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Theorem mbflimlem 25622

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 7058	. . . . 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 15413	. . . . . . . 8 ⊢ Rel ⇝
9	8	releldmi 5895	. . . . . . 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 15592	. . . . . 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 15598	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)))
14		climuni 15473	. . . 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 5189	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵))) = (𝑥 ∈ 𝐴 ↦ 𝐶))
17		eqid 2734	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵))) = (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)))
18		eqid 2734	. . 3 ⊢ (𝑚 ∈ ℝ ↦ sup((((𝑛 ∈ 𝑍 ↦ 𝐵) “ (𝑚[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑚 ∈ ℝ ↦ sup((((𝑛 ∈ 𝑍 ↦ 𝐵) “ (𝑚[,)+∞)) ∩ ℝ^), ℝ^, < ))
19	4	ffvelcdmda 7027	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℝ)
20	1, 6, 13, 19	climrecl 15504	. . 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 25621	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵))) ∈ MblFn)
23	16, 22	eqeltrrd 2835	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3049 ∃wrex 3058 ∩ cin 3898 class class class wbr 5096 ↦ cmpt 5177 dom cdm 5622 “ cima 5625 ‘cfv 6490 (class class class)co 7356 supcsup 9341 ℝcr 11023 +∞cpnf 11161 ℝ^*cxr 11163 < clt 11164 − cmin 11362 ℤcz 12486 ℤ_≥cuz 12749 ℝ⁺crp 12903 [,)cico 13261 abscabs 15155 lim supclsp 15391 ⇝ cli 15405 MblFncmbf 25569

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cc 10343 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-disj 5064 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8633 df-map 8763 df-pm 8764 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-sup 9343 df-inf 9344 df-oi 9413 df-dju 9811 df-card 9849 df-acn 9852 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-n0 12400 df-z 12487 df-uz 12750 df-q 12860 df-rp 12904 df-xadd 13025 df-ioo 13263 df-ioc 13264 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-fl 13710 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-limsup 15392 df-clim 15409 df-rlim 15410 df-sum 15608 df-xmet 21300 df-met 21301 df-ovol 25419 df-vol 25420 df-mbf 25574

This theorem is referenced by: mbflim 25623

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