esumlef - Mathbox for Thierry Arnoux

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Theorem esumlef 34026

Description: If all of the terms of an extended sums compare, so do the sums. (Contributed by Thierry Arnoux, 8-Jun-2017.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
esumlef	⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ≤ Σ^𝑘 ∈ 𝐴𝐶)

Distinct variable group: 𝑘,𝑉 Allowed substitution hints: 𝜑(𝑘) 𝐴(𝑘) 𝐵(𝑘) 𝐶(𝑘)

**Proof of Theorem esumlef**
Step	Hyp	Ref	Expression
1		iccssxr 13490	. . . . 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 412	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞)))
6	3, 5	ralrimi 3263	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞))
7		esumaddf.a	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
8	7	esumcl 33994	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞))
9	2, 6, 8	syl2anc 583	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞))
10	1, 9	sselid 4006	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ∈ ℝ^)
11		esumaddf.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))
12	1, 11	sselid 4006	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ^*)
13	1, 4	sselid 4006	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ^*)
14	13	xnegcld 13362	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -_𝑒𝐵 ∈ ℝ^*)
15	12, 14	xaddcld 13363	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +_𝑒 -_𝑒𝐵) ∈ ℝ^*)
16		esumlef.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶)
17		xsubge0 13323	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ (𝐶 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐶))
18	12, 13, 17	syl2anc 583	. . . . . . . . . 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 13193	. . . . . . . . . 10 ⊢ ((𝐶 +_𝑒 -_𝑒𝐵) ∈ ℝ^* → (𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞)
21	15, 20	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞)
22		0xr 11337	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
23		pnfxr 11344	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
24		elicc1 13451	. . . . . . . . . 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 1343	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
27	26	ex 412	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞)))
28	3, 27	ralrimi 3263	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
29	7	esumcl 33994	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
30	2, 28, 29	syl2anc 583	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
31	1, 30	sselid 4006	. . . 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 13474	. . . . . . 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 1371	. . . . . 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 232	. . . . 5 ⊢ (𝜑 → (0 ≤ Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) ∧ Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞))
37	36	simpld 494	. . . 4 ⊢ (𝜑 → 0 ≤ Σ^*𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵))
38		xraddge02 32763	. . . . 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 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)))
40	10, 31, 37, 39	syl21anc 837	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ≤ (Σ^𝑘 ∈ 𝐴𝐵 +_𝑒 Σ^*𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵)))
41		xaddcom 13302	. . . 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 583	. . 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 5192	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ≤ (Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 Σ^*𝑘 ∈ 𝐴𝐵))
44	3, 7, 2, 26, 4	esumaddf 34025	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = (Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 Σ^*𝑘 ∈ 𝐴𝐵))
45		xrge0npcan 33006	. . . . . . 7 ⊢ ((𝐶 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐶) → ((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐶)
46	11, 4, 16, 45	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐶)
47	46	ex 412	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → ((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐶))
48	3, 47	ralrimi 3263	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐶)
49	3, 48	esumeq2d 34001	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = Σ^𝑘 ∈ 𝐴𝐶)
50	44, 49	eqtr3d 2782	. 2 ⊢ (𝜑 → (Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 Σ^𝑘 ∈ 𝐴𝐵) = Σ^*𝑘 ∈ 𝐴𝐶)
51	43, 50	breqtrd 5192	1 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ≤ Σ^𝑘 ∈ 𝐴𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 Ⅎwnf 1781 ∈ wcel 2108 Ⅎwnfc 2893 ∀wral 3067 class class class wbr 5166 (class class class)co 7448 0cc0 11184 +∞cpnf 11321 ℝ^*cxr 11323 ≤ cle 11325 -_𝑒cxne 13172 +_𝑒 cxad 13173 [,]cicc 13410 Σ^*cesum 33991

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-ordt 17561 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-ps 18636 df-tsr 18637 df-plusf 18677 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-subrng 20572 df-subrg 20597 df-abv 20832 df-lmod 20882 df-scaf 20883 df-sra 21195 df-rgmod 21196 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-tmd 24101 df-tgp 24102 df-tsms 24156 df-trg 24189 df-xms 24351 df-ms 24352 df-tms 24353 df-nm 24616 df-ngp 24617 df-nrg 24619 df-nlm 24620 df-ii 24922 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616 df-esum 33992

This theorem is referenced by: esumpinfval 34037 esumpinfsum 34041 esum2d 34057 omssubadd 34265

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