itggt0 - Metamath Proof Explorer

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Theorem itggt0 25899

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 25822	. . . . 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 25691	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom vol)
6		itggt0.1	. . 3 ⊢ (𝜑 → 0 < (vol‘𝐴))
7	4	rpred 13099	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
8	4	rpge0d 13103	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵)
9		elrege0 13514	. . . . . . 7 ⊢ (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))
10	7, 8, 9	sylanbrc 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))
11		0e0icopnf 13518	. . . . . . 7 ⊢ 0 ∈ (0[,)+∞)
12	11	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
13	10, 12	ifclda 4583	. . . . 5 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞))
14	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞))
15	14	fmpttd 7149	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)):ℝ⟶(0[,)+∞))
16		mblss 25585	. . . . 5 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
17	5, 16	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
18		rembl 25594	. . . . 5 ⊢ ℝ ∈ dom vol
19	18	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ dom vol)
20	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞))
21		eldifn 4155	. . . . . 6 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
22	21	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴)
23	22	iffalsed 4559	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → if(𝑥 ∈ 𝐴, 𝐵, 0) = 0)
24		iftrue 4554	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, 𝐵, 0) = 𝐵)
25	24	mpteq2ia 5269	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) = (𝑥 ∈ 𝐴 ↦ 𝐵)
26	25, 3	eqeltrid 2848	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) ∈ MblFn)
27	17, 19, 20, 23, 26	mbfss 25700	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) ∈ MblFn)
28	4	rpgt0d 13102	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 < 𝐵)
29	17	sselda 4008	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
30	24	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, 𝐵, 0) = 𝐵)
31	30, 4	eqeltrd 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ ℝ⁺)
32		eqid 2740	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))
33	32	fvmpt2 7040	. . . . . . . . 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 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥) = if(𝑥 ∈ 𝐴, 𝐵, 0))
35	34, 30	eqtrd 2780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥) = 𝐵)
36	28, 35	breqtrrd 5194	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥))
37	36	ralrimiva 3152	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥))
38		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑥0
39		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑥 <
40		nffvmpt1 6931	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦)
41	38, 39, 40	nfbr 5213	. . . . . 6 ⊢ Ⅎ𝑥0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦)
42		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑦0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥)
43		fveq2 6920	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥))
44	43	breq2d 5178	. . . . . 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 3312	. . . . 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 234	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦))
47	46	r19.21bi 3257	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦))
48	5, 6, 15, 27, 47	itg2gt0 25815	. 2 ⊢ (𝜑 → 0 < (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))))
49	7, 1, 8	itgposval 25851	. 2 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))))
50	48, 49	breqtrrd 5194	1 ⊢ (𝜑 → 0 < ∫𝐴𝐵 d𝑥)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∖ cdif 3973 ⊆ wss 3976 ifcif 4548 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 0cc0 11184 +∞cpnf 11321 < clt 11324 ≤ cle 11325 ℝ⁺crp 13057 [,)cico 13409 volcvol 25517 MblFncmbf 25668 ∫₂citg2 25670 𝐿¹cibl 25671 ∫citg 25672

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cc 10504 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-n0 12554 df-z 12640 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-rlim 15535 df-sum 15735 df-rest 17482 df-topgen 17503 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-top 22921 df-topon 22938 df-bases 22974 df-cmp 23416 df-cncf 24923 df-ovol 25518 df-vol 25519 df-mbf 25673 df-itg1 25674 df-itg2 25675 df-ibl 25676 df-itg 25677 df-0p 25724

This theorem is referenced by: ftc1lem4 26100 fdvposlt 34576

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