Theorem meaiunincf 46404

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 1913	. . . . . 6 ⊢ Ⅎ𝑛 𝑘 ∈ 𝑍
7	5, 6	nfan 1898	. . . . 5 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑘 ∈ 𝑍)
8		meaiunincf.f	. . . . . . 7 ⊢ Ⅎ𝑛𝐸
9		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑛𝑘
10	8, 9	nffv 6930	. . . . . 6 ⊢ Ⅎ𝑛(𝐸‘𝑘)
11		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑛(𝑘 + 1)
12	8, 11	nffv 6930	. . . . . 6 ⊢ Ⅎ𝑛(𝐸‘(𝑘 + 1))
13	10, 12	nfss 4001	. . . . 5 ⊢ Ⅎ𝑛(𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))
14	7, 13	nfim 1895	. . . 4 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))
15		eleq1w 2827	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ 𝑍 ↔ 𝑘 ∈ 𝑍))
16	15	anbi2d 629	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑘 ∈ 𝑍)))
17		fveq2 6920	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐸‘𝑛) = (𝐸‘𝑘))
18		fvoveq1 7471	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑘 + 1)))
19	17, 18	sseq12d 4042	. . . . 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 5170	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ (𝑀‘(𝐸‘𝑛)) ≤ 𝑦))
25	24	ralbidv 3184	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑦))
26		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑀‘(𝐸‘𝑛)) ≤ 𝑦
27		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑛𝑀
28	27, 10	nffv 6930	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑀‘(𝐸‘𝑘))
29		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑛 ≤
30		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑦
31	28, 29, 30	nfbr 5213	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑀‘(𝐸‘𝑘)) ≤ 𝑦
32		2fveq3 6925	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑘)))
33	32	breq1d 5176	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ((𝑀‘(𝐸‘𝑛)) ≤ 𝑦 ↔ (𝑀‘(𝐸‘𝑘)) ≤ 𝑦))
34	26, 31, 33	cbvralw 3312	. . . . . . 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 3244	. . . 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 2908	. . . . 5 ⊢ Ⅎ𝑘(𝑀‘(𝐸‘𝑛))
41	40, 28, 32	cbvmpt 5277	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑘 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑘)))
42	39, 41	eqtri 2768	. . 3 ⊢ 𝑆 = (𝑘 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑘)))
43	1, 2, 3, 4, 22, 38, 42	meaiuninc 46402	. 2 ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘)))
44		nfcv 2908	. . . 4 ⊢ Ⅎ𝑘(𝐸‘𝑛)
45		fveq2 6920	. . . 4 ⊢ (𝑘 = 𝑛 → (𝐸‘𝑘) = (𝐸‘𝑛))
46	10, 44, 45	cbviun 5059	. . 3 ⊢ ∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘) = ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)
47	46	fveq2i 6923	. 2 ⊢ (𝑀‘∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘)) = (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
48	43, 47	breqtrdi 5207	1 ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2108  Ⅎwnfc 2893  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  ∪ ciun 5015   class class class wbr 5166   ↦ cmpt 5249  dom cdm 5700  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  ℝcr 11183  1c1 11185   + caddc 11187   ≤ cle 11325  ℤcz 12639  ℤ_≥cuz 12903   ⇝ cli 15530  Meascmea 46370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-disj 5134  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-omul 8527  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-oi 9579  df-card 10008  df-acn 10011  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-xadd 13176  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735  df-salg 46230  df-sumge0 46284  df-mea 46371

This theorem is referenced by:  meaiuninc3v  46405