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Mirrors > Home > ILE Home > Th. List > ser3le | GIF version |
Description: Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Jim Kingdon, 23-Apr-2023.) |
Ref | Expression |
---|---|
ser3ge0.1 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
ser3ge0.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ) |
ser3le.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℝ) |
serle.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ≤ (𝐺‘𝑘)) |
Ref | Expression |
---|---|
ser3le | ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ (seq𝑀( + , 𝐺)‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ser3ge0.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | vex 2740 | . . . . . 6 ⊢ 𝑘 ∈ V | |
3 | ser3le.3 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℝ) | |
4 | ser3ge0.2 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ) | |
5 | 3, 4 | resubcld 8336 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝐺‘𝑘) − (𝐹‘𝑘)) ∈ ℝ) |
6 | fveq2 5515 | . . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝐺‘𝑥) = (𝐺‘𝑘)) | |
7 | fveq2 5515 | . . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝐹‘𝑥) = (𝐹‘𝑘)) | |
8 | 6, 7 | oveq12d 5892 | . . . . . . 7 ⊢ (𝑥 = 𝑘 → ((𝐺‘𝑥) − (𝐹‘𝑥)) = ((𝐺‘𝑘) − (𝐹‘𝑘))) |
9 | eqid 2177 | . . . . . . 7 ⊢ (𝑥 ∈ V ↦ ((𝐺‘𝑥) − (𝐹‘𝑥))) = (𝑥 ∈ V ↦ ((𝐺‘𝑥) − (𝐹‘𝑥))) | |
10 | 8, 9 | fvmptg 5592 | . . . . . 6 ⊢ ((𝑘 ∈ V ∧ ((𝐺‘𝑘) − (𝐹‘𝑘)) ∈ ℝ) → ((𝑥 ∈ V ↦ ((𝐺‘𝑥) − (𝐹‘𝑥)))‘𝑘) = ((𝐺‘𝑘) − (𝐹‘𝑘))) |
11 | 2, 5, 10 | sylancr 414 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝑥 ∈ V ↦ ((𝐺‘𝑥) − (𝐹‘𝑥)))‘𝑘) = ((𝐺‘𝑘) − (𝐹‘𝑘))) |
12 | 11, 5 | eqeltrd 2254 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝑥 ∈ V ↦ ((𝐺‘𝑥) − (𝐹‘𝑥)))‘𝑘) ∈ ℝ) |
13 | elfzuz 10018 | . . . . . . 7 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
14 | serle.4 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ≤ (𝐺‘𝑘)) | |
15 | 13, 14 | sylan2 286 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ≤ (𝐺‘𝑘)) |
16 | 3, 4 | subge0d 8490 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (0 ≤ ((𝐺‘𝑘) − (𝐹‘𝑘)) ↔ (𝐹‘𝑘) ≤ (𝐺‘𝑘))) |
17 | 13, 16 | sylan2 286 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (0 ≤ ((𝐺‘𝑘) − (𝐹‘𝑘)) ↔ (𝐹‘𝑘) ≤ (𝐺‘𝑘))) |
18 | 15, 17 | mpbird 167 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ≤ ((𝐺‘𝑘) − (𝐹‘𝑘))) |
19 | 13, 11 | sylan2 286 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑥 ∈ V ↦ ((𝐺‘𝑥) − (𝐹‘𝑥)))‘𝑘) = ((𝐺‘𝑘) − (𝐹‘𝑘))) |
20 | 18, 19 | breqtrrd 4031 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ≤ ((𝑥 ∈ V ↦ ((𝐺‘𝑥) − (𝐹‘𝑥)))‘𝑘)) |
21 | 1, 12, 20 | ser3ge0 10514 | . . 3 ⊢ (𝜑 → 0 ≤ (seq𝑀( + , (𝑥 ∈ V ↦ ((𝐺‘𝑥) − (𝐹‘𝑥))))‘𝑁)) |
22 | 3 | recnd 7984 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ) |
23 | 4 | recnd 7984 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ) |
24 | 1, 22, 23, 11 | ser3sub 10503 | . . 3 ⊢ (𝜑 → (seq𝑀( + , (𝑥 ∈ V ↦ ((𝐺‘𝑥) − (𝐹‘𝑥))))‘𝑁) = ((seq𝑀( + , 𝐺)‘𝑁) − (seq𝑀( + , 𝐹)‘𝑁))) |
25 | 21, 24 | breqtrd 4029 | . 2 ⊢ (𝜑 → 0 ≤ ((seq𝑀( + , 𝐺)‘𝑁) − (seq𝑀( + , 𝐹)‘𝑁))) |
26 | eqid 2177 | . . . . 5 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
27 | eluzel2 9531 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
28 | 1, 27 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
29 | 26, 28, 3 | serfre 10472 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐺):(ℤ≥‘𝑀)⟶ℝ) |
30 | 29, 1 | ffvelcdmd 5652 | . . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐺)‘𝑁) ∈ ℝ) |
31 | 26, 28, 4 | serfre 10472 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶ℝ) |
32 | 31, 1 | ffvelcdmd 5652 | . . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ ℝ) |
33 | 30, 32 | subge0d 8490 | . 2 ⊢ (𝜑 → (0 ≤ ((seq𝑀( + , 𝐺)‘𝑁) − (seq𝑀( + , 𝐹)‘𝑁)) ↔ (seq𝑀( + , 𝐹)‘𝑁) ≤ (seq𝑀( + , 𝐺)‘𝑁))) |
34 | 25, 33 | mpbid 147 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ (seq𝑀( + , 𝐺)‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 Vcvv 2737 class class class wbr 4003 ↦ cmpt 4064 ‘cfv 5216 (class class class)co 5874 ℝcr 7809 0cc0 7810 + caddc 7813 ≤ cle 7991 − cmin 8126 ℤcz 9251 ℤ≥cuz 9526 ...cfz 10006 seqcseq 10442 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-inn 8918 df-n0 9175 df-z 9252 df-uz 9527 df-fz 10007 df-fzo 10140 df-seqfrec 10443 |
This theorem is referenced by: iserle 11345 cvgcmpub 11479 |
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