esumlef - Mathbox for Thierry Arnoux

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Theorem esumlef 32464

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 13301	. . . . 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 413	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞)))
6	3, 5	ralrimi 3238	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞))
7		esumaddf.a	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
8	7	esumcl 32432	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞))
9	2, 6, 8	syl2anc 584	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞))
10	1, 9	sselid 3940	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ∈ ℝ^)
11		esumaddf.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))
12	1, 11	sselid 3940	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ^*)
13	1, 4	sselid 3940	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ^*)
14	13	xnegcld 13173	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -_𝑒𝐵 ∈ ℝ^*)
15	12, 14	xaddcld 13174	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +_𝑒 -_𝑒𝐵) ∈ ℝ^*)
16		esumlef.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶)
17		xsubge0 13134	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ (𝐶 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐶))
18	12, 13, 17	syl2anc 584	. . . . . . . . . 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 13005	. . . . . . . . . 10 ⊢ ((𝐶 +_𝑒 -_𝑒𝐵) ∈ ℝ^* → (𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞)
21	15, 20	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞)
22		0xr 11160	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
23		pnfxr 11167	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
24		elicc1 13262	. . . . . . . . . 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 1343	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
27	26	ex 413	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞)))
28	3, 27	ralrimi 3238	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
29	7	esumcl 32432	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
30	2, 28, 29	syl2anc 584	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
31	1, 30	sselid 3940	. . . 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 13285	. . . . . . 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 231	. . . . 5 ⊢ (𝜑 → (0 ≤ Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) ∧ Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞))
37	36	simpld 495	. . . 4 ⊢ (𝜑 → 0 ≤ Σ^*𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵))
38		xraddge02 31485	. . . . 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 407	. . . 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 13113	. . . 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 584	. . 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 5129	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ≤ (Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 Σ^*𝑘 ∈ 𝐴𝐵))
44	3, 7, 2, 26, 4	esumaddf 32463	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = (Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 Σ^*𝑘 ∈ 𝐴𝐵))
45		xrge0npcan 31709	. . . . . . 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 413	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → ((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐶))
48	3, 47	ralrimi 3238	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐶)
49	3, 48	esumeq2d 32439	. . 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 5129	1 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ≤ Σ^𝑘 ∈ 𝐴𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2885 ∀wral 3062 class class class wbr 5103 (class class class)co 7351 0cc0 11009 +∞cpnf 11144 ℝ^*cxr 11146 ≤ cle 11148 -_𝑒cxne 12984 +_𝑒 cxad 12985 [,]cicc 13221 Σ^*cesum 32429

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-inf2 9535 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-er 8606 df-map 8725 df-pm 8726 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-fi 9305 df-sup 9336 df-inf 9337 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-q 12828 df-rp 12870 df-xneg 12987 df-xadd 12988 df-xmul 12989 df-ioo 13222 df-ioc 13223 df-ico 13224 df-icc 13225 df-fz 13379 df-fzo 13522 df-fl 13651 df-mod 13729 df-seq 13861 df-exp 13922 df-fac 14127 df-bc 14156 df-hash 14184 df-shft 14911 df-cj 14943 df-re 14944 df-im 14945 df-sqrt 15079 df-abs 15080 df-limsup 15312 df-clim 15329 df-rlim 15330 df-sum 15530 df-ef 15909 df-sin 15911 df-cos 15912 df-pi 15914 df-struct 16978 df-sets 16995 df-slot 17013 df-ndx 17025 df-base 17043 df-ress 17072 df-plusg 17105 df-mulr 17106 df-starv 17107 df-sca 17108 df-vsca 17109 df-ip 17110 df-tset 17111 df-ple 17112 df-ds 17114 df-unif 17115 df-hom 17116 df-cco 17117 df-rest 17263 df-topn 17264 df-0g 17282 df-gsum 17283 df-topgen 17284 df-pt 17285 df-prds 17288 df-ordt 17342 df-xrs 17343 df-qtop 17348 df-imas 17349 df-xps 17351 df-mre 17425 df-mrc 17426 df-acs 17428 df-ps 18414 df-tsr 18415 df-plusf 18455 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-mhm 18560 df-submnd 18561 df-grp 18710 df-minusg 18711 df-sbg 18712 df-mulg 18831 df-subg 18883 df-cntz 19055 df-cmn 19522 df-abl 19523 df-mgp 19855 df-ur 19872 df-ring 19919 df-cring 19920 df-subrg 20172 df-abv 20228 df-lmod 20276 df-scaf 20277 df-sra 20585 df-rgmod 20586 df-psmet 20740 df-xmet 20741 df-met 20742 df-bl 20743 df-mopn 20744 df-fbas 20745 df-fg 20746 df-cnfld 20749 df-top 22194 df-topon 22211 df-topsp 22233 df-bases 22247 df-cld 22321 df-ntr 22322 df-cls 22323 df-nei 22400 df-lp 22438 df-perf 22439 df-cn 22529 df-cnp 22530 df-haus 22617 df-tx 22864 df-hmeo 23057 df-fil 23148 df-fm 23240 df-flim 23241 df-flf 23242 df-tmd 23374 df-tgp 23375 df-tsms 23429 df-trg 23462 df-xms 23624 df-ms 23625 df-tms 23626 df-nm 23889 df-ngp 23890 df-nrg 23892 df-nlm 23893 df-ii 24191 df-cncf 24192 df-limc 25181 df-dv 25182 df-log 25863 df-esum 32430

This theorem is referenced by: esumpinfval 32475 esumpinfsum 32479 esum2d 32495 omssubadd 32703

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