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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 11590 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 − 𝐵) ∈ ℝ) |
5 | fsumle.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶) | |
6 | 2, 3 | subge0d 11752 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (0 ≤ (𝐶 − 𝐵) ↔ 𝐵 ≤ 𝐶)) |
7 | 5, 6 | mpbird 257 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ (𝐶 − 𝐵)) |
8 | 1, 4, 7 | fsumge0 15687 | . . 3 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝐴 (𝐶 − 𝐵)) |
9 | 2 | recnd 11190 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
10 | 3 | recnd 11190 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
11 | 1, 9, 10 | fsumsub 15680 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐶 − 𝐵) = (Σ𝑘 ∈ 𝐴 𝐶 − Σ𝑘 ∈ 𝐴 𝐵)) |
12 | 8, 11 | breqtrd 5136 | . 2 ⊢ (𝜑 → 0 ≤ (Σ𝑘 ∈ 𝐴 𝐶 − Σ𝑘 ∈ 𝐴 𝐵)) |
13 | 1, 2 | fsumrecl 15626 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 ∈ ℝ) |
14 | 1, 3 | fsumrecl 15626 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ) |
15 | 13, 14 | subge0d 11752 | . 2 ⊢ (𝜑 → (0 ≤ (Σ𝑘 ∈ 𝐴 𝐶 − Σ𝑘 ∈ 𝐴 𝐵) ↔ Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐶)) |
16 | 12, 15 | mpbid 231 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 class class class wbr 5110 (class class class)co 7362 Fincfn 8890 ℝcr 11057 0cc0 11058 ≤ cle 11197 − cmin 11392 Σcsu 15577 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-ico 13277 df-fz 13432 df-fzo 13575 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 |
This theorem is referenced by: o1fsum 15705 climcndslem1 15741 climcndslem2 15742 mertenslem1 15776 ovoliunlem1 24882 ovolicc2lem4 24900 uniioombllem4 24966 dvfsumle 25401 dvfsumabs 25403 mtest 25779 mtestbdd 25780 abelthlem7 25813 birthdaylem3 26319 fsumharmonic 26377 ftalem1 26438 ftalem5 26442 basellem8 26453 chtleppi 26574 chpub 26584 logfaclbnd 26586 bposlem1 26648 chebbnd1lem1 26833 chtppilimlem1 26837 vmadivsum 26846 rplogsumlem1 26848 rplogsumlem2 26849 rpvmasumlem 26851 dchrisumlem2 26854 dchrmusum2 26858 dchrvmasumlem3 26863 dchrvmasumiflem1 26865 dchrisum0fno1 26875 dchrisum0lem1 26880 dchrisum0lem2a 26881 mudivsum 26894 mulogsumlem 26895 mulog2sumlem2 26899 vmalogdivsum2 26902 2vmadivsumlem 26904 selberglem2 26910 selbergb 26913 selberg2b 26916 chpdifbndlem1 26917 logdivbnd 26920 selberg3lem1 26921 selberg4lem1 26924 pntrlog2bndlem1 26941 pntrlog2bndlem2 26942 pntrlog2bndlem3 26943 pntrlog2bndlem5 26945 pntrlog2bndlem6 26947 pntpbnd2 26951 pntlemj 26967 reprlt 33272 reprgt 33274 hgt750lemf 33306 hgt750lemb 33309 knoppndvlem11 35014 geomcau 36247 lcmineqlem17 40531 fltnltalem 41029 stoweidlem11 44326 stoweidlem26 44341 stoweidlem38 44353 stirlinglem12 44400 etransclem23 44572 etransclem32 44581 sge0le 44722 hoidmvlelem2 44911 |
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