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Mirrors > Home > ILE Home > Th. List > fsumlessfi | GIF version |
Description: A shorter sum of nonnegative terms is no greater than a longer one. (Contributed by NM, 26-Dec-2005.) (Revised by Jim Kingdon, 12-Oct-2022.) |
Ref | Expression |
---|---|
fsumge0.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumge0.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) |
fsumge0.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) |
fsumless.4 | ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
fsumlessfi.c | ⊢ (𝜑 → 𝐶 ∈ Fin) |
Ref | Expression |
---|---|
fsumlessfi | ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumge0.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
2 | fsumlessfi.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ Fin) | |
3 | fsumless.4 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
4 | diffifi 6841 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ∈ Fin ∧ 𝐶 ⊆ 𝐴) → (𝐴 ∖ 𝐶) ∈ Fin) | |
5 | 1, 2, 3, 4 | syl3anc 1220 | . . . 4 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ∈ Fin) |
6 | eldifi 3230 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∖ 𝐶) → 𝑘 ∈ 𝐴) | |
7 | fsumge0.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
8 | 6, 7 | sylan2 284 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐶)) → 𝐵 ∈ ℝ) |
9 | fsumge0.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) | |
10 | 6, 9 | sylan2 284 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐶)) → 0 ≤ 𝐵) |
11 | 5, 8, 10 | fsumge0 11367 | . . 3 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ (𝐴 ∖ 𝐶)𝐵) |
12 | 3 | sselda 3128 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝑘 ∈ 𝐴) |
13 | 12, 7 | syldan 280 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐵 ∈ ℝ) |
14 | 2, 13 | fsumrecl 11309 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ∈ ℝ) |
15 | 5, 8 | fsumrecl 11309 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∖ 𝐶)𝐵 ∈ ℝ) |
16 | 14, 15 | addge01d 8412 | . . 3 ⊢ (𝜑 → (0 ≤ Σ𝑘 ∈ (𝐴 ∖ 𝐶)𝐵 ↔ Σ𝑘 ∈ 𝐶 𝐵 ≤ (Σ𝑘 ∈ 𝐶 𝐵 + Σ𝑘 ∈ (𝐴 ∖ 𝐶)𝐵))) |
17 | 11, 16 | mpbid 146 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ≤ (Σ𝑘 ∈ 𝐶 𝐵 + Σ𝑘 ∈ (𝐴 ∖ 𝐶)𝐵)) |
18 | disjdif 3467 | . . . 4 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐶)) = ∅ | |
19 | 18 | a1i 9 | . . 3 ⊢ (𝜑 → (𝐶 ∩ (𝐴 ∖ 𝐶)) = ∅) |
20 | undiffi 6871 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ∈ Fin ∧ 𝐶 ⊆ 𝐴) → 𝐴 = (𝐶 ∪ (𝐴 ∖ 𝐶))) | |
21 | 1, 2, 3, 20 | syl3anc 1220 | . . 3 ⊢ (𝜑 → 𝐴 = (𝐶 ∪ (𝐴 ∖ 𝐶))) |
22 | 7 | recnd 7908 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
23 | 19, 21, 1, 22 | fsumsplit 11315 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (Σ𝑘 ∈ 𝐶 𝐵 + Σ𝑘 ∈ (𝐴 ∖ 𝐶)𝐵)) |
24 | 17, 23 | breqtrrd 3994 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1335 ∈ wcel 2128 ∖ cdif 3099 ∪ cun 3100 ∩ cin 3101 ⊆ wss 3102 ∅c0 3395 class class class wbr 3967 (class class class)co 5826 Fincfn 6687 ℝcr 7733 0cc0 7734 + caddc 7737 ≤ cle 7915 Σcsu 11261 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4081 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-mulrcl 7833 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-precex 7844 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-apti 7849 ax-pre-ltadd 7850 ax-pre-mulgt0 7851 ax-pre-mulext 7852 ax-arch 7853 ax-caucvg 7854 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-id 4255 df-po 4258 df-iso 4259 df-iord 4328 df-on 4330 df-ilim 4331 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-isom 5181 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-recs 6254 df-irdg 6319 df-frec 6340 df-1o 6365 df-oadd 6369 df-er 6482 df-en 6688 df-dom 6689 df-fin 6690 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-reap 8454 df-ap 8461 df-div 8550 df-inn 8839 df-2 8897 df-3 8898 df-4 8899 df-n0 9096 df-z 9173 df-uz 9445 df-q 9535 df-rp 9567 df-ico 9804 df-fz 9919 df-fzo 10051 df-seqfrec 10354 df-exp 10428 df-ihash 10661 df-cj 10753 df-re 10754 df-im 10755 df-rsqrt 10909 df-abs 10910 df-clim 11187 df-sumdc 11262 |
This theorem is referenced by: fsumge1 11369 fsum00 11370 |
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