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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 5284 | . . . . 5 ⊢ (𝜑 → (𝐶 ∖ 𝐴) ∈ V) |
3 | esummono.f | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
4 | simpr 485 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐶 ∖ 𝐴)) → 𝑘 ∈ (𝐶 ∖ 𝐴)) | |
5 | 4 | eldifad 3920 | . . . . . . . 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 3238 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ (0[,]+∞)) |
10 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑘(𝐶 ∖ 𝐴) | |
11 | 10 | esumcl 32402 | . . . . 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 13302 | . . . . 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 13275 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
17 | esummono.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
18 | 1, 17 | ssexd 5279 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V) |
19 | 17 | sselda 3942 | . . . . . . . . 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 3238 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
23 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑘𝐴 | |
24 | 23 | esumcl 32402 | . . . . . 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 3940 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) |
27 | 16, 12 | sselid 3940 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ ℝ*) |
28 | xraddge02 31455 | . . . 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 4429 | . . . . 5 ⊢ (𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅ | |
32 | 31 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅) |
33 | 3, 23, 10, 18, 2, 32, 20, 7 | esumsplit 32425 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐴 ∪ (𝐶 ∖ 𝐴))𝐵 = (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵)) |
34 | undif 4439 | . . . . 5 ⊢ (𝐴 ⊆ 𝐶 ↔ (𝐴 ∪ (𝐶 ∖ 𝐴)) = 𝐶) | |
35 | 17, 34 | sylib 217 | . . . 4 ⊢ (𝜑 → (𝐴 ∪ (𝐶 ∖ 𝐴)) = 𝐶) |
36 | 3, 35 | esumeq1d 32407 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐴 ∪ (𝐶 ∖ 𝐴))𝐵 = Σ*𝑘 ∈ 𝐶𝐵) |
37 | 33, 36 | eqtr3d 2779 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵) = Σ*𝑘 ∈ 𝐶𝐵) |
38 | 30, 37 | breqtrd 5129 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ Σ*𝑘 ∈ 𝐶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 ∀wral 3062 Vcvv 3443 ∖ cdif 3905 ∪ cun 3906 ∩ cin 3907 ⊆ wss 3908 ∅c0 4280 class class class wbr 5103 (class class class)co 7349 0cc0 10984 +∞cpnf 11119 ℝ*cxr 11121 ≤ cle 11123 +𝑒 cxad 12959 [,]cicc 13195 Σ*cesum 32399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-inf2 9510 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 ax-pre-sup 11062 ax-addf 11063 ax-mulf 11064 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-se 5586 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7607 df-om 7793 df-1st 7911 df-2nd 7912 df-supp 8060 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-1o 8379 df-2o 8380 df-er 8581 df-map 8700 df-pm 8701 df-ixp 8769 df-en 8817 df-dom 8818 df-sdom 8819 df-fin 8820 df-fsupp 9239 df-fi 9280 df-sup 9311 df-inf 9312 df-oi 9379 df-card 9808 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-div 11746 df-nn 12087 df-2 12149 df-3 12150 df-4 12151 df-5 12152 df-6 12153 df-7 12154 df-8 12155 df-9 12156 df-n0 12347 df-z 12433 df-dec 12551 df-uz 12696 df-q 12802 df-rp 12844 df-xneg 12961 df-xadd 12962 df-xmul 12963 df-ioo 13196 df-ioc 13197 df-ico 13198 df-icc 13199 df-fz 13353 df-fzo 13496 df-fl 13625 df-mod 13703 df-seq 13835 df-exp 13896 df-fac 14101 df-bc 14130 df-hash 14158 df-shft 14885 df-cj 14917 df-re 14918 df-im 14919 df-sqrt 15053 df-abs 15054 df-limsup 15287 df-clim 15304 df-rlim 15305 df-sum 15505 df-ef 15884 df-sin 15886 df-cos 15887 df-pi 15889 df-struct 16953 df-sets 16970 df-slot 16988 df-ndx 17000 df-base 17018 df-ress 17047 df-plusg 17080 df-mulr 17081 df-starv 17082 df-sca 17083 df-vsca 17084 df-ip 17085 df-tset 17086 df-ple 17087 df-ds 17089 df-unif 17090 df-hom 17091 df-cco 17092 df-rest 17238 df-topn 17239 df-0g 17257 df-gsum 17258 df-topgen 17259 df-pt 17260 df-prds 17263 df-ordt 17317 df-xrs 17318 df-qtop 17323 df-imas 17324 df-xps 17326 df-mre 17400 df-mrc 17401 df-acs 17403 df-ps 18389 df-tsr 18390 df-plusf 18430 df-mgm 18431 df-sgrp 18480 df-mnd 18491 df-mhm 18535 df-submnd 18536 df-grp 18685 df-minusg 18686 df-sbg 18687 df-mulg 18806 df-subg 18857 df-cntz 19029 df-cmn 19493 df-abl 19494 df-mgp 19826 df-ur 19843 df-ring 19890 df-cring 19891 df-subrg 20143 df-abv 20199 df-lmod 20247 df-scaf 20248 df-sra 20556 df-rgmod 20557 df-psmet 20711 df-xmet 20712 df-met 20713 df-bl 20714 df-mopn 20715 df-fbas 20716 df-fg 20717 df-cnfld 20720 df-top 22165 df-topon 22182 df-topsp 22204 df-bases 22218 df-cld 22292 df-ntr 22293 df-cls 22294 df-nei 22371 df-lp 22409 df-perf 22410 df-cn 22500 df-cnp 22501 df-haus 22588 df-tx 22835 df-hmeo 23028 df-fil 23119 df-fm 23211 df-flim 23212 df-flf 23213 df-tmd 23345 df-tgp 23346 df-tsms 23400 df-trg 23433 df-xms 23595 df-ms 23596 df-tms 23597 df-nm 23860 df-ngp 23861 df-nrg 23863 df-nlm 23864 df-ii 24162 df-cncf 24163 df-limc 25152 df-dv 25153 df-log 25834 df-esum 32400 |
This theorem is referenced by: esumpad2 32428 esumrnmpt2 32440 esumfsup 32442 esum2d 32465 esumiun 32466 omssubadd 32673 carsggect 32691 |
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