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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumlef | 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 |
---|---|
esumaddf.0 | ⊢ Ⅎ𝑘𝜑 |
esumaddf.a | ⊢ Ⅎ𝑘𝐴 |
esumaddf.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
esumaddf.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
esumaddf.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) |
esumlef.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶) |
Ref | Expression |
---|---|
esumlef | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ Σ*𝑘 ∈ 𝐴𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssxr 13354 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | esumaddf.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | esumaddf.0 | . . . . . . 7 ⊢ Ⅎ𝑘𝜑 | |
4 | esumaddf.2 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
5 | 4 | ex 414 | . . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞))) |
6 | 3, 5 | ralrimi 3243 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
7 | esumaddf.a | . . . . . . 7 ⊢ Ⅎ𝑘𝐴 | |
8 | 7 | esumcl 32669 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
9 | 2, 6, 8 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
10 | 1, 9 | sselid 3947 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) |
11 | esumaddf.3 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) | |
12 | 1, 11 | sselid 3947 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ*) |
13 | 1, 4 | sselid 3947 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
14 | 13 | xnegcld 13226 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -𝑒𝐵 ∈ ℝ*) |
15 | 12, 14 | xaddcld 13227 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +𝑒 -𝑒𝐵) ∈ ℝ*) |
16 | esumlef.3 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶) | |
17 | xsubge0 13187 | . . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ (𝐶 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐶)) | |
18 | 12, 13, 17 | syl2anc 585 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (0 ≤ (𝐶 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐶)) |
19 | 16, 18 | mpbird 257 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ (𝐶 +𝑒 -𝑒𝐵)) |
20 | pnfge 13058 | . . . . . . . . . 10 ⊢ ((𝐶 +𝑒 -𝑒𝐵) ∈ ℝ* → (𝐶 +𝑒 -𝑒𝐵) ≤ +∞) | |
21 | 15, 20 | syl 17 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +𝑒 -𝑒𝐵) ≤ +∞) |
22 | 0xr 11209 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ* | |
23 | pnfxr 11216 | . . . . . . . . . 10 ⊢ +∞ ∈ ℝ* | |
24 | elicc1 13315 | . . . . . . . . . 10 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞) ↔ ((𝐶 +𝑒 -𝑒𝐵) ∈ ℝ* ∧ 0 ≤ (𝐶 +𝑒 -𝑒𝐵) ∧ (𝐶 +𝑒 -𝑒𝐵) ≤ +∞))) | |
25 | 22, 23, 24 | mp2an 691 | . . . . . . . . 9 ⊢ ((𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞) ↔ ((𝐶 +𝑒 -𝑒𝐵) ∈ ℝ* ∧ 0 ≤ (𝐶 +𝑒 -𝑒𝐵) ∧ (𝐶 +𝑒 -𝑒𝐵) ≤ +∞)) |
26 | 15, 19, 21, 25 | syl3anbrc 1344 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) |
27 | 26 | ex 414 | . . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → (𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞))) |
28 | 3, 27 | ralrimi 3243 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) |
29 | 7 | esumcl 32669 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 (𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) |
30 | 2, 28, 29 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) |
31 | 1, 30 | sselid 3947 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∈ ℝ*) |
32 | 22 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ*) |
33 | 23 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ*) |
34 | elicc4 13338 | . . . . . . 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) ≤ +∞))) | |
35 | 32, 33, 31, 34 | syl3anc 1372 | . . . . . 6 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞) ↔ (0 ≤ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∧ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ≤ +∞))) |
36 | 30, 35 | mpbid 231 | . . . . 5 ⊢ (𝜑 → (0 ≤ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∧ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ≤ +∞)) |
37 | 36 | simpld 496 | . . . 4 ⊢ (𝜑 → 0 ≤ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵)) |
38 | xraddge02 31703 | . . . . 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)))) | |
39 | 38 | imp 408 | . . . 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))) |
40 | 10, 31, 37, 39 | syl21anc 837 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵))) |
41 | xaddcom 13166 | . . . 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)) | |
42 | 10, 31, 41 | syl2anc 585 | . . 3 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵)) = (Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) +𝑒 Σ*𝑘 ∈ 𝐴𝐵)) |
43 | 40, 42 | breqtrd 5136 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ (Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) +𝑒 Σ*𝑘 ∈ 𝐴𝐵)) |
44 | 3, 7, 2, 26, 4 | esumaddf 32700 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴((𝐶 +𝑒 -𝑒𝐵) +𝑒 𝐵) = (Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) +𝑒 Σ*𝑘 ∈ 𝐴𝐵)) |
45 | xrge0npcan 31927 | . . . . . . 7 ⊢ ((𝐶 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐶) → ((𝐶 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐶) | |
46 | 11, 4, 16, 45 | syl3anc 1372 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐶 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐶) |
47 | 46 | ex 414 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → ((𝐶 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐶)) |
48 | 3, 47 | ralrimi 3243 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((𝐶 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐶) |
49 | 3, 48 | esumeq2d 32676 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴((𝐶 +𝑒 -𝑒𝐵) +𝑒 𝐵) = Σ*𝑘 ∈ 𝐴𝐶) |
50 | 44, 49 | eqtr3d 2779 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) +𝑒 Σ*𝑘 ∈ 𝐴𝐵) = Σ*𝑘 ∈ 𝐴𝐶) |
51 | 43, 50 | breqtrd 5136 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ Σ*𝑘 ∈ 𝐴𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 Ⅎwnfc 2888 ∀wral 3065 class class class wbr 5110 (class class class)co 7362 0cc0 11058 +∞cpnf 11193 ℝ*cxr 11195 ≤ cle 11197 -𝑒cxne 13037 +𝑒 cxad 13038 [,]cicc 13274 Σ*cesum 32666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13275 df-ioc 13276 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14959 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-limsup 15360 df-clim 15377 df-rlim 15378 df-sum 15578 df-ef 15957 df-sin 15959 df-cos 15960 df-pi 15962 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-hom 17164 df-cco 17165 df-rest 17311 df-topn 17312 df-0g 17330 df-gsum 17331 df-topgen 17332 df-pt 17333 df-prds 17336 df-ordt 17390 df-xrs 17391 df-qtop 17396 df-imas 17397 df-xps 17399 df-mre 17473 df-mrc 17474 df-acs 17476 df-ps 18462 df-tsr 18463 df-plusf 18503 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-mhm 18608 df-submnd 18609 df-grp 18758 df-minusg 18759 df-sbg 18760 df-mulg 18880 df-subg 18932 df-cntz 19104 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-cring 19974 df-subrg 20236 df-abv 20292 df-lmod 20340 df-scaf 20341 df-sra 20649 df-rgmod 20650 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-lp 22503 df-perf 22504 df-cn 22594 df-cnp 22595 df-haus 22682 df-tx 22929 df-hmeo 23122 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-tmd 23439 df-tgp 23440 df-tsms 23494 df-trg 23527 df-xms 23689 df-ms 23690 df-tms 23691 df-nm 23954 df-ngp 23955 df-nrg 23957 df-nlm 23958 df-ii 24256 df-cncf 24257 df-limc 25246 df-dv 25247 df-log 25928 df-esum 32667 |
This theorem is referenced by: esumpinfval 32712 esumpinfsum 32716 esum2d 32732 omssubadd 32940 |
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