itggt0 - Metamath Proof Explorer

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Theorem itggt0 24695

Description: The integral of a strictly positive function is positive. (Contributed by Mario Carneiro, 30-Aug-2014.)

**Hypotheses**
Ref	Expression
itggt0.1	⊢ (𝜑 → 0 < (vol‘𝐴))
itggt0.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
itggt0.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ⁺)

**Assertion**
Ref	Expression
itggt0	⊢ (𝜑 → 0 < ∫𝐴𝐵 d𝑥)

Distinct variable groups: 𝑥,𝐴 𝜑,𝑥 Allowed substitution hint: 𝐵(𝑥)

Proof of Theorem itggt0 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		itggt0.2	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
2		iblmbf 24619	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
4		itggt0.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ⁺)
5	3, 4	mbfdm2 24488	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom vol)
6		itggt0.1	. . 3 ⊢ (𝜑 → 0 < (vol‘𝐴))
7	4	rpred 12593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
8	4	rpge0d 12597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵)
9		elrege0 13007	. . . . . . 7 ⊢ (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))
10	7, 8, 9	sylanbrc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))
11		0e0icopnf 13011	. . . . . . 7 ⊢ 0 ∈ (0[,)+∞)
12	11	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
13	10, 12	ifclda 4460	. . . . 5 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞))
14	13	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞))
15	14	fmpttd 6910	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)):ℝ⟶(0[,)+∞))
16		mblss 24382	. . . . 5 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
17	5, 16	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
18		rembl 24391	. . . . 5 ⊢ ℝ ∈ dom vol
19	18	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ dom vol)
20	13	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞))
21		eldifn 4028	. . . . . 6 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
22	21	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴)
23	22	iffalsed 4436	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → if(𝑥 ∈ 𝐴, 𝐵, 0) = 0)
24		iftrue 4431	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, 𝐵, 0) = 𝐵)
25	24	mpteq2ia 5131	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) = (𝑥 ∈ 𝐴 ↦ 𝐵)
26	25, 3	eqeltrid 2835	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) ∈ MblFn)
27	17, 19, 20, 23, 26	mbfss 24497	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) ∈ MblFn)
28	4	rpgt0d 12596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 < 𝐵)
29	17	sselda 3887	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
30	24	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, 𝐵, 0) = 𝐵)
31	30, 4	eqeltrd 2831	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ ℝ⁺)
32		eqid 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))
33	32	fvmpt2 6807	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ ℝ⁺) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥) = if(𝑥 ∈ 𝐴, 𝐵, 0))
34	29, 31, 33	syl2anc 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥) = if(𝑥 ∈ 𝐴, 𝐵, 0))
35	34, 30	eqtrd 2771	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥) = 𝐵)
36	28, 35	breqtrrd 5067	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥))
37	36	ralrimiva 3095	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥))
38		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑥0
39		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑥 <
40		nffvmpt1 6706	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦)
41	38, 39, 40	nfbr 5086	. . . . . 6 ⊢ Ⅎ𝑥0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦)
42		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑦0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥)
43		fveq2 6695	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥))
44	43	breq2d 5051	. . . . . 6 ⊢ (𝑦 = 𝑥 → (0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦) ↔ 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥)))
45	41, 42, 44	cbvralw 3339	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦) ↔ ∀𝑥 ∈ 𝐴 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥))
46	37, 45	sylibr 237	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦))
47	46	r19.21bi 3120	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦))
48	5, 6, 15, 27, 47	itg2gt0 24612	. 2 ⊢ (𝜑 → 0 < (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))))
49	7, 1, 8	itgposval 24647	. 2 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))))
50	48, 49	breqtrrd 5067	1 ⊢ (𝜑 → 0 < ∫𝐴𝐵 d𝑥)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∖ cdif 3850 ⊆ wss 3853 ifcif 4425 class class class wbr 5039 ↦ cmpt 5120 dom cdm 5536 ‘cfv 6358 (class class class)co 7191 ℝcr 10693 0cc0 10694 +∞cpnf 10829 < clt 10832 ≤ cle 10833 ℝ⁺crp 12551 [,)cico 12902 volcvol 24314 MblFncmbf 24465 ∫₂citg2 24467 𝐿¹cibl 24468 ∫citg 24469

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cc 10014 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-disj 5005 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-ofr 7448 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fi 9005 df-sup 9036 df-inf 9037 df-oi 9104 df-dju 9482 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-n0 12056 df-z 12142 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-ioo 12904 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-fl 13332 df-mod 13408 df-seq 13540 df-exp 13601 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-clim 15014 df-rlim 15015 df-sum 15215 df-rest 16881 df-topgen 16902 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-top 21745 df-topon 21762 df-bases 21797 df-cmp 22238 df-cncf 23729 df-ovol 24315 df-vol 24316 df-mbf 24470 df-itg1 24471 df-itg2 24472 df-ibl 24473 df-itg 24474 df-0p 24521

This theorem is referenced by: ftc1lem4 24890 fdvposlt 32245

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