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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumle | Structured version Visualization version GIF version |
Description: If all of the terms of an extended sums compare, so do the sums. (Contributed by Thierry Arnoux, 8-Jun-2017.) |
Ref | Expression |
---|---|
esumadd.0 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
esumadd.1 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
esumadd.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) |
esumle.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶) |
Ref | Expression |
---|---|
esumle | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ Σ*𝑘 ∈ 𝐴𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssxr 13440 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | esumadd.0 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | esumadd.1 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
4 | 3 | ralrimiva 3143 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
5 | nfcv 2899 | . . . . . . 7 ⊢ Ⅎ𝑘𝐴 | |
6 | 5 | esumcl 33649 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
7 | 2, 4, 6 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
8 | 1, 7 | sselid 3978 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) |
9 | esumadd.2 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) | |
10 | 1, 9 | sselid 3978 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ*) |
11 | 1, 3 | sselid 3978 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
12 | 11 | xnegcld 13312 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -𝑒𝐵 ∈ ℝ*) |
13 | 10, 12 | xaddcld 13313 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +𝑒 -𝑒𝐵) ∈ ℝ*) |
14 | esumle.3 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶) | |
15 | xsubge0 13273 | . . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ (𝐶 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐶)) | |
16 | 10, 11, 15 | syl2anc 583 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (0 ≤ (𝐶 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐶)) |
17 | 14, 16 | mpbird 257 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ (𝐶 +𝑒 -𝑒𝐵)) |
18 | pnfge 13143 | . . . . . . . . 9 ⊢ ((𝐶 +𝑒 -𝑒𝐵) ∈ ℝ* → (𝐶 +𝑒 -𝑒𝐵) ≤ +∞) | |
19 | 13, 18 | syl 17 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +𝑒 -𝑒𝐵) ≤ +∞) |
20 | 0xr 11292 | . . . . . . . . 9 ⊢ 0 ∈ ℝ* | |
21 | pnfxr 11299 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
22 | elicc1 13401 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞) ↔ ((𝐶 +𝑒 -𝑒𝐵) ∈ ℝ* ∧ 0 ≤ (𝐶 +𝑒 -𝑒𝐵) ∧ (𝐶 +𝑒 -𝑒𝐵) ≤ +∞))) | |
23 | 20, 21, 22 | mp2an 691 | . . . . . . . 8 ⊢ ((𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞) ↔ ((𝐶 +𝑒 -𝑒𝐵) ∈ ℝ* ∧ 0 ≤ (𝐶 +𝑒 -𝑒𝐵) ∧ (𝐶 +𝑒 -𝑒𝐵) ≤ +∞)) |
24 | 13, 17, 19, 23 | syl3anbrc 1341 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) |
25 | 24 | ralrimiva 3143 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) |
26 | 5 | esumcl 33649 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 (𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) |
27 | 2, 25, 26 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) |
28 | 1, 27 | sselid 3978 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∈ ℝ*) |
29 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ*) |
30 | 21 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ*) |
31 | elicc4 13424 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∈ ℝ*) → (Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞) ↔ (0 ≤ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∧ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ≤ +∞))) | |
32 | 29, 30, 28, 31 | syl3anc 1369 | . . . . . 6 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞) ↔ (0 ≤ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∧ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ≤ +∞))) |
33 | 27, 32 | mpbid 231 | . . . . 5 ⊢ (𝜑 → (0 ≤ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∧ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ≤ +∞)) |
34 | 33 | simpld 494 | . . . 4 ⊢ (𝜑 → 0 ≤ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵)) |
35 | xraddge02 32539 | . . . . 5 ⊢ ((Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ* ∧ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∈ ℝ*) → (0 ≤ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) → Σ*𝑘 ∈ 𝐴𝐵 ≤ (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵)))) | |
36 | 35 | imp 406 | . . . 4 ⊢ (((Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ* ∧ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∈ ℝ*) ∧ 0 ≤ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵)) → Σ*𝑘 ∈ 𝐴𝐵 ≤ (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵))) |
37 | 8, 28, 34, 36 | syl21anc 837 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵))) |
38 | xaddcom 13252 | . . . 4 ⊢ ((Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ* ∧ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∈ ℝ*) → (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵)) = (Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) +𝑒 Σ*𝑘 ∈ 𝐴𝐵)) | |
39 | 8, 28, 38 | syl2anc 583 | . . 3 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵)) = (Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) +𝑒 Σ*𝑘 ∈ 𝐴𝐵)) |
40 | 37, 39 | breqtrd 5174 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ (Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) +𝑒 Σ*𝑘 ∈ 𝐴𝐵)) |
41 | 2, 24, 3 | esumadd 33676 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴((𝐶 +𝑒 -𝑒𝐵) +𝑒 𝐵) = (Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) +𝑒 Σ*𝑘 ∈ 𝐴𝐵)) |
42 | xrge0npcan 32763 | . . . . 5 ⊢ ((𝐶 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐶) → ((𝐶 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐶) | |
43 | 9, 3, 14, 42 | syl3anc 1369 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐶 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐶) |
44 | 43 | esumeq2dv 33657 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴((𝐶 +𝑒 -𝑒𝐵) +𝑒 𝐵) = Σ*𝑘 ∈ 𝐴𝐶) |
45 | 41, 44 | eqtr3d 2770 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) +𝑒 Σ*𝑘 ∈ 𝐴𝐵) = Σ*𝑘 ∈ 𝐴𝐶) |
46 | 40, 45 | breqtrd 5174 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ Σ*𝑘 ∈ 𝐴𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3058 class class class wbr 5148 (class class class)co 7420 0cc0 11139 +∞cpnf 11276 ℝ*cxr 11278 ≤ cle 11280 -𝑒cxne 13122 +𝑒 cxad 13123 [,]cicc 13360 Σ*cesum 33646 |
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 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ioc 13362 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-mod 13868 df-seq 14000 df-exp 14060 df-fac 14266 df-bc 14295 df-hash 14323 df-shft 15047 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-limsup 15448 df-clim 15465 df-rlim 15466 df-sum 15666 df-ef 16044 df-sin 16046 df-cos 16047 df-pi 16049 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-pt 17426 df-prds 17429 df-ordt 17483 df-xrs 17484 df-qtop 17489 df-imas 17490 df-xps 17492 df-mre 17566 df-mrc 17567 df-acs 17569 df-ps 18558 df-tsr 18559 df-plusf 18599 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-mulg 19024 df-subg 19078 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-subrng 20483 df-subrg 20508 df-abv 20697 df-lmod 20745 df-scaf 20746 df-sra 21058 df-rgmod 21059 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-perf 23054 df-cn 23144 df-cnp 23145 df-haus 23232 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-tmd 23989 df-tgp 23990 df-tsms 24044 df-trg 24077 df-xms 24239 df-ms 24240 df-tms 24241 df-nm 24504 df-ngp 24505 df-nrg 24507 df-nlm 24508 df-ii 24810 df-cncf 24811 df-limc 25808 df-dv 25809 df-log 26503 df-esum 33647 |
This theorem is referenced by: measiun 33837 |
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