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Theorem isumless 15797

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 3977	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑍)
3		isumless.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)
4	3	recnd 11246	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
5	2, 4	syldan 590	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
6	5	ralrimiva 3140	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
7		isumless.1	. . . . . 6 ⊢ 𝑍 = (ℤ_≥‘𝑀)
8	7	eqimssi 4037	. . . . 5 ⊢ 𝑍 ⊆ (ℤ_≥‘𝑀)
9	8	orci 862	. . . 4 ⊢ (𝑍 ⊆ (ℤ_≥‘𝑀) ∨ 𝑍 ∈ Fin)
10	9	a1i 11	. . 3 ⊢ (𝜑 → (𝑍 ⊆ (ℤ_≥‘𝑀) ∨ 𝑍 ∈ Fin))
11		sumss2 15678	. . 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 835	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝑍 if(𝑘 ∈ 𝐴, 𝐵, 0))
13		isumless.2	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
14		eleq1w 2810	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
15		fveq2 6885	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
16	14, 15	ifbieq1d 4547	. . . . . 6 ⊢ (𝑗 = 𝑘 → if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0) = if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0))
17		eqid 2726	. . . . . 6 ⊢ (𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0)) = (𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))
18		fvex 6898	. . . . . . 7 ⊢ (𝐹‘𝑘) ∈ V
19		c0ex 11212	. . . . . . 7 ⊢ 0 ∈ V
20	18, 19	ifex 4573	. . . . . 6 ⊢ if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0) ∈ V
21	16, 17, 20	fvmpt 6992	. . . . 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 4542	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0) = if(𝑘 ∈ 𝐴, 𝐵, 0))
25	22, 24	eqtrd 2766	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))
26		0re 11220	. . . 4 ⊢ 0 ∈ ℝ
27		ifcl 4568	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℝ)
28	3, 26, 27	sylancl 585	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℝ)
29		isumless.7	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐵)
30		leid 11314	. . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≤ 𝐵)
31		breq1 5144	. . . . . 6 ⊢ (𝐵 = if(𝑘 ∈ 𝐴, 𝐵, 0) → (𝐵 ≤ 𝐵 ↔ if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵))
32		breq1 5144	. . . . . 6 ⊢ (0 = if(𝑘 ∈ 𝐴, 𝐵, 0) → (0 ≤ 𝐵 ↔ if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵))
33	31, 32	ifboth 4562	. . . . 5 ⊢ ((𝐵 ≤ 𝐵 ∧ 0 ≤ 𝐵) → if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵)
34	30, 33	sylan 579	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) → if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵)
35	3, 29, 34	syl2anc 583	. . 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 15681	. . 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 15796	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ Σ𝑘 ∈ 𝑍 𝐵)
40	12, 39	eqbrtrd 5163	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝑍 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ⊆ wss 3943 ifcif 4523 class class class wbr 5141 ↦ cmpt 5224 dom cdm 5669 ‘cfv 6537 Fincfn 8941 ℂcc 11110 ℝcr 11111 0cc0 11112 + caddc 11115 ≤ cle 11253 ℤcz 12562 ℤ_≥cuz 12826 seqcseq 13972 ⇝ cli 15434 Σcsu 15638

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 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 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-rlim 15439 df-sum 15639

This theorem is referenced by: isumltss 15800 climcnds 15803 harmonic 15811 mertenslem1 15836 prmreclem5 16862 ovoliunlem1 25386 ovoliun2 25390 esumpcvgval 33606 eulerpartlems 33889 geomcau 37140

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