itggt0 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > itggt0		Structured version Visualization version GIF version

Theorem itggt0 26194

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 26118	. . . . 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 25988	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom vol)
6		itggt0.1	. . 3 ⊢ (𝜑 → 0 < (vol‘𝐴))
7	4	rpred 13191	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
8	4	rpge0d 13195	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵)
9		elrege0 13612	. . . . . . 7 ⊢ (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))
10	7, 8, 9	sylanbrc 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))
11		0e0icopnf 13616	. . . . . . 7 ⊢ 0 ∈ (0[,)+∞)
12	11	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
13	10, 12	ifclda 4521	. . . . 5 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞))
14	13	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞))
15	14	fmpttd 7139	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)):ℝ⟶(0[,)+∞))
16		mblss 25882	. . . . 5 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
17	5, 16	syl 18	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
18		rembl 25891	. . . . 5 ⊢ ℝ ∈ dom vol
19	18	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ dom vol)
20	13	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞))
21		eldifn 4071	. . . . . 6 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
22	21	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴)
23	22	iffalsed 4496	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → if(𝑥 ∈ 𝐴, 𝐵, 0) = 0)
24		iftrue 4491	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, 𝐵, 0) = 𝐵)
25	24	mpteq2ia 5203	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) = (𝑥 ∈ 𝐴 ↦ 𝐵)
26	25, 3	eqeltrid 2868	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) ∈ MblFn)
27	17, 19, 20, 23, 26	mbfss 25997	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) ∈ MblFn)
28	4	rpgt0d 13194	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 < 𝐵)
29	17	sselda 3913	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
30	24	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, 𝐵, 0) = 𝐵)
31	30, 4	eqeltrd 2864	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ ℝ⁺)
32		eqid 2763	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))
33	32	fvmpt2 7028	. . . . . . . . 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 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥) = if(𝑥 ∈ 𝐴, 𝐵, 0))
35	34, 30	eqtrd 2798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥) = 𝐵)
36	28, 35	breqtrrd 5136	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥))
37	36	ralrimiva 3158	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥))
38		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑥0
39		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑥 <
40		nffvmpt1 6919	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦)
41	38, 39, 40	nfbr 5155	. . . . . 6 ⊢ Ⅎ𝑥0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦)
42		nfv 1949	. . . . . 6 ⊢ Ⅎ𝑦0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥)
43		fveq2 6908	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑥))
44	43	breq2d 5118	. . . . . 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 3308	. . . . 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 3258	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 0 < ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))‘𝑦))
48	5, 6, 15, 27, 47	itg2gt0 26111	. 2 ⊢ (𝜑 → 0 < (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))))
49	7, 1, 8	itgposval 26146	. 2 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))))
50	48, 49	breqtrrd 5136	1 ⊢ (𝜑 → 0 < ∫𝐴𝐵 d𝑥)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ⊆ wss 3900 ∖ cdif 4048 ifcif 4485 class class class wbr 5106 ↦ cmpt 5189 dom cdm 5655 ‘cfv 6558 (class class class)co 7443 ℝcr 11226 0cc0 11227 +∞cpnf 11367 < clt 11370 ≤ cle 11371 ℝ⁺crp 13147 [,)cico 13505 volcvol 25814 MblFncmbf 25965 ∫₂citg2 25967 𝐿¹cibl 25968 ∫citg 25969

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-inf2 9669 ax-cc 10540 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305 ax-addf 11306

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-disj 5074 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 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 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-ofr 7718 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-er 8745 df-map 8877 df-pm 8878 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fi 9430 df-sup 9461 df-inf 9462 df-oi 9531 df-dju 10009 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-n0 12634 df-z 12721 df-uz 12993 df-q 13103 df-rp 13148 df-xneg 13268 df-xadd 13269 df-xmul 13270 df-ioo 13507 df-ico 13509 df-icc 13510 df-fz 13667 df-fzo 13816 df-fl 13959 df-mod 14037 df-seq 14172 df-exp 14232 df-hash 14502 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432 df-clim 15684 df-rlim 15685 df-sum 15883 df-rest 17622 df-topgen 17643 df-psmet 21700 df-xmet 21701 df-met 21702 df-bl 21703 df-mopn 21704 df-top 23242 df-topon 23259 df-bases 23294 df-cmp 23735 df-cncf 25229 df-ovol 25815 df-vol 25816 df-mbf 25970 df-itg1 25971 df-itg2 25972 df-ibl 25973 df-itg 25974 df-0p 26021

This theorem is used by: ftc1lem4 26389 fdvposlt 35298

W3C validator