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Theorem mbflimlem 25896

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 668	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ)
4	3	fmpttd 7109	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ)
5		mbflim.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
6	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ ℤ)
7		mbflim.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶)
8		climrel 15580	. . . . . . . 8 ⊢ Rel ⇝
9	8	releldmi 5932	. . . . . . 7 ⊢ ((𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶 → (𝑛 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )
10	7, 9	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )
11	1	climcau 15759	. . . . . 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 596	. . . . 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 15765	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)))
14		climuni 15640	. . . 4 ⊢ (((𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ∧ (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) = 𝐶)
15	13, 7, 14	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) = 𝐶)
16	15	mpteq2dva 5198	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵))) = (𝑥 ∈ 𝐴 ↦ 𝐶))
17		eqid 2760	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵))) = (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)))
18		eqid 2760	. . 3 ⊢ (𝑚 ∈ ℝ ↦ sup((((𝑛 ∈ 𝑍 ↦ 𝐵) “ (𝑚[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑚 ∈ ℝ ↦ sup((((𝑛 ∈ 𝑍 ↦ 𝐵) “ (𝑚[,)+∞)) ∩ ℝ^), ℝ^, < ))
19	4	ffvelcdmda 7078	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℝ)
20	1, 6, 13, 19	climrecl 15671	. . 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 25895	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵))) ∈ MblFn)
23	16, 22	eqeltrrd 2861	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∃wrex 3086 ∩ cin 3898 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5655 “ cima 5658 ‘cfv 6533 (class class class)co 7414 supcsup 9411 ℝcr 11124 +∞cpnf 11265 ℝ^*cxr 11267 < clt 11268 − cmin 11466 ℤcz 12616 ℤ_≥cuz 12888 ℝ⁺crp 13043 [,)cico 13401 abscabs 15322 lim supclsp 15558 ⇝ cli 15572 MblFncmbf 25843

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9621 ax-cc 10438 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-oadd 8460 df-omul 8461 df-er 8697 df-map 8829 df-pm 8830 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9413 df-inf 9414 df-oi 9483 df-dju 9907 df-card 9945 df-acn 9948 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-n0 12530 df-z 12617 df-uz 12889 df-q 12999 df-rp 13044 df-xadd 13165 df-ioo 13403 df-ioc 13404 df-ico 13405 df-icc 13406 df-fz 13563 df-fzo 13711 df-fl 13854 df-seq 14067 df-exp 14127 df-hash 14396 df-cj 15187 df-re 15188 df-im 15189 df-sqrt 15323 df-abs 15324 df-limsup 15559 df-clim 15576 df-rlim 15577 df-sum 15775 df-xmet 21579 df-met 21580 df-ovol 25693 df-vol 25694 df-mbf 25848

This theorem is used by: mbflim 25897

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