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Mirrors > Home > MPE Home > Th. List > serge0 | Structured version Visualization version GIF version |
Description: A finite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 8-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.) |
Ref | Expression |
---|---|
serge0.1 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
serge0.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) |
serge0.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹‘𝑘)) |
Ref | Expression |
---|---|
serge0 | ⊢ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | serge0.1 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | breq2 4927 | . . . 4 ⊢ (𝑥 = (𝐹‘𝑘) → (0 ≤ 𝑥 ↔ 0 ≤ (𝐹‘𝑘))) | |
3 | serge0.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) | |
4 | serge0.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹‘𝑘)) | |
5 | 2, 3, 4 | elrabd 3592 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥}) |
6 | breq2 4927 | . . . . . 6 ⊢ (𝑥 = 𝑘 → (0 ≤ 𝑥 ↔ 0 ≤ 𝑘)) | |
7 | 6 | elrab 3589 | . . . . 5 ⊢ (𝑘 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ (𝑘 ∈ ℝ ∧ 0 ≤ 𝑘)) |
8 | breq2 4927 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (0 ≤ 𝑥 ↔ 0 ≤ 𝑦)) | |
9 | 8 | elrab 3589 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) |
10 | breq2 4927 | . . . . . 6 ⊢ (𝑥 = (𝑘 + 𝑦) → (0 ≤ 𝑥 ↔ 0 ≤ (𝑘 + 𝑦))) | |
11 | readdcl 10410 | . . . . . . 7 ⊢ ((𝑘 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑘 + 𝑦) ∈ ℝ) | |
12 | 11 | ad2ant2r 734 | . . . . . 6 ⊢ (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) → (𝑘 + 𝑦) ∈ ℝ) |
13 | addge0 10922 | . . . . . . 7 ⊢ (((𝑘 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (0 ≤ 𝑘 ∧ 0 ≤ 𝑦)) → 0 ≤ (𝑘 + 𝑦)) | |
14 | 13 | an4s 647 | . . . . . 6 ⊢ (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) → 0 ≤ (𝑘 + 𝑦)) |
15 | 10, 12, 14 | elrabd 3592 | . . . . 5 ⊢ (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) → (𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥}) |
16 | 7, 9, 15 | syl2anb 588 | . . . 4 ⊢ ((𝑘 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ∧ 𝑦 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥}) → (𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥}) |
17 | 16 | adantl 474 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ∧ 𝑦 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})) → (𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥}) |
18 | 1, 5, 17 | seqcl 13198 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥}) |
19 | breq2 4927 | . . . 4 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑁) → (0 ≤ 𝑥 ↔ 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))) | |
20 | 19 | elrab 3589 | . . 3 ⊢ ((seq𝑀( + , 𝐹)‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ ((seq𝑀( + , 𝐹)‘𝑁) ∈ ℝ ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))) |
21 | 20 | simprbi 489 | . 2 ⊢ ((seq𝑀( + , 𝐹)‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁)) |
22 | 18, 21 | syl 17 | 1 ⊢ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∈ wcel 2048 {crab 3086 class class class wbr 4923 ‘cfv 6182 (class class class)co 6970 ℝcr 10326 0cc0 10327 + caddc 10330 ≤ cle 10467 ℤ≥cuz 12051 ...cfz 12701 seqcseq 13177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-n0 11701 df-z 11787 df-uz 12052 df-fz 12702 df-seq 13178 |
This theorem is referenced by: serle 13233 |
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