9431 ⇝ cli 10117

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 ax-caucvg 7096

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-n0 8289 df-z 8352 df-uz 8620 df-rp 8735 df-fz 9030 df-fzo 9153 df-iseq 9432 df-iexp 9476 df-cj 9729 df-re 9730 df-im 9731 df-rsqrt 9884 df-abs 9885 df-clim 10118

This theorem is referenced by: iserige0 10181

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