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Theorem mbflimlem 25638

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 7115	. . . . 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 15510	. . . . . . . 8 ⊢ Rel ⇝
9	8	releldmi 5939	. . . . . . 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 15689	. . . . . 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 15695	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)))
14		climuni 15570	. . . 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 5222	. 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 7084	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℝ)
20	1, 6, 13, 19	climrecl 15601	. . 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 25637	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵))) ∈ MblFn)
23	16, 22	eqeltrrd 2834	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3050 ∃wrex 3059 ∩ cin 3930 class class class wbr 5123 ↦ cmpt 5205 dom cdm 5665 “ cima 5668 ‘cfv 6541 (class class class)co 7413 supcsup 9462 ℝcr 11136 +∞cpnf 11274 ℝ^*cxr 11276 < clt 11277 − cmin 11474 ℤcz 12596 ℤ_≥cuz 12860 ℝ⁺crp 13016 [,)cico 13371 abscabs 15255 lim supclsp 15488 ⇝ cli 15502 MblFncmbf 25585

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-inf2 9663 ax-cc 10457 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214 ax-pre-sup 11215

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-int 4927 df-iun 4973 df-disj 5091 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-se 5618 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7870 df-1st 7996 df-2nd 7997 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-oadd 8492 df-omul 8493 df-er 8727 df-map 8850 df-pm 8851 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-sup 9464 df-inf 9465 df-oi 9532 df-dju 9923 df-card 9961 df-acn 9964 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-div 11903 df-nn 12249 df-2 12311 df-3 12312 df-n0 12510 df-z 12597 df-uz 12861 df-q 12973 df-rp 13017 df-xadd 13137 df-ioo 13373 df-ioc 13374 df-ico 13375 df-icc 13376 df-fz 13530 df-fzo 13677 df-fl 13814 df-seq 14025 df-exp 14085 df-hash 14352 df-cj 15120 df-re 15121 df-im 15122 df-sqrt 15256 df-abs 15257 df-limsup 15489 df-clim 15506 df-rlim 15507 df-sum 15705 df-xmet 21319 df-met 21320 df-ovol 25435 df-vol 25436 df-mbf 25590

This theorem is referenced by: mbflim 25639

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