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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 5337 | . . . . 5 ⊢ (𝜑 → (𝐶 ∖ 𝐴) ∈ V) |
3 | esummono.f | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
4 | simpr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐶 ∖ 𝐴)) → 𝑘 ∈ (𝐶 ∖ 𝐴)) | |
5 | 4 | eldifad 3975 | . . . . . . . 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 412 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ (𝐶 ∖ 𝐴) → 𝐵 ∈ (0[,]+∞))) |
9 | 3, 8 | ralrimi 3255 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ (0[,]+∞)) |
10 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑘(𝐶 ∖ 𝐴) | |
11 | 10 | esumcl 34011 | . . . . 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 13494 | . . . . 5 ⊢ (Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ (0[,]+∞) ↔ (Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ ℝ* ∧ 0 ≤ Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵)) | |
14 | 13 | simprbi 496 | . . . 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 13467 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
17 | esummono.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
18 | 1, 17 | ssexd 5330 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V) |
19 | 17 | sselda 3995 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐶) |
20 | 19, 6 | syldan 591 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
21 | 20 | ex 412 | . . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞))) |
22 | 3, 21 | ralrimi 3255 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
23 | nfcv 2903 | . . . . . . 7 ⊢ Ⅎ𝑘𝐴 | |
24 | 23 | esumcl 34011 | . . . . . 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 3993 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) |
27 | 16, 12 | sselid 3993 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵 ∈ ℝ*) |
28 | xraddge02 32767 | . . . 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 4478 | . . . . 5 ⊢ (𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅ | |
32 | 31 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅) |
33 | 3, 23, 10, 18, 2, 32, 20, 7 | esumsplit 34034 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐴 ∪ (𝐶 ∖ 𝐴))𝐵 = (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵)) |
34 | undif 4488 | . . . . 5 ⊢ (𝐴 ⊆ 𝐶 ↔ (𝐴 ∪ (𝐶 ∖ 𝐴)) = 𝐶) | |
35 | 17, 34 | sylib 218 | . . . 4 ⊢ (𝜑 → (𝐴 ∪ (𝐶 ∖ 𝐴)) = 𝐶) |
36 | 3, 35 | esumeq1d 34016 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐴 ∪ (𝐶 ∖ 𝐴))𝐵 = Σ*𝑘 ∈ 𝐶𝐵) |
37 | 33, 36 | eqtr3d 2777 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ (𝐶 ∖ 𝐴)𝐵) = Σ*𝑘 ∈ 𝐶𝐵) |
38 | 30, 37 | breqtrd 5174 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ Σ*𝑘 ∈ 𝐶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 class class class wbr 5148 (class class class)co 7431 0cc0 11153 +∞cpnf 11290 ℝ*cxr 11292 ≤ cle 11294 +𝑒 cxad 13150 [,]cicc 13387 Σ*cesum 34008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 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 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-ordt 17548 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-ps 18624 df-tsr 18625 df-plusf 18665 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-abv 20827 df-lmod 20877 df-scaf 20878 df-sra 21190 df-rgmod 21191 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-tmd 24096 df-tgp 24097 df-tsms 24151 df-trg 24184 df-xms 24346 df-ms 24347 df-tms 24348 df-nm 24611 df-ngp 24612 df-nrg 24614 df-nlm 24615 df-ii 24917 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-esum 34009 |
This theorem is referenced by: esumpad2 34037 esumrnmpt2 34049 esumfsup 34051 esum2d 34074 esumiun 34075 omssubadd 34282 carsggect 34300 |
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