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Theorem bddibl 25720

Description: A bounded function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)

**Assertion**
Ref	Expression
bddibl	⊢ ((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) → 𝐹 ∈ 𝐿¹)

Distinct variable group: 𝑥,𝑦,𝐹

Proof of Theorem bddibl Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfdm 25506	. . . 4 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)
2	1	3ad2ant1 1130	. . 3 ⊢ ((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) → dom 𝐹 ∈ dom vol)
3		mbff 25505	. . . . 5 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)
4	3	3ad2ant1 1130	. . . 4 ⊢ ((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) → 𝐹:dom 𝐹⟶ℂ)
5	4	ffnd 6711	. . 3 ⊢ ((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) → 𝐹 Fn dom 𝐹)
6		1cnd 11210	. . . 4 ⊢ ((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) → 1 ∈ ℂ)
7		fnconstg 6772	. . . 4 ⊢ (1 ∈ ℂ → (dom 𝐹 × {1}) Fn dom 𝐹)
8	6, 7	syl 17	. . 3 ⊢ ((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) → (dom 𝐹 × {1}) Fn dom 𝐹)
9		eqidd 2727	. . 3 ⊢ (((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) = (𝐹‘𝑧))
10		1ex 11211	. . . . 5 ⊢ 1 ∈ V
11	10	fvconst2 7200	. . . 4 ⊢ (𝑧 ∈ dom 𝐹 → ((dom 𝐹 × {1})‘𝑧) = 1)
12	11	adantl 481	. . 3 ⊢ (((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) ∧ 𝑧 ∈ dom 𝐹) → ((dom 𝐹 × {1})‘𝑧) = 1)
13	4	ffvelcdmda 7079	. . . 4 ⊢ (((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ ℂ)
14	13	mulridd 11232	. . 3 ⊢ (((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) ∧ 𝑧 ∈ dom 𝐹) → ((𝐹‘𝑧) · 1) = (𝐹‘𝑧))
15	2, 5, 8, 5, 9, 12, 14	offveq 7690	. 2 ⊢ ((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) → (𝐹 ∘_f · (dom 𝐹 × {1})) = 𝐹)
16		simp2 1134	. . . 4 ⊢ ((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) → (vol‘dom 𝐹) ∈ ℝ)
17		iblconst 25698	. . . 4 ⊢ ((dom 𝐹 ∈ dom vol ∧ (vol‘dom 𝐹) ∈ ℝ ∧ 1 ∈ ℂ) → (dom 𝐹 × {1}) ∈ 𝐿¹)
18	2, 16, 6, 17	syl3anc 1368	. . 3 ⊢ ((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) → (dom 𝐹 × {1}) ∈ 𝐿¹)
19		bddmulibl 25719	. . 3 ⊢ ((𝐹 ∈ MblFn ∧ (dom 𝐹 × {1}) ∈ 𝐿¹ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) → (𝐹 ∘_f · (dom 𝐹 × {1})) ∈ 𝐿¹)
20	18, 19	syld3an2 1408	. 2 ⊢ ((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) → (𝐹 ∘_f · (dom 𝐹 × {1})) ∈ 𝐿¹)
21	15, 20	eqeltrrd 2828	1 ⊢ ((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) → 𝐹 ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 {csn 4623 class class class wbr 5141 × cxp 5667 dom cdm 5669 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 ∘_f cof 7664 ℂcc 11107 ℝcr 11108 1c1 11110 · cmul 11114 ≤ cle 11250 abscabs 15185 volcvol 25343 MblFncmbf 25494 𝐿¹cibl 25497

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cc 10429 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-oadd 8468 df-omul 8469 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-acn 9936 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ioc 13332 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14031 df-hash 14294 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19231 df-cmn 19700 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-cnfld 21237 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-cn 23082 df-cnp 23083 df-cmp 23242 df-tx 23417 df-hmeo 23610 df-xms 24177 df-ms 24178 df-tms 24179 df-cncf 24749 df-ovol 25344 df-vol 25345 df-mbf 25499 df-itg1 25500 df-itg2 25501 df-ibl 25502 df-0p 25550

This theorem is referenced by: cniccibl 25721 iblulm 26294 ftc2re 34139 cnioobibld 42520 cnbdibl 45231

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