esumlef - Mathbox for Thierry Arnoux

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Theorem esumlef 30938

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 12669	. . . . 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 3183	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞))
7		esumaddf.a	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
8	7	esumcl 30906	. . . . . 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	sseldi 3887	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ∈ ℝ^)
11		esumaddf.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))
12	1, 11	sseldi 3887	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ^*)
13	1, 4	sseldi 3887	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ^*)
14	13	xnegcld 12543	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -_𝑒𝐵 ∈ ℝ^*)
15	12, 14	xaddcld 12544	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +_𝑒 -_𝑒𝐵) ∈ ℝ^*)
16		esumlef.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶)
17		xsubge0 12504	. . . . . . . . . . 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 258	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ (𝐶 +_𝑒 -_𝑒𝐵))
20		pnfge 12375	. . . . . . . . . 10 ⊢ ((𝐶 +_𝑒 -_𝑒𝐵) ∈ ℝ^* → (𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞)
21	15, 20	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞)
22		0xr 10534	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
23		pnfxr 10541	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
24		elicc1 12632	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → ((𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞) ↔ ((𝐶 +_𝑒 -_𝑒𝐵) ∈ ℝ^ ∧ 0 ≤ (𝐶 +_𝑒 -_𝑒𝐵) ∧ (𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞)))
25	22, 23, 24	mp2an 688	. . . . . . . . 9 ⊢ ((𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞) ↔ ((𝐶 +_𝑒 -_𝑒𝐵) ∈ ℝ^* ∧ 0 ≤ (𝐶 +_𝑒 -_𝑒𝐵) ∧ (𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞))
26	15, 19, 21, 25	syl3anbrc 1336	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
27	26	ex 413	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞)))
28	3, 27	ralrimi 3183	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
29	7	esumcl 30906	. . . . . 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	sseldi 3887	. . . 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 12653	. . . . . . 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 1364	. . . . . 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 233	. . . . 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 30168	. . . . 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 834	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ≤ (Σ^𝑘 ∈ 𝐴𝐵 +_𝑒 Σ^*𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵)))
41		xaddcom 12483	. . . 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 4988	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ≤ (Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 Σ^*𝑘 ∈ 𝐴𝐵))
44	3, 7, 2, 26, 4	esumaddf 30937	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = (Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 Σ^*𝑘 ∈ 𝐴𝐵))
45		xrge0npcan 30355	. . . . . . 7 ⊢ ((𝐶 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐶) → ((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐶)
46	11, 4, 16, 45	syl3anc 1364	. . . . . 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 3183	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐶)
49	3, 48	esumeq2d 30913	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = Σ^𝑘 ∈ 𝐴𝐶)
50	44, 49	eqtr3d 2833	. 2 ⊢ (𝜑 → (Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 Σ^𝑘 ∈ 𝐴𝐵) = Σ^*𝑘 ∈ 𝐴𝐶)
51	43, 50	breqtrd 4988	1 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ≤ Σ^𝑘 ∈ 𝐴𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 Ⅎwnf 1765 ∈ wcel 2081 Ⅎwnfc 2933 ∀wral 3105 class class class wbr 4962 (class class class)co 7016 0cc0 10383 +∞cpnf 10518 ℝ^*cxr 10520 ≤ cle 10522 -_𝑒cxne 12354 +_𝑒 cxad 12355 [,]cicc 12591 Σ^*cesum 30903

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-om 7437 df-1st 7545 df-2nd 7546 df-supp 7682 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-er 8139 df-map 8258 df-pm 8259 df-ixp 8311 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-fsupp 8680 df-fi 8721 df-sup 8752 df-inf 8753 df-oi 8820 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-ioo 12592 df-ioc 12593 df-ico 12594 df-icc 12595 df-fz 12743 df-fzo 12884 df-fl 13012 df-mod 13088 df-seq 13220 df-exp 13280 df-fac 13484 df-bc 13513 df-hash 13541 df-shft 14260 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-limsup 14662 df-clim 14679 df-rlim 14680 df-sum 14877 df-ef 15254 df-sin 15256 df-cos 15257 df-pi 15259 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-sca 16410 df-vsca 16411 df-ip 16412 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-hom 16418 df-cco 16419 df-rest 16525 df-topn 16526 df-0g 16544 df-gsum 16545 df-topgen 16546 df-pt 16547 df-prds 16550 df-ordt 16603 df-xrs 16604 df-qtop 16609 df-imas 16610 df-xps 16612 df-mre 16686 df-mrc 16687 df-acs 16689 df-ps 17639 df-tsr 17640 df-plusf 17680 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-mhm 17774 df-submnd 17775 df-grp 17864 df-minusg 17865 df-sbg 17866 df-mulg 17982 df-subg 18030 df-cntz 18188 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-ring 18989 df-cring 18990 df-subrg 19223 df-abv 19278 df-lmod 19326 df-scaf 19327 df-sra 19634 df-rgmod 19635 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-fbas 20224 df-fg 20225 df-cnfld 20228 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-cld 21311 df-ntr 21312 df-cls 21313 df-nei 21390 df-lp 21428 df-perf 21429 df-cn 21519 df-cnp 21520 df-haus 21607 df-tx 21854 df-hmeo 22047 df-fil 22138 df-fm 22230 df-flim 22231 df-flf 22232 df-tmd 22364 df-tgp 22365 df-tsms 22418 df-trg 22451 df-xms 22613 df-ms 22614 df-tms 22615 df-nm 22875 df-ngp 22876 df-nrg 22878 df-nlm 22879 df-ii 23168 df-cncf 23169 df-limc 24147 df-dv 24148 df-log 24821 df-esum 30904

This theorem is referenced by: esumpinfval 30949 esumpinfsum 30953 esum2d 30969 omssubadd 31175

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