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Theorem isumless 15770

Description: A finite sum of nonnegative numbers is less than or equal to its limit. (Contributed by Mario Carneiro, 24-Apr-2014.)

**Hypotheses**
Ref	Expression
isumless.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
isumless.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
isumless.3	⊢ (𝜑 → 𝐴 ∈ Fin)
isumless.4	⊢ (𝜑 → 𝐴 ⊆ 𝑍)
isumless.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)
isumless.6	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)
isumless.7	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐵)
isumless.8	⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

**Assertion**
Ref	Expression
isumless	⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝑍 𝐵)

Distinct variable groups: 𝐴,𝑘 𝑘,𝐹 𝑘,𝑀 𝜑,𝑘 𝑘,𝑍 Allowed substitution hint: 𝐵(𝑘)

Proof of Theorem isumless Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isumless.4	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑍)
2	1	sselda 3937	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑍)
3		isumless.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)
4	3	recnd 11162	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
5	2, 4	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
6	5	ralrimiva 3121	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
7		isumless.1	. . . . . 6 ⊢ 𝑍 = (ℤ_≥‘𝑀)
8	7	eqimssi 3998	. . . . 5 ⊢ 𝑍 ⊆ (ℤ_≥‘𝑀)
9	8	orci 865	. . . 4 ⊢ (𝑍 ⊆ (ℤ_≥‘𝑀) ∨ 𝑍 ∈ Fin)
10	9	a1i 11	. . 3 ⊢ (𝜑 → (𝑍 ⊆ (ℤ_≥‘𝑀) ∨ 𝑍 ∈ Fin))
11		sumss2 15651	. . 3 ⊢ (((𝐴 ⊆ 𝑍 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) ∧ (𝑍 ⊆ (ℤ_≥‘𝑀) ∨ 𝑍 ∈ Fin)) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝑍 if(𝑘 ∈ 𝐴, 𝐵, 0))
12	1, 6, 10, 11	syl21anc 837	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝑍 if(𝑘 ∈ 𝐴, 𝐵, 0))
13		isumless.2	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
14		eleq1w 2811	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
15		fveq2 6826	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
16	14, 15	ifbieq1d 4503	. . . . . 6 ⊢ (𝑗 = 𝑘 → if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0) = if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0))
17		eqid 2729	. . . . . 6 ⊢ (𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0)) = (𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))
18		fvex 6839	. . . . . . 7 ⊢ (𝐹‘𝑘) ∈ V
19		c0ex 11128	. . . . . . 7 ⊢ 0 ∈ V
20	18, 19	ifex 4529	. . . . . 6 ⊢ if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0) ∈ V
21	16, 17, 20	fvmpt 6934	. . . . 5 ⊢ (𝑘 ∈ 𝑍 → ((𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))‘𝑘) = if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0))
22	21	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))‘𝑘) = if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0))
23		isumless.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)
24	23	ifeq1d 4498	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0) = if(𝑘 ∈ 𝐴, 𝐵, 0))
25	22, 24	eqtrd 2764	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))
26		0re 11136	. . . 4 ⊢ 0 ∈ ℝ
27		ifcl 4524	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℝ)
28	3, 26, 27	sylancl 586	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℝ)
29		isumless.7	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐵)
30		leid 11230	. . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≤ 𝐵)
31		breq1 5098	. . . . . 6 ⊢ (𝐵 = if(𝑘 ∈ 𝐴, 𝐵, 0) → (𝐵 ≤ 𝐵 ↔ if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵))
32		breq1 5098	. . . . . 6 ⊢ (0 = if(𝑘 ∈ 𝐴, 𝐵, 0) → (0 ≤ 𝐵 ↔ if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵))
33	31, 32	ifboth 4518	. . . . 5 ⊢ ((𝐵 ≤ 𝐵 ∧ 0 ≤ 𝐵) → if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵)
34	30, 33	sylan 580	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) → if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵)
35	3, 29, 34	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵)
36		isumless.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
37	7, 13, 36, 1, 25, 5	fsumcvg3 15654	. . 3 ⊢ (𝜑 → seq𝑀( + , (𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))) ∈ dom ⇝ )
38		isumless.8	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
39	7, 13, 25, 28, 23, 3, 35, 37, 38	isumle 15769	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ Σ𝑘 ∈ 𝑍 𝐵)
40	12, 39	eqbrtrd 5117	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝑍 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ⊆ wss 3905 ifcif 4478 class class class wbr 5095 ↦ cmpt 5176 dom cdm 5623 ‘cfv 6486 Fincfn 8879 ℂcc 11026 ℝcr 11027 0cc0 11028 + caddc 11031 ≤ cle 11169 ℤcz 12489 ℤ_≥cuz 12753 seqcseq 13926 ⇝ cli 15409 Σcsu 15611

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-pm 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9351 df-inf 9352 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-n0 12403 df-z 12490 df-uz 12754 df-rp 12912 df-fz 13429 df-fzo 13576 df-fl 13714 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-rlim 15414 df-sum 15612

This theorem is referenced by: isumltss 15773 climcnds 15776 harmonic 15784 mertenslem1 15809 prmreclem5 16850 ovoliunlem1 25419 ovoliun2 25423 esumpcvgval 34044 eulerpartlems 34327 geomcau 37738

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