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Mathbox for Thierry Arnoux |
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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 13447 | . . . . 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 411 | . . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞))) |
6 | 3, 5 | ralrimi 3252 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
7 | esumaddf.a | . . . . . . 7 ⊢ Ⅎ𝑘𝐴 | |
8 | 7 | esumcl 33682 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
9 | 2, 6, 8 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
10 | 1, 9 | sselid 3980 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) |
11 | esumaddf.3 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) | |
12 | 1, 11 | sselid 3980 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ*) |
13 | 1, 4 | sselid 3980 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
14 | 13 | xnegcld 13319 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -𝑒𝐵 ∈ ℝ*) |
15 | 12, 14 | xaddcld 13320 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +𝑒 -𝑒𝐵) ∈ ℝ*) |
16 | esumlef.3 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶) | |
17 | xsubge0 13280 | . . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ (𝐶 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐶)) | |
18 | 12, 13, 17 | syl2anc 582 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (0 ≤ (𝐶 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐶)) |
19 | 16, 18 | mpbird 256 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ (𝐶 +𝑒 -𝑒𝐵)) |
20 | pnfge 13150 | . . . . . . . . . 10 ⊢ ((𝐶 +𝑒 -𝑒𝐵) ∈ ℝ* → (𝐶 +𝑒 -𝑒𝐵) ≤ +∞) | |
21 | 15, 20 | syl 17 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +𝑒 -𝑒𝐵) ≤ +∞) |
22 | 0xr 11299 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ* | |
23 | pnfxr 11306 | . . . . . . . . . 10 ⊢ +∞ ∈ ℝ* | |
24 | elicc1 13408 | . . . . . . . . . 10 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞) ↔ ((𝐶 +𝑒 -𝑒𝐵) ∈ ℝ* ∧ 0 ≤ (𝐶 +𝑒 -𝑒𝐵) ∧ (𝐶 +𝑒 -𝑒𝐵) ≤ +∞))) | |
25 | 22, 23, 24 | mp2an 690 | . . . . . . . . 9 ⊢ ((𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞) ↔ ((𝐶 +𝑒 -𝑒𝐵) ∈ ℝ* ∧ 0 ≤ (𝐶 +𝑒 -𝑒𝐵) ∧ (𝐶 +𝑒 -𝑒𝐵) ≤ +∞)) |
26 | 15, 19, 21, 25 | syl3anbrc 1340 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) |
27 | 26 | ex 411 | . . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → (𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞))) |
28 | 3, 27 | ralrimi 3252 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) |
29 | 7 | esumcl 33682 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 (𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) |
30 | 2, 28, 29 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) |
31 | 1, 30 | sselid 3980 | . . . 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 13431 | . . . . . . 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 1368 | . . . . . 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 493 | . . . 4 ⊢ (𝜑 → 0 ≤ Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵)) |
38 | xraddge02 32547 | . . . . 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 405 | . . . 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 836 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵))) |
41 | xaddcom 13259 | . . . 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 582 | . . 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 5178 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ (Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) +𝑒 Σ*𝑘 ∈ 𝐴𝐵)) |
44 | 3, 7, 2, 26, 4 | esumaddf 33713 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴((𝐶 +𝑒 -𝑒𝐵) +𝑒 𝐵) = (Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) +𝑒 Σ*𝑘 ∈ 𝐴𝐵)) |
45 | xrge0npcan 32771 | . . . . . . 7 ⊢ ((𝐶 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐶) → ((𝐶 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐶) | |
46 | 11, 4, 16, 45 | syl3anc 1368 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐶 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐶) |
47 | 46 | ex 411 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → ((𝐶 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐶)) |
48 | 3, 47 | ralrimi 3252 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((𝐶 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐶) |
49 | 3, 48 | esumeq2d 33689 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴((𝐶 +𝑒 -𝑒𝐵) +𝑒 𝐵) = Σ*𝑘 ∈ 𝐴𝐶) |
50 | 44, 49 | eqtr3d 2770 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴(𝐶 +𝑒 -𝑒𝐵) +𝑒 Σ*𝑘 ∈ 𝐴𝐵) = Σ*𝑘 ∈ 𝐴𝐶) |
51 | 43, 50 | breqtrd 5178 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ≤ Σ*𝑘 ∈ 𝐴𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2879 ∀wral 3058 class class class wbr 5152 (class class class)co 7426 0cc0 11146 +∞cpnf 11283 ℝ*cxr 11285 ≤ cle 11287 -𝑒cxne 13129 +𝑒 cxad 13130 [,]cicc 13367 Σ*cesum 33679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 ax-mulf 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 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 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-ef 16051 df-sin 16053 df-cos 16054 df-pi 16056 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-ordt 17490 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-ps 18565 df-tsr 18566 df-plusf 18606 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-cntz 19275 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-subrng 20490 df-subrg 20515 df-abv 20704 df-lmod 20752 df-scaf 20753 df-sra 21065 df-rgmod 21066 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-perf 23061 df-cn 23151 df-cnp 23152 df-haus 23239 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-tmd 23996 df-tgp 23997 df-tsms 24051 df-trg 24084 df-xms 24246 df-ms 24247 df-tms 24248 df-nm 24511 df-ngp 24512 df-nrg 24514 df-nlm 24515 df-ii 24817 df-cncf 24818 df-limc 25815 df-dv 25816 df-log 26510 df-esum 33680 |
This theorem is referenced by: esumpinfval 33725 esumpinfsum 33729 esum2d 33745 omssubadd 33953 |
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