itggt0 - Metamath Proof Explorer

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Theorem itggt0 25964

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 25887	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
4		itggt0.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ⁺)
5	3, 4	mbfdm2 25757	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom vol)
6		itggt0.1	. . 3 ⊢ (𝜑 → 0 < (vol‘𝐴))
7	4	rpred 13051	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
8	4	rpge0d 13055	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵)
9		elrege0 13472	. . . . . . 7 ⊢ (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))
10	7, 8, 9	sylanbrc 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))
11		0e0icopnf 13476	. . . . . . 7 ⊢ 0 ∈ (0[,)+∞)
12	11	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
13	10, 12	ifclda 4519	. . . . 5 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞))
14	13	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞))
15	14	fmpttd 7100	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)):ℝ⟶(0[,)+∞))
16		mblss 25651	. . . . 5 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
17	5, 16	syl 18	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
18		rembl 25660	. . . . 5 ⊢ ℝ ∈ dom vol
19	18	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ dom vol)
20	13	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞))
21		eldifn 4088	. . . . . 6 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
22	21	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴)
23	22	iffalsed 4494	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → if(𝑥 ∈ 𝐴, 𝐵, 0) = 0)
24		iftrue 4489	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, 𝐵, 0) = 𝐵)
25	24	mpteq2ia 5200	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) = (𝑥 ∈ 𝐴 ↦ 𝐵)
26	25, 3	eqeltrid 2869	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) ∈ MblFn)
27	17, 19, 20, 23, 26	mbfss 25766	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) ∈ MblFn)
28	4	rpgt0d 13054	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 < 𝐵)
29	17	sselda 3939	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
30	24	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, 𝐵, 0) = 𝐵)
31	30, 4	eqeltrd 2865	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ ℝ⁺)
32		eqid 2765	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))
33	32	fvmpt2 6991	. . . . . . . . 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 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥) = if(𝑥 ∈ 𝐴, 𝐵, 0))
35	34, 30	eqtrd 2800	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥) = 𝐵)
36	28, 35	breqtrrd 5133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥))
37	36	ralrimiva 3157	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥))
38		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑥0
39		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑥 <
40		nffvmpt1 6882	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦)
41	38, 39, 40	nfbr 5152	. . . . . 6 ⊢ Ⅎ𝑥0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦)
42		nfv 1937	. . . . . 6 ⊢ Ⅎ𝑦0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥)
43		fveq2 6871	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥))
44	43	breq2d 5117	. . . . . 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 3307	. . . . 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 3257	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦))
48	5, 6, 15, 27, 47	itg2gt0 25880	. 2 ⊢ (𝜑 → 0 < (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))))
49	7, 1, 8	itgposval 25916	. 2 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))))
50	48, 49	breqtrrd 5133	1 ⊢ (𝜑 → 0 < ∫𝐴𝐵 d𝑥)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ∀wral 3079 ∖ cdif 3904 ⊆ wss 3907 ifcif 4483 class class class wbr 5105 ↦ cmpt 5186 dom cdm 5652 ‘cfv 6525 (class class class)co 7400 ℝcr 11087 0cc0 11088 +∞cpnf 11228 < clt 11231 ≤ cle 11232 ℝ⁺crp 13007 [,)cico 13365 volcvol 25583 MblFncmbf 25734 ∫₂citg2 25736 𝐿¹cibl 25737 ∫citg 25738

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-inf2 9598 ax-cc 10407 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-disj 5073 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-ofr 7665 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-pm 8815 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-dju 9875 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-n0 12496 df-z 12583 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13367 df-ico 13369 df-icc 13370 df-fz 13527 df-fzo 13674 df-fl 13816 df-mod 13894 df-seq 14029 df-exp 14089 df-hash 14358 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-clim 15529 df-rlim 15530 df-sum 15728 df-rest 17465 df-topgen 17486 df-psmet 21474 df-xmet 21475 df-met 21476 df-bl 21477 df-mopn 21478 df-top 23012 df-topon 23029 df-bases 23064 df-cmp 23505 df-cncf 24998 df-ovol 25584 df-vol 25585 df-mbf 25739 df-itg1 25740 df-itg2 25741 df-ibl 25742 df-itg 25743 df-0p 25790

This theorem is referenced by: ftc1lem4 26159 fdvposlt 34903

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