esumlef - Mathbox for Thierry Arnoux

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Theorem esumlef 34559

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 13485	. . . . 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 418	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞)))
6	3, 5	ralrimi 3262	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞))
7		esumaddf.a	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
8	7	esumcl 34527	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞))
9	2, 6, 8	syl2anc 596	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞))
10	1, 9	sselid 3932	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ∈ ℝ^)
11		esumaddf.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))
12	1, 11	sselid 3932	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ^*)
13	1, 4	sselid 3932	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ^*)
14	13	xnegcld 13354	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → -_𝑒𝐵 ∈ ℝ^*)
15	12, 14	xaddcld 13355	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +_𝑒 -_𝑒𝐵) ∈ ℝ^*)
16		esumlef.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶)
17		xsubge0 13315	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ (𝐶 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐶))
18	12, 13, 17	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (0 ≤ (𝐶 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐶))
19	16, 18	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ (𝐶 +_𝑒 -_𝑒𝐵))
20		pnfge 13183	. . . . . . . . . 10 ⊢ ((𝐶 +_𝑒 -_𝑒𝐵) ∈ ℝ^* → (𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞)
21	15, 20	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞)
22		0xr 11283	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
23		pnfxr 11290	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
24		elicc1 13444	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → ((𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞) ↔ ((𝐶 +_𝑒 -_𝑒𝐵) ∈ ℝ^ ∧ 0 ≤ (𝐶 +_𝑒 -_𝑒𝐵) ∧ (𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞)))
25	22, 23, 24	mp2an 705	. . . . . . . . 9 ⊢ ((𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞) ↔ ((𝐶 +_𝑒 -_𝑒𝐵) ∈ ℝ^* ∧ 0 ≤ (𝐶 +_𝑒 -_𝑒𝐵) ∧ (𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞))
26	15, 19, 21, 25	syl3anbrc 1362	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
27	26	ex 418	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞)))
28	3, 27	ralrimi 3262	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
29	7	esumcl 34527	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 (𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
30	2, 28, 29	syl2anc 596	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) ∈ (0[,]+∞))
31	1, 30	sselid 3932	. . . 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 13468	. . . . . . 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 1398	. . . . . 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 235	. . . . 5 ⊢ (𝜑 → (0 ≤ Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) ∧ Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) ≤ +∞))
37	36	simpld 500	. . . 4 ⊢ (𝜑 → 0 ≤ Σ^*𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵))
38		xraddge02 33215	. . . . 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 412	. . . 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 851	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ≤ (Σ^𝑘 ∈ 𝐴𝐵 +_𝑒 Σ^*𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵)))
41		xaddcom 13294	. . . 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 596	. . 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 5135	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ≤ (Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 Σ^*𝑘 ∈ 𝐴𝐵))
44	3, 7, 2, 26, 4	esumaddf 34558	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = (Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 Σ^*𝑘 ∈ 𝐴𝐵))
45		xrge0npcan 33447	. . . . . . 7 ⊢ ((𝐶 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐶) → ((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐶)
46	11, 4, 16, 45	syl3anc 1398	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐶)
47	46	ex 418	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → ((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐶))
48	3, 47	ralrimi 3262	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐶)
49	3, 48	esumeq2d 34534	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴((𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = Σ^𝑘 ∈ 𝐴𝐶)
50	44, 49	eqtr3d 2799	. 2 ⊢ (𝜑 → (Σ^𝑘 ∈ 𝐴(𝐶 +_𝑒 -_𝑒𝐵) +_𝑒 Σ^𝑘 ∈ 𝐴𝐵) = Σ^*𝑘 ∈ 𝐴𝐶)
51	43, 50	breqtrd 5135	1 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ≤ Σ^𝑘 ∈ 𝐴𝐶)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 Ⅎwnf 1816 ∈ wcel 2145 Ⅎwnfc 2909 ∀wral 3078 class class class wbr 5107 (class class class)co 7416 0cc0 11127 +∞cpnf 11267 ℝ^*cxr 11269 ≤ cle 11271 -_𝑒cxne 13162 +_𝑒 cxad 13163 [,]cicc 13403 Σ^*cesum 34524

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206 ax-mulf 11207

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-fi 9384 df-sup 9415 df-inf 9416 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-q 13001 df-rp 13045 df-xneg 13165 df-xadd 13166 df-xmul 13167 df-ioo 13404 df-ioc 13405 df-ico 13406 df-icc 13407 df-fz 13564 df-fzo 13712 df-fl 13855 df-mod 13933 df-seq 14068 df-exp 14128 df-fac 14340 df-bc 14369 df-hash 14397 df-shft 15142 df-cj 15188 df-re 15189 df-im 15190 df-sqrt 15324 df-abs 15325 df-limsup 15560 df-clim 15577 df-rlim 15578 df-sum 15776 df-ef 16157 df-sin 16159 df-cos 16160 df-pi 16162 df-struct 17243 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-plusg 17359 df-mulr 17360 df-starv 17361 df-sca 17362 df-vsca 17363 df-ip 17364 df-tset 17365 df-ple 17366 df-ds 17368 df-unif 17369 df-hom 17370 df-cco 17371 df-rest 17511 df-topn 17512 df-0g 17530 df-gsum 17531 df-topgen 17532 df-pt 17533 df-prds 17536 df-ordt 17591 df-xrs 17592 df-qtop 17597 df-imas 17598 df-xps 17600 df-mre 17674 df-mrc 17675 df-acs 17677 df-ps 18658 df-tsr 18659 df-plusf 18733 df-mgm 18734 df-sgrp 18823 df-mnd 18839 df-mhm 18892 df-submnd 18893 df-grp 19061 df-minusg 19062 df-sbg 19063 df-mulg 19192 df-subg 19247 df-cntz 19445 df-cmn 19910 df-abl 19911 df-mgp 20275 df-rng 20289 df-ur 20322 df-ring 20375 df-cring 20376 df-subrng 20709 df-subrg 20733 df-abv 20976 df-lmod 21047 df-scaf 21048 df-sra 21358 df-rgmod 21359 df-psmet 21578 df-xmet 21579 df-met 21580 df-bl 21581 df-mopn 21582 df-fbas 21583 df-fg 21584 df-cnfld 21587 df-top 23120 df-topon 23137 df-topsp 23159 df-bases 23172 df-cld 23245 df-ntr 23246 df-cls 23247 df-nei 23324 df-lp 23362 df-perf 23363 df-cn 23453 df-cnp 23454 df-haus 23541 df-tx 23789 df-hmeo 23982 df-fil 24073 df-fm 24165 df-flim 24166 df-flf 24167 df-tmd 24299 df-tgp 24300 df-tsms 24354 df-trg 24387 df-xms 24547 df-ms 24548 df-tms 24549 df-nm 24809 df-ngp 24810 df-nrg 24812 df-nlm 24813 df-ii 25106 df-cncf 25107 df-limc 26095 df-dv 26096 df-log 26791 df-esum 34525

This theorem is used by: esumpinfval 34570 esumpinfsum 34574 esum2d 34590 omssubadd 34798

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