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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 2239 | . . . 4 ⊢ (𝑘 = 𝑀 → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ)) |
4 | fsumge0.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
5 | 4 | recnd 7935 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
6 | 5 | ralrimiva 2543 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
7 | 3, 6, 1 | rspcdva 2839 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
8 | 2 | sumsn 11361 | . . 3 ⊢ ((𝑀 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐵 = 𝐶) |
9 | 1, 7, 8 | syl2anc 409 | . 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 3723 | . . 3 ⊢ (𝜑 → {𝑀} ⊆ 𝐴) |
13 | snfig 6788 | . . . 4 ⊢ (𝑀 ∈ 𝐴 → {𝑀} ∈ Fin) | |
14 | 1, 13 | syl 14 | . . 3 ⊢ (𝜑 → {𝑀} ∈ Fin) |
15 | 10, 4, 11, 12, 14 | fsumlessfi 11410 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
16 | 9, 15 | eqbrtrrd 4011 | 1 ⊢ (𝜑 → 𝐶 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 {csn 3581 class class class wbr 3987 Fincfn 6714 ℂcc 7759 ℝcr 7760 0cc0 7761 ≤ cle 7942 Σcsu 11303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879 ax-arch 7880 ax-caucvg 7881 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-irdg 6346 df-frec 6367 df-1o 6392 df-oadd 6396 df-er 6509 df-en 6715 df-dom 6716 df-fin 6717 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 df-inn 8866 df-2 8924 df-3 8925 df-4 8926 df-n0 9123 df-z 9200 df-uz 9475 df-q 9566 df-rp 9598 df-ico 9838 df-fz 9953 df-fzo 10086 df-seqfrec 10389 df-exp 10463 df-ihash 10697 df-cj 10793 df-re 10794 df-im 10795 df-rsqrt 10949 df-abs 10950 df-clim 11229 df-sumdc 11304 |
This theorem is referenced by: mertenslemub 11484 |
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