isumlessdc - Intuitionistic Logic Explorer

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Theorem isumlessdc 12210

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.dc	⊢ (𝜑 → ∀𝑘 ∈ 𝑍 DECID 𝑘 ∈ 𝐴)
isumless.6	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)
isumless.7	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐵)
isumless.8	⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

**Assertion**
Ref	Expression
isumlessdc	⊢ (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 isumlessdc Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isumless.4	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑍)
2		isumless.dc	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 DECID 𝑘 ∈ 𝐴)
3	1	sselda 3242	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑍)
4		isumless.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)
5	4	recnd 8318	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
6	3, 5	syldan 282	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
7	6	ralrimiva 2617	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
8		isumless.2	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
9		isumless.1	. . . . . . 7 ⊢ 𝑍 = (ℤ_≥‘𝑀)
10	9	eqimssi 3298	. . . . . 6 ⊢ 𝑍 ⊆ (ℤ_≥‘𝑀)
11	10	a1i 9	. . . . 5 ⊢ (𝜑 → 𝑍 ⊆ (ℤ_≥‘𝑀))
12	9	eleq2i 2301	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ_≥‘𝑀))
13	12	biimpri 133	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑘 ∈ 𝑍)
14	13	orcd 741	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝑘 ∈ 𝑍 ∨ ¬ 𝑘 ∈ 𝑍))
15		df-dc 843	. . . . . . . 8 ⊢ (DECID 𝑘 ∈ 𝑍 ↔ (𝑘 ∈ 𝑍 ∨ ¬ 𝑘 ∈ 𝑍))
16	14, 15	sylibr 134	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → DECID 𝑘 ∈ 𝑍)
17	16	rgen 2597	. . . . . 6 ⊢ ∀𝑘 ∈ (ℤ_≥‘𝑀)DECID 𝑘 ∈ 𝑍
18	17	a1i 9	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (ℤ_≥‘𝑀)DECID 𝑘 ∈ 𝑍)
19	8, 11, 18	3jca 1204	. . . 4 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑍 ⊆ (ℤ_≥‘𝑀) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑀)DECID 𝑘 ∈ 𝑍))
20	19	orcd 741	. . 3 ⊢ (𝜑 → ((𝑀 ∈ ℤ ∧ 𝑍 ⊆ (ℤ_≥‘𝑀) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑀)DECID 𝑘 ∈ 𝑍) ∨ 𝑍 ∈ Fin))
21	1, 2, 7, 20	isumss2 12107	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝑍 if(𝑘 ∈ 𝐴, 𝐵, 0))
22		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
23		isumless.5	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)
24	23, 4	eqeltrd 2311	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
25	24	adantr 276	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ℝ)
26		0red 8291	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ¬ 𝑘 ∈ 𝐴) → 0 ∈ ℝ)
27	2	r19.21bi 2632	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → DECID 𝑘 ∈ 𝐴)
28	25, 26, 27	ifcldadc 3656	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0) ∈ ℝ)
29		eleq1w 2295	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
30		fveq2 5675	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
31	29, 30	ifbieq1d 3649	. . . . . 6 ⊢ (𝑗 = 𝑘 → if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0) = if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0))
32		eqid 2234	. . . . . 6 ⊢ (𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0)) = (𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))
33	31, 32	fvmptg 5758	. . . . 5 ⊢ ((𝑘 ∈ 𝑍 ∧ if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0) ∈ ℝ) → ((𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))‘𝑘) = if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0))
34	22, 28, 33	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))‘𝑘) = if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0))
35	23	ifeq1d 3644	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → if(𝑘 ∈ 𝐴, (𝐹‘𝑘), 0) = if(𝑘 ∈ 𝐴, 𝐵, 0))
36	34, 35	eqtrd 2267	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))
37	35, 28	eqeltrrd 2312	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℝ)
38	4	leidd 8806	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ≤ 𝐵)
39		isumless.7	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐵)
40		breq1 4117	. . . . 5 ⊢ (𝐵 = if(𝑘 ∈ 𝐴, 𝐵, 0) → (𝐵 ≤ 𝐵 ↔ if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵))
41		breq1 4117	. . . . 5 ⊢ (0 = if(𝑘 ∈ 𝐴, 𝐵, 0) → (0 ≤ 𝐵 ↔ if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵))
42	40, 41	ifbothdc 3661	. . . 4 ⊢ ((𝐵 ≤ 𝐵 ∧ 0 ≤ 𝐵 ∧ DECID 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵)
43	38, 39, 27, 42	syl3anc 1274	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ 𝐵)
44		isumless.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
45	13, 27	sylan2 286	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)
46	9, 8, 44, 1, 45, 36, 6	fsum3cvg3 12110	. . 3 ⊢ (𝜑 → seq𝑀( + , (𝑗 ∈ 𝑍 ↦ if(𝑗 ∈ 𝐴, (𝐹‘𝑗), 0))) ∈ dom ⇝ )
47		isumless.8	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
48	9, 8, 36, 37, 23, 4, 43, 46, 47	isumle 12209	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 if(𝑘 ∈ 𝐴, 𝐵, 0) ≤ Σ𝑘 ∈ 𝑍 𝐵)
49	21, 48	eqbrtrd 4136	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝑍 𝐵)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 716 DECID wdc 842 ∧ w3a 1005 = wceq 1398 ∈ wcel 2205 ∀wral 2522 ⊆ wss 3214 ifcif 3624 class class class wbr 4114 ↦ cmpt 4176 dom cdm 4754 ‘cfv 5357 Fincfn 6988 ℂcc 8141 ℝcr 8142 0cc0 8143 + caddc 8146 ≤ cle 8325 ℤcz 9597 ℤ_≥cuz 9874 seqcseq 10836 ⇝ cli 11991 Σcsu 12066

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-oadd 6664 df-er 6780 df-en 6989 df-dom 6990 df-fin 6991 df-sup 7288 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8463 df-neg 8464 df-reap 8867 df-ap 8874 df-div 8967 df-inn 9258 df-2 9316 df-3 9317 df-4 9318 df-n0 9517 df-z 9598 df-uz 9875 df-q 9973 df-rp 10008 df-fz 10365 df-fzo 10502 df-seqfrec 10837 df-exp 10928 df-ihash 11167 df-cj 11555 df-re 11556 df-im 11557 df-rsqrt 11711 df-abs 11712 df-clim 11992 df-sumdc 12067

This theorem is referenced by: mertenslemi1 12249

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