esumlef - Mathbox for Thierry Arnoux

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Theorem esumlef 34258

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 13378	. . . . 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 3239	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞))
7		esumaddf.a	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
8	7	esumcl 34226	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞))
9	2, 6, 8	syl2anc 591	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞))
10	1, 9	sselid 3915	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ∈ ℝ^)
11		esumaddf.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))
12	1, 11	sselid 3915	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ^*)
13	1, 4	sselid 3915	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ^*)
14	13	xnegcld 13247	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -_𝑒𝐵 ∈ ℝ^*)
15	12, 14	xaddcld 13248	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +_𝑒 -_𝑒𝐵) ∈ ℝ^*)
16		esumlef.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶)
17		xsubge0 13208	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ (𝐶 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐶))
18	12, 13, 17	syl2anc 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (0 ≤ (𝐶 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐶))
19	16, 18	mpbird 259	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ (𝐶 +_𝑒 -_𝑒𝐵))
20		pnfge 13076	. . . . . . . . . 10 ⊢ ((𝐶 +_𝑒 -_𝑒𝐵) ∈ ℝ^* → (𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞)
21	15, 20	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞)
22		0xr 11187	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
23		pnfxr 11194	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
24		elicc1 13337	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → ((𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞) ↔ ((𝐶 +_𝑒 -_𝑒𝐵) ∈ ℝ^ ∧ 0 ≤ (𝐶 +_𝑒 -_𝑒𝐵) ∧ (𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞)))
25	22, 23, 24	mp2an 699	. . . . . . . . 9 ⊢ ((𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞) ↔ ((𝐶 +_𝑒 -_𝑒𝐵) ∈ ℝ^* ∧ 0 ≤ (𝐶 +_𝑒 -_𝑒𝐵) ∧ (𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞))
26	15, 19, 21, 25	syl3anbrc 1351	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
27	26	ex 414	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞)))
28	3, 27	ralrimi 3239	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
29	7	esumcl 34226	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
30	2, 28, 29	syl2anc 591	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
31	1, 30	sselid 3915	. . . 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 13361	. . . . . . 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 1380	. . . . . 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 234	. . . . 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 32853	. . . . 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 844	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ≤ (Σ^𝑘 ∈ 𝐴𝐵 +_𝑒 Σ^*𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵)))
41		xaddcom 13187	. . . 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 591	. . 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 5101	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ≤ (Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 Σ^*𝑘 ∈ 𝐴𝐵))
44	3, 7, 2, 26, 4	esumaddf 34257	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = (Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 Σ^*𝑘 ∈ 𝐴𝐵))
45		xrge0npcan 33103	. . . . . . 7 ⊢ ((𝐶 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐶) → ((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐶)
46	11, 4, 16, 45	syl3anc 1380	. . . . . 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 3239	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐶)
49	3, 48	esumeq2d 34233	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = Σ^𝑘 ∈ 𝐴𝐶)
50	44, 49	eqtr3d 2778	. 2 ⊢ (𝜑 → (Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 Σ^𝑘 ∈ 𝐴𝐵) = Σ^*𝑘 ∈ 𝐴𝐶)
51	43, 50	breqtrd 5101	1 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ≤ Σ^𝑘 ∈ 𝐴𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 397 ∧ w3a 1093 = wceq 1548 Ⅎwnf 1791 ∈ wcel 2121 Ⅎwnfc 2888 ∀wral 3055 class class class wbr 5075 (class class class)co 7360 0cc0 11033 +∞cpnf 11171 ℝ^*cxr 11173 ≤ cle 11175 -_𝑒cxne 13055 +_𝑒 cxad 13056 [,]cicc 13296 Σ^*cesum 34223

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112 ax-mulf 11113

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-ioc 13298 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15024 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-limsup 15428 df-clim 15445 df-rlim 15446 df-sum 15644 df-ef 16027 df-sin 16029 df-cos 16030 df-pi 16032 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-ordt 17460 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-ps 18527 df-tsr 18528 df-plusf 18602 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-submnd 18747 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19039 df-subg 19094 df-cntz 19287 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-cring 20212 df-subrng 20522 df-subrg 20546 df-abv 20785 df-lmod 20856 df-scaf 20857 df-sra 21167 df-rgmod 21168 df-psmet 21343 df-xmet 21344 df-met 21345 df-bl 21346 df-mopn 21347 df-fbas 21348 df-fg 21349 df-cnfld 21352 df-top 22881 df-topon 22898 df-topsp 22920 df-bases 22933 df-cld 23006 df-ntr 23007 df-cls 23008 df-nei 23085 df-lp 23123 df-perf 23124 df-cn 23214 df-cnp 23215 df-haus 23302 df-tx 23549 df-hmeo 23742 df-fil 23833 df-fm 23925 df-flim 23926 df-flf 23927 df-tmd 24059 df-tgp 24060 df-tsms 24114 df-trg 24147 df-xms 24307 df-ms 24308 df-tms 24309 df-nm 24569 df-ngp 24570 df-nrg 24572 df-nlm 24573 df-ii 24866 df-cncf 24867 df-limc 25855 df-dv 25856 df-log 26542 df-esum 34224

This theorem is referenced by: esumpinfval 34269 esumpinfsum 34273 esum2d 34289 omssubadd 34496

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