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Mirrors > Home > MPE Home > Th. List > fsumle | Structured version Visualization version GIF version |
Description: If all of the terms of finite sums compare, so do the sums. (Contributed by NM, 11-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
fsumle.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumle.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) |
fsumle.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ) |
fsumle.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶) |
Ref | Expression |
---|---|
fsumle | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumle.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
2 | fsumle.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ) | |
3 | fsumle.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
4 | 2, 3 | resubcld 11068 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 − 𝐵) ∈ ℝ) |
5 | fsumle.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶) | |
6 | 2, 3 | subge0d 11230 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (0 ≤ (𝐶 − 𝐵) ↔ 𝐵 ≤ 𝐶)) |
7 | 5, 6 | mpbird 259 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ (𝐶 − 𝐵)) |
8 | 1, 4, 7 | fsumge0 15150 | . . 3 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝐴 (𝐶 − 𝐵)) |
9 | 2 | recnd 10669 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
10 | 3 | recnd 10669 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
11 | 1, 9, 10 | fsumsub 15143 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐶 − 𝐵) = (Σ𝑘 ∈ 𝐴 𝐶 − Σ𝑘 ∈ 𝐴 𝐵)) |
12 | 8, 11 | breqtrd 5092 | . 2 ⊢ (𝜑 → 0 ≤ (Σ𝑘 ∈ 𝐴 𝐶 − Σ𝑘 ∈ 𝐴 𝐵)) |
13 | 1, 2 | fsumrecl 15091 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 ∈ ℝ) |
14 | 1, 3 | fsumrecl 15091 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ) |
15 | 13, 14 | subge0d 11230 | . 2 ⊢ (𝜑 → (0 ≤ (Σ𝑘 ∈ 𝐴 𝐶 − Σ𝑘 ∈ 𝐴 𝐵) ↔ Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐶)) |
16 | 12, 15 | mpbid 234 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 Fincfn 8509 ℝcr 10536 0cc0 10537 ≤ cle 10676 − cmin 10870 Σcsu 15042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-ico 12745 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 |
This theorem is referenced by: o1fsum 15168 climcndslem1 15204 climcndslem2 15205 mertenslem1 15240 ovoliunlem1 24103 ovolicc2lem4 24121 uniioombllem4 24187 dvfsumle 24618 dvfsumabs 24620 mtest 24992 mtestbdd 24993 abelthlem7 25026 birthdaylem3 25531 fsumharmonic 25589 ftalem1 25650 ftalem5 25654 basellem8 25665 chtleppi 25786 chpub 25796 logfaclbnd 25798 bposlem1 25860 chebbnd1lem1 26045 chtppilimlem1 26049 vmadivsum 26058 rplogsumlem1 26060 rplogsumlem2 26061 rpvmasumlem 26063 dchrisumlem2 26066 dchrmusum2 26070 dchrvmasumlem3 26075 dchrvmasumiflem1 26077 dchrisum0fno1 26087 dchrisum0lem1 26092 dchrisum0lem2a 26093 mudivsum 26106 mulogsumlem 26107 mulog2sumlem2 26111 vmalogdivsum2 26114 2vmadivsumlem 26116 selberglem2 26122 selbergb 26125 selberg2b 26128 chpdifbndlem1 26129 logdivbnd 26132 selberg3lem1 26133 selberg4lem1 26136 pntrlog2bndlem1 26153 pntrlog2bndlem2 26154 pntrlog2bndlem3 26155 pntrlog2bndlem5 26157 pntrlog2bndlem6 26159 pntpbnd2 26163 pntlemj 26179 reprlt 31890 reprgt 31892 hgt750lemf 31924 hgt750lemb 31927 knoppndvlem11 33861 geomcau 35049 fltnltalem 39323 stoweidlem11 42345 stoweidlem26 42360 stoweidlem38 42372 stirlinglem12 42419 etransclem23 42591 etransclem32 42600 sge0le 42738 hoidmvlelem2 42927 |
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