![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 5336 | . . . . 5 ⊢ (𝜑 → (𝐶 ∖ 𝐴) ∈ V) |
3 | esummono.f | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
4 | simpr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐶 ∖ 𝐴)) → 𝑘 ∈ (𝐶 ∖ 𝐴)) | |
5 | 4 | eldifad 3959 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐶 ∖ 𝐴)) → 𝑘 ∈ 𝐶) |
6 | esummono.b | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐵 ∈ (0[,]+∞)) | |
7 | 5, 6 | syldan 589 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐶 ∖ 𝐴)) → 𝐵 ∈ (0[,]+∞)) |
8 | 7 | ex 411 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ (𝐶 ∖ 𝐴) → 𝐵 ∈ (0[,]+∞))) |
9 | 3, 8 | ralrimi 3245 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ (0[,]+∞)) |
10 | nfcv 2892 | . . . . . 6 ⊢ Ⅎ𝑘(𝐶 ∖ 𝐴) | |
11 | 10 | esumcl 33863 | . . . . 5 ⊢ (((𝐶 ∖ 𝐴) ∈ V ∧ ∀𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ (0[,]+∞)) |
12 | 2, 9, 11 | syl2anc 582 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ (0[,]+∞)) |
13 | elxrge0 13488 | . . . . 5 ⊢ (Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ (0[,]+∞) ↔ (Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ ℝ* ∧ 0 ≤ Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵)) | |
14 | 13 | simprbi 495 | . . . 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 13461 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
17 | esummono.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
18 | 1, 17 | ssexd 5329 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V) |
19 | 17 | sselda 3979 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐶) |
20 | 19, 6 | syldan 589 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
21 | 20 | ex 411 | . . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞))) |
22 | 3, 21 | ralrimi 3245 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
23 | nfcv 2892 | . . . . . . 7 ⊢ Ⅎ𝑘𝐴 | |
24 | 23 | esumcl 33863 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
25 | 18, 22, 24 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
26 | 16, 25 | sselid 3977 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) |
27 | 16, 12 | sselid 3977 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ ℝ*) |
28 | xraddge02 32660 | . . . 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 582 | . . 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 4476 | . . . . 5 ⊢ (𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅ | |
32 | 31 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅) |
33 | 3, 23, 10, 18, 2, 32, 20, 7 | esumsplit 33886 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐴 ∪ (𝐶 ∖ 𝐴))𝐵 = (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵)) |
34 | undif 4486 | . . . . 5 ⊢ (𝐴 ⊆ 𝐶 ↔ (𝐴 ∪ (𝐶 ∖ 𝐴)) = 𝐶) | |
35 | 17, 34 | sylib 217 | . . . 4 ⊢ (𝜑 → (𝐴 ∪ (𝐶 ∖ 𝐴)) = 𝐶) |
36 | 3, 35 | esumeq1d 33868 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐴 ∪ (𝐶 ∖ 𝐴))𝐵 = Σ*𝑘 ∈ 𝐶𝐵) |
37 | 33, 36 | eqtr3d 2768 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵) = Σ*𝑘 ∈ 𝐶𝐵) |
38 | 30, 37 | breqtrd 5179 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ Σ*𝑘 ∈ 𝐶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 ∀wral 3051 Vcvv 3462 ∖ cdif 3944 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 ∅c0 4325 class class class wbr 5153 (class class class)co 7424 0cc0 11158 +∞cpnf 11295 ℝ*cxr 11297 ≤ cle 11299 +𝑒 cxad 13144 [,]cicc 13381 Σ*cesum 33860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237 ax-mulf 11238 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-mod 13890 df-seq 14022 df-exp 14082 df-fac 14291 df-bc 14320 df-hash 14348 df-shft 15072 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-limsup 15473 df-clim 15490 df-rlim 15491 df-sum 15691 df-ef 16069 df-sin 16071 df-cos 16072 df-pi 16074 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-rest 17437 df-topn 17438 df-0g 17456 df-gsum 17457 df-topgen 17458 df-pt 17459 df-prds 17462 df-ordt 17516 df-xrs 17517 df-qtop 17522 df-imas 17523 df-xps 17525 df-mre 17599 df-mrc 17600 df-acs 17602 df-ps 18591 df-tsr 18592 df-plusf 18632 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-mhm 18773 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-mulg 19062 df-subg 19117 df-cntz 19311 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-cring 20219 df-subrng 20528 df-subrg 20553 df-abv 20788 df-lmod 20838 df-scaf 20839 df-sra 21151 df-rgmod 21152 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-fbas 21340 df-fg 21341 df-cnfld 21344 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-cld 23014 df-ntr 23015 df-cls 23016 df-nei 23093 df-lp 23131 df-perf 23132 df-cn 23222 df-cnp 23223 df-haus 23310 df-tx 23557 df-hmeo 23750 df-fil 23841 df-fm 23933 df-flim 23934 df-flf 23935 df-tmd 24067 df-tgp 24068 df-tsms 24122 df-trg 24155 df-xms 24317 df-ms 24318 df-tms 24319 df-nm 24582 df-ngp 24583 df-nrg 24585 df-nlm 24586 df-ii 24888 df-cncf 24889 df-limc 25886 df-dv 25887 df-log 26583 df-esum 33861 |
This theorem is referenced by: esumpad2 33889 esumrnmpt2 33901 esumfsup 33903 esum2d 33926 esumiun 33927 omssubadd 34134 carsggect 34152 |
Copyright terms: Public domain | W3C validator |