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Mirrors > Home > MPE Home > Th. List > cvgcmpub | Structured version Visualization version GIF version |
Description: An upper bound for the limit of a real infinite series. This theorem can also be used to compare two infinite series. (Contributed by Mario Carneiro, 24-Mar-2014.) |
Ref | Expression |
---|---|
cvgcmp.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
cvgcmp.2 | ⊢ (𝜑 → 𝑁 ∈ 𝑍) |
cvgcmp.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
cvgcmp.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ) |
cvgcmpub.5 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) |
cvgcmpub.6 | ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵) |
cvgcmpub.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ≤ (𝐹‘𝑘)) |
Ref | Expression |
---|---|
cvgcmpub | ⊢ (𝜑 → 𝐵 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvgcmp.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | cvgcmp.2 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑍) | |
3 | 2, 1 | eleqtrdi 2848 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
4 | eluzel2 12443 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
6 | cvgcmpub.6 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵) | |
7 | cvgcmpub.5 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) | |
8 | cvgcmp.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ) | |
9 | 1, 5, 8 | serfre 13605 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℝ) |
10 | 9 | ffvelrnda 6904 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ) |
11 | cvgcmp.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) | |
12 | 1, 5, 11 | serfre 13605 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ) |
13 | 12 | ffvelrnda 6904 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ) |
14 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍) | |
15 | 14, 1 | eleqtrdi 2848 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (ℤ≥‘𝑀)) |
16 | simpl 486 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝜑) | |
17 | elfzuz 13108 | . . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
18 | 17, 1 | eleqtrrdi 2849 | . . . 4 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍) |
19 | 16, 18, 8 | syl2an 599 | . . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) ∈ ℝ) |
20 | 16, 18, 11 | syl2an 599 | . . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℝ) |
21 | cvgcmpub.7 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ≤ (𝐹‘𝑘)) | |
22 | 16, 18, 21 | syl2an 599 | . . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) ≤ (𝐹‘𝑘)) |
23 | 15, 19, 20, 22 | serle 13631 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ≤ (seq𝑀( + , 𝐹)‘𝑛)) |
24 | 1, 5, 6, 7, 10, 13, 23 | climle 15201 | 1 ⊢ (𝜑 → 𝐵 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 ℝcr 10728 + caddc 10732 ≤ cle 10868 ℤcz 12176 ℤ≥cuz 12438 ...cfz 13095 seqcseq 13574 ⇝ cli 15045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-sup 9058 df-inf 9059 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-rlim 15050 |
This theorem is referenced by: (None) |
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