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Theorem mbflimlem 25568

Description: The pointwise limit of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)

**Hypotheses**
Ref	Expression
mbflim.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
mbflim.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
mbflim.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶)
mbflim.5	⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
mbflimlem.6	⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)

**Assertion**
Ref	Expression
mbflimlem	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)

Distinct variable groups: 𝑥,𝑛,𝐴 𝜑,𝑛,𝑥 𝑛,𝑍,𝑥 Allowed substitution hints: 𝐵(𝑥,𝑛) 𝐶(𝑥,𝑛) 𝑀(𝑥,𝑛)

Proof of Theorem mbflimlem Dummy variables 𝑗 𝑘 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbflim.1	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑀)
2		mbflimlem.6	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)
3	2	anass1rs 655	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ)
4	3	fmpttd 7087	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ)
5		mbflim.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ ℤ)
7		mbflim.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶)
8		climrel 15458	. . . . . . . 8 ⊢ Rel ⇝
9	8	releldmi 5912	. . . . . . 7 ⊢ ((𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶 → (𝑛 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )
10	7, 9	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )
11	1	climcau 15637	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ (𝑛 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ) → ∀𝑦 ∈ ℝ⁺ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑘)(abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑗) − ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘))) < 𝑦)
12	6, 10, 11	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ ℝ⁺ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑘)(abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑗) − ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘))) < 𝑦)
13	1, 4, 12	caurcvg 15643	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)))
14		climuni 15518	. . . 4 ⊢ (((𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ∧ (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) = 𝐶)
15	13, 7, 14	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) = 𝐶)
16	15	mpteq2dva 5200	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵))) = (𝑥 ∈ 𝐴 ↦ 𝐶))
17		eqid 2729	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵))) = (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)))
18		eqid 2729	. . 3 ⊢ (𝑚 ∈ ℝ ↦ sup((((𝑛 ∈ 𝑍 ↦ 𝐵) “ (𝑚[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑚 ∈ ℝ ↦ sup((((𝑛 ∈ 𝑍 ↦ 𝐵) “ (𝑚[,)+∞)) ∩ ℝ^), ℝ^, < ))
19	4	ffvelcdmda 7056	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℝ)
20	1, 6, 13, 19	climrecl 15549	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)
21		mbflim.5	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
22	1, 17, 18, 5, 20, 21, 2	mbflimsup 25567	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵))) ∈ MblFn)
23	16, 22	eqeltrrd 2829	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 ∩ cin 3913 class class class wbr 5107 ↦ cmpt 5188 dom cdm 5638 “ cima 5641 ‘cfv 6511 (class class class)co 7387 supcsup 9391 ℝcr 11067 +∞cpnf 11205 ℝ^*cxr 11207 < clt 11208 − cmin 11405 ℤcz 12529 ℤ_≥cuz 12793 ℝ⁺crp 12951 [,)cico 13308 abscabs 15200 lim supclsp 15436 ⇝ cli 15450 MblFncmbf 25515

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-cc 10388 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-disj 5075 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-of 7653 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-2o 8435 df-oadd 8438 df-omul 8439 df-er 8671 df-map 8801 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-dju 9854 df-card 9892 df-acn 9895 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-q 12908 df-rp 12952 df-xadd 13073 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 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-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-xmet 21257 df-met 21258 df-ovol 25365 df-vol 25366 df-mbf 25520

This theorem is referenced by: mbflim 25569

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