Theorem meaiunincf 46484

Description: Measures are continuous from below (bounded case): if 𝐸 is a sequence of nondecreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is Proposition 112C (e) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 13-Feb-2022.)

Hypotheses

Ref	Expression
meaiunincf.p	⊢ Ⅎ𝑛𝜑
meaiunincf.f	⊢ Ⅎ𝑛𝐸
meaiunincf.m	⊢ (𝜑 → 𝑀 ∈ Meas)
meaiunincf.n	⊢ (𝜑 → 𝑁 ∈ ℤ)
meaiunincf.z	⊢ 𝑍 = (ℤ≥‘𝑁)
meaiunincf.e	⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀)
meaiunincf.i	⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)))
meaiunincf.x	⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
meaiunincf.s	⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))

Assertion

Ref	Expression
meaiunincf	⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))

Distinct variable groups:   𝑥,𝐸   𝑛,𝑀,𝑥   𝑛,𝑍,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑛)   𝑆(𝑥,𝑛)   𝐸(𝑛)   𝑁(𝑥,𝑛)

Proof of Theorem meaiunincf

Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		meaiunincf.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ Meas)
2		meaiunincf.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
3		meaiunincf.z	. . 3 ⊢ 𝑍 = (ℤ≥‘𝑁)
4		meaiunincf.e	. . 3 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀)
5		meaiunincf.p	. . . . . 6 ⊢ Ⅎ𝑛𝜑
6		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑛 𝑘 ∈ 𝑍
7	5, 6	nfan 1899	. . . . 5 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑘 ∈ 𝑍)
8		meaiunincf.f	. . . . . . 7 ⊢ Ⅎ𝑛𝐸
9		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑛𝑘
10	8, 9	nffv 6832	. . . . . 6 ⊢ Ⅎ𝑛(𝐸‘𝑘)
11		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑛(𝑘 + 1)
12	8, 11	nffv 6832	. . . . . 6 ⊢ Ⅎ𝑛(𝐸‘(𝑘 + 1))
13	10, 12	nfss 3928	. . . . 5 ⊢ Ⅎ𝑛(𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))
14	7, 13	nfim 1896	. . . 4 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))
15		eleq1w 2811	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ 𝑍 ↔ 𝑘 ∈ 𝑍))
16	15	anbi2d 630	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑘 ∈ 𝑍)))
17		fveq2 6822	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐸‘𝑛) = (𝐸‘𝑘))
18		fvoveq1 7372	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑘 + 1)))
19	17, 18	sseq12d 3969	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)) ↔ (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))))
20	16, 19	imbi12d 344	. . . 4 ⊢ (𝑛 = 𝑘 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) ↔ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))))
21		meaiunincf.i	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)))
22	14, 20, 21	chvarfv 2241	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))
23		meaiunincf.x	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
24		breq2 5096	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ (𝑀‘(𝐸‘𝑛)) ≤ 𝑦))
25	24	ralbidv 3152	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑦))
26		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑀‘(𝐸‘𝑛)) ≤ 𝑦
27		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑛𝑀
28	27, 10	nffv 6832	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑀‘(𝐸‘𝑘))
29		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑛 ≤
30		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑦
31	28, 29, 30	nfbr 5139	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑀‘(𝐸‘𝑘)) ≤ 𝑦
32		2fveq3 6827	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑘)))
33	32	breq1d 5102	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ((𝑀‘(𝐸‘𝑛)) ≤ 𝑦 ↔ (𝑀‘(𝐸‘𝑘)) ≤ 𝑦))
34	26, 31, 33	cbvralw 3271	. . . . . . 7 ⊢ (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑦 ↔ ∀𝑘 ∈ 𝑍 (𝑀‘(𝐸‘𝑘)) ≤ 𝑦)
35	34	a1i 11	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑦 ↔ ∀𝑘 ∈ 𝑍 (𝑀‘(𝐸‘𝑘)) ≤ 𝑦))
36	25, 35	bitrd 279	. . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 (𝑀‘(𝐸‘𝑘)) ≤ 𝑦))
37	36	cbvrexvw 3208	. . . 4 ⊢ (∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝑀‘(𝐸‘𝑘)) ≤ 𝑦)
38	23, 37	sylib 218	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝑀‘(𝐸‘𝑘)) ≤ 𝑦)
39		meaiunincf.s	. . . 4 ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
40		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑘(𝑀‘(𝐸‘𝑛))
41	40, 28, 32	cbvmpt 5194	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑘 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑘)))
42	39, 41	eqtri 2752	. . 3 ⊢ 𝑆 = (𝑘 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑘)))
43	1, 2, 3, 4, 22, 38, 42	meaiuninc 46482	. 2 ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘)))
44		nfcv 2891	. . . 4 ⊢ Ⅎ𝑘(𝐸‘𝑛)
45		fveq2 6822	. . . 4 ⊢ (𝑘 = 𝑛 → (𝐸‘𝑘) = (𝐸‘𝑛))
46	10, 44, 45	cbviun 4985	. . 3 ⊢ ∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘) = ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)
47	46	fveq2i 6825	. 2 ⊢ (𝑀‘∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘)) = (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
48	43, 47	breqtrdi 5133	1 ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  Ⅎwnfc 2876  ∀wral 3044  ∃wrex 3053   ⊆ wss 3903  ∪ ciun 4941   class class class wbr 5092   ↦ cmpt 5173  dom cdm 5619  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  ℝcr 11008  1c1 11010   + caddc 11012   ≤ cle 11150  ℤcz 12471  ℤ_≥cuz 12735   ⇝ cli 15391  Meascmea 46450

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-disj 5060  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-omul 8393  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-oi 9402  df-card 9835  df-acn 9838  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-z 12472  df-uz 12736  df-rp 12894  df-xadd 13015  df-ico 13254  df-icc 13255  df-fz 13411  df-fzo 13558  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-sum 15594  df-salg 46310  df-sumge0 46364  df-mea 46451

This theorem is referenced by:  meaiuninc3v  46485