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Mirrors > Home > MPE Home > Th. List > Mathboxes > esummono | Structured version Visualization version GIF version |
Description: Extended sum is monotonic. (Contributed by Thierry Arnoux, 19-Oct-2017.) |
Ref | Expression |
---|---|
esummono.f | ⊢ Ⅎ𝑘𝜑 |
esummono.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
esummono.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐵 ∈ (0[,]+∞)) |
esummono.a | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Ref | Expression |
---|---|
esummono | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ Σ*𝑘 ∈ 𝐶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esummono.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
2 | 1 | difexd 5257 | . . . . 5 ⊢ (𝜑 → (𝐶 ∖ 𝐴) ∈ V) |
3 | esummono.f | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
4 | simpr 485 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐶 ∖ 𝐴)) → 𝑘 ∈ (𝐶 ∖ 𝐴)) | |
5 | 4 | eldifad 3904 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐶 ∖ 𝐴)) → 𝑘 ∈ 𝐶) |
6 | esummono.b | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐵 ∈ (0[,]+∞)) | |
7 | 5, 6 | syldan 591 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐶 ∖ 𝐴)) → 𝐵 ∈ (0[,]+∞)) |
8 | 7 | ex 413 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ (𝐶 ∖ 𝐴) → 𝐵 ∈ (0[,]+∞))) |
9 | 3, 8 | ralrimi 3142 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ (0[,]+∞)) |
10 | nfcv 2909 | . . . . . 6 ⊢ Ⅎ𝑘(𝐶 ∖ 𝐴) | |
11 | 10 | esumcl 31994 | . . . . 5 ⊢ (((𝐶 ∖ 𝐴) ∈ V ∧ ∀𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ (0[,]+∞)) |
12 | 2, 9, 11 | syl2anc 584 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ (0[,]+∞)) |
13 | elxrge0 13188 | . . . . 5 ⊢ (Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ (0[,]+∞) ↔ (Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ ℝ* ∧ 0 ≤ Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵)) | |
14 | 13 | simprbi 497 | . . . 4 ⊢ (Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ (0[,]+∞) → 0 ≤ Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵) |
15 | 12, 14 | syl 17 | . . 3 ⊢ (𝜑 → 0 ≤ Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵) |
16 | iccssxr 13161 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
17 | esummono.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
18 | 1, 17 | ssexd 5252 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V) |
19 | 17 | sselda 3926 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐶) |
20 | 19, 6 | syldan 591 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
21 | 20 | ex 413 | . . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞))) |
22 | 3, 21 | ralrimi 3142 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
23 | nfcv 2909 | . . . . . . 7 ⊢ Ⅎ𝑘𝐴 | |
24 | 23 | esumcl 31994 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
25 | 18, 22, 24 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
26 | 16, 25 | sselid 3924 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) |
27 | 16, 12 | sselid 3924 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ ℝ*) |
28 | xraddge02 31075 | . . . 4 ⊢ ((Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ* ∧ Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ ℝ*) → (0 ≤ Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 → Σ*𝑘 ∈ 𝐴𝐵 ≤ (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵))) | |
29 | 26, 27, 28 | syl2anc 584 | . . 3 ⊢ (𝜑 → (0 ≤ Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 → Σ*𝑘 ∈ 𝐴𝐵 ≤ (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵))) |
30 | 15, 29 | mpd 15 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵)) |
31 | disjdif 4411 | . . . . 5 ⊢ (𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅ | |
32 | 31 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅) |
33 | 3, 23, 10, 18, 2, 32, 20, 7 | esumsplit 32017 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐴 ∪ (𝐶 ∖ 𝐴))𝐵 = (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵)) |
34 | undif 4421 | . . . . 5 ⊢ (𝐴 ⊆ 𝐶 ↔ (𝐴 ∪ (𝐶 ∖ 𝐴)) = 𝐶) | |
35 | 17, 34 | sylib 217 | . . . 4 ⊢ (𝜑 → (𝐴 ∪ (𝐶 ∖ 𝐴)) = 𝐶) |
36 | 3, 35 | esumeq1d 31999 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐴 ∪ (𝐶 ∖ 𝐴))𝐵 = Σ*𝑘 ∈ 𝐶𝐵) |
37 | 33, 36 | eqtr3d 2782 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵) = Σ*𝑘 ∈ 𝐶𝐵) |
38 | 30, 37 | breqtrd 5105 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ Σ*𝑘 ∈ 𝐶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 Ⅎwnf 1790 ∈ wcel 2110 ∀wral 3066 Vcvv 3431 ∖ cdif 3889 ∪ cun 3890 ∩ cin 3891 ⊆ wss 3892 ∅c0 4262 class class class wbr 5079 (class class class)co 7271 0cc0 10872 +∞cpnf 11007 ℝ*cxr 11009 ≤ cle 11011 +𝑒 cxad 12845 [,]cicc 13081 Σ*cesum 31991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-er 8481 df-map 8600 df-pm 8601 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-fi 9148 df-sup 9179 df-inf 9180 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-ioo 13082 df-ioc 13083 df-ico 13084 df-icc 13085 df-fz 13239 df-fzo 13382 df-fl 13510 df-mod 13588 df-seq 13720 df-exp 13781 df-fac 13986 df-bc 14015 df-hash 14043 df-shft 14776 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-limsup 15178 df-clim 15195 df-rlim 15196 df-sum 15396 df-ef 15775 df-sin 15777 df-cos 15778 df-pi 15780 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-0g 17150 df-gsum 17151 df-topgen 17152 df-pt 17153 df-prds 17156 df-ordt 17210 df-xrs 17211 df-qtop 17216 df-imas 17217 df-xps 17219 df-mre 17293 df-mrc 17294 df-acs 17296 df-ps 18282 df-tsr 18283 df-plusf 18323 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-mulg 18699 df-subg 18750 df-cntz 18921 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-cring 19784 df-subrg 20020 df-abv 20075 df-lmod 20123 df-scaf 20124 df-sra 20432 df-rgmod 20433 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-fbas 20592 df-fg 20593 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cld 22168 df-ntr 22169 df-cls 22170 df-nei 22247 df-lp 22285 df-perf 22286 df-cn 22376 df-cnp 22377 df-haus 22464 df-tx 22711 df-hmeo 22904 df-fil 22995 df-fm 23087 df-flim 23088 df-flf 23089 df-tmd 23221 df-tgp 23222 df-tsms 23276 df-trg 23309 df-xms 23471 df-ms 23472 df-tms 23473 df-nm 23736 df-ngp 23737 df-nrg 23739 df-nlm 23740 df-ii 24038 df-cncf 24039 df-limc 25028 df-dv 25029 df-log 25710 df-esum 31992 |
This theorem is referenced by: esumpad2 32020 esumrnmpt2 32032 esumfsup 32034 esum2d 32057 esumiun 32058 omssubadd 32263 carsggect 32281 |
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