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Theorem isumless 15811

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 3946	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑍)
3		isumless.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)
4	3	recnd 11202	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
5	2, 4	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
6	5	ralrimiva 3125	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
7		isumless.1	. . . . . 6 ⊢ 𝑍 = (ℤ_≥‘𝑀)
8	7	eqimssi 4007	. . . . 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 15692	. . 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 6858	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
16	14, 15	ifbieq1d 4513	. . . . . 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 6871	. . . . . . 7 ⊢ (𝐹‘𝑘) ∈ V
19		c0ex 11168	. . . . . . 7 ⊢ 0 ∈ V
20	18, 19	ifex 4539	. . . . . 6 ⊢ if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0) ∈ V
21	16, 17, 20	fvmpt 6968	. . . . 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 4508	. . . 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 11176	. . . 4 ⊢ 0 ∈ ℝ
27		ifcl 4534	. . . 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 11270	. . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≤ 𝐵)
31		breq1 5110	. . . . . 6 ⊢ (𝐵 = if(𝑘 ∈ 𝐴, 𝐵, 0) → (𝐵 ≤ 𝐵 ↔ if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵))
32		breq1 5110	. . . . . 6 ⊢ (0 = if(𝑘 ∈ 𝐴, 𝐵, 0) → (0 ≤ 𝐵 ↔ if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵))
33	31, 32	ifboth 4528	. . . . 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 15695	. . 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 15810	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ Σ𝑘 ∈ 𝑍 𝐵)
40	12, 39	eqbrtrd 5129	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 3914 ifcif 4488 class class class wbr 5107 ↦ cmpt 5188 dom cdm 5638 ‘cfv 6511 Fincfn 8918 ℂcc 11066 ℝcr 11067 0cc0 11068 + caddc 11071 ≤ cle 11209 ℤcz 12529 ℤ_≥cuz 12793 seqcseq 13966 ⇝ cli 15450 Σcsu 15652

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 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

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 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-fz 13469 df-fzo 13616 df-fl 13754 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-rlim 15455 df-sum 15653

This theorem is referenced by: isumltss 15814 climcnds 15817 harmonic 15825 mertenslem1 15850 prmreclem5 16891 ovoliunlem1 25403 ovoliun2 25407 esumpcvgval 34068 eulerpartlems 34351 geomcau 37753

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