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Mirrors > Home > ILE Home > Th. List > fsumge1 | GIF version |
Description: A sum of nonnegative numbers is greater than or equal to any one of its terms. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 4-Jun-2014.) |
Ref | Expression |
---|---|
fsumge0.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumge0.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) |
fsumge0.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) |
fsumge1.4 | ⊢ (𝑘 = 𝑀 → 𝐵 = 𝐶) |
fsumge1.5 | ⊢ (𝜑 → 𝑀 ∈ 𝐴) |
Ref | Expression |
---|---|
fsumge1 | ⊢ (𝜑 → 𝐶 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumge1.5 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝐴) | |
2 | fsumge1.4 | . . . . 5 ⊢ (𝑘 = 𝑀 → 𝐵 = 𝐶) | |
3 | 2 | eleq1d 2157 | . . . 4 ⊢ (𝑘 = 𝑀 → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ)) |
4 | fsumge0.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
5 | 4 | recnd 7577 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
6 | 5 | ralrimiva 2447 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
7 | 3, 6, 1 | rspcdva 2728 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
8 | 2 | sumsn 10866 | . . 3 ⊢ ((𝑀 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐵 = 𝐶) |
9 | 1, 7, 8 | syl2anc 404 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐵 = 𝐶) |
10 | fsumge0.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
11 | fsumge0.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) | |
12 | 1 | snssd 3588 | . . 3 ⊢ (𝜑 → {𝑀} ⊆ 𝐴) |
13 | snfig 6585 | . . . 4 ⊢ (𝑀 ∈ 𝐴 → {𝑀} ∈ Fin) | |
14 | 1, 13 | syl 14 | . . 3 ⊢ (𝜑 → {𝑀} ∈ Fin) |
15 | 10, 4, 11, 12, 14 | fsumlessfi 10915 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
16 | 9, 15 | eqbrtrrd 3873 | 1 ⊢ (𝜑 → 𝐶 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1290 ∈ wcel 1439 {csn 3450 class class class wbr 3851 Fincfn 6511 ℂcc 7409 ℝcr 7410 0cc0 7411 ≤ cle 7584 Σcsu 10803 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-mulrcl 7505 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-precex 7516 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-apti 7521 ax-pre-ltadd 7522 ax-pre-mulgt0 7523 ax-pre-mulext 7524 ax-arch 7525 ax-caucvg 7526 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-if 3398 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-po 4132 df-iso 4133 df-iord 4202 df-on 4204 df-ilim 4205 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-isom 5037 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-frec 6170 df-1o 6195 df-oadd 6199 df-er 6306 df-en 6512 df-dom 6513 df-fin 6514 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-reap 8113 df-ap 8120 df-div 8201 df-inn 8484 df-2 8542 df-3 8543 df-4 8544 df-n0 8735 df-z 8812 df-uz 9081 df-q 9166 df-rp 9196 df-ico 9373 df-fz 9486 df-fzo 9615 df-iseq 9914 df-seq3 9915 df-exp 10016 df-ihash 10245 df-cj 10337 df-re 10338 df-im 10339 df-rsqrt 10492 df-abs 10493 df-clim 10728 df-isum 10804 |
This theorem is referenced by: mertenslemub 10989 |
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