itg11 - Metamath Proof Explorer

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Theorem itg11 25199

Description: The integral of an indicator function is the volume of the set. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

**Hypothesis**
Ref	Expression
i1f1.1	⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 1, 0))

**Assertion**
Ref	Expression
itg11	⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (∫₁‘𝐹) = (vol‘𝐴))

Distinct variable group: 𝑥,𝐴 Allowed substitution hint: 𝐹(𝑥)

Proof of Theorem itg11 Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovol0 25001	. . . . 5 ⊢ (vol*‘∅) = 0
2		0mbl 25047	. . . . . 6 ⊢ ∅ ∈ dom vol
3		mblvol 25038	. . . . . 6 ⊢ (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
4	2, 3	ax-mp 5	. . . . 5 ⊢ (vol‘∅) = (vol*‘∅)
5		itg10 25196	. . . . 5 ⊢ (∫₁‘(ℝ × {0})) = 0
6	1, 4, 5	3eqtr4ri 2771	. . . 4 ⊢ (∫₁‘(ℝ × {0})) = (vol‘∅)
7		noel 4329	. . . . . . . . 9 ⊢ ¬ 𝑥 ∈ ∅
8		eleq2 2822	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
9	7, 8	mtbiri 326	. . . . . . . 8 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
10	9	iffalsed 4538	. . . . . . 7 ⊢ (𝐴 = ∅ → if(𝑥 ∈ 𝐴, 1, 0) = 0)
11	10	mpteq2dv 5249	. . . . . 6 ⊢ (𝐴 = ∅ → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 1, 0)) = (𝑥 ∈ ℝ ↦ 0))
12		i1f1.1	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 1, 0))
13		fconstmpt 5736	. . . . . 6 ⊢ (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0)
14	11, 12, 13	3eqtr4g 2797	. . . . 5 ⊢ (𝐴 = ∅ → 𝐹 = (ℝ × {0}))
15	14	fveq2d 6892	. . . 4 ⊢ (𝐴 = ∅ → (∫₁‘𝐹) = (∫₁‘(ℝ × {0})))
16		fveq2 6888	. . . 4 ⊢ (𝐴 = ∅ → (vol‘𝐴) = (vol‘∅))
17	6, 15, 16	3eqtr4a 2798	. . 3 ⊢ (𝐴 = ∅ → (∫₁‘𝐹) = (vol‘𝐴))
18	17	a1i 11	. 2 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (𝐴 = ∅ → (∫₁‘𝐹) = (vol‘𝐴)))
19		n0 4345	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴)
20	12	i1f1 25198	. . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹 ∈ dom ∫₁)
21	20	adantr 481	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 𝐹 ∈ dom ∫₁)
22		itg1val 25191	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫₁ → (∫₁‘𝐹) = Σ𝑧 ∈ (ran 𝐹 ∖ {0})(𝑧 · (vol‘(^◡𝐹 “ {𝑧}))))
23	21, 22	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (∫₁‘𝐹) = Σ𝑧 ∈ (ran 𝐹 ∖ {0})(𝑧 · (vol‘(^◡𝐹 “ {𝑧}))))
24	12	i1f1lem 25197	. . . . . . . . . . . . . 14 ⊢ (𝐹:ℝ⟶{0, 1} ∧ (𝐴 ∈ dom vol → (^◡𝐹 “ {1}) = 𝐴))
25	24	simpli 484	. . . . . . . . . . . . 13 ⊢ 𝐹:ℝ⟶{0, 1}
26		frn 6721	. . . . . . . . . . . . 13 ⊢ (𝐹:ℝ⟶{0, 1} → ran 𝐹 ⊆ {0, 1})
27	25, 26	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran 𝐹 ⊆ {0, 1}
28		ssdif 4138	. . . . . . . . . . . 12 ⊢ (ran 𝐹 ⊆ {0, 1} → (ran 𝐹 ∖ {0}) ⊆ ({0, 1} ∖ {0}))
29	27, 28	ax-mp 5	. . . . . . . . . . 11 ⊢ (ran 𝐹 ∖ {0}) ⊆ ({0, 1} ∖ {0})
30		difprsnss 4801	. . . . . . . . . . 11 ⊢ ({0, 1} ∖ {0}) ⊆ {1}
31	29, 30	sstri 3990	. . . . . . . . . 10 ⊢ (ran 𝐹 ∖ {0}) ⊆ {1}
32	31	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (ran 𝐹 ∖ {0}) ⊆ {1})
33		mblss 25039	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
34	33	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐴 ⊆ ℝ)
35	34	sselda 3981	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
36		eleq1w 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
37	36	ifbid 4550	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → if(𝑥 ∈ 𝐴, 1, 0) = if(𝑦 ∈ 𝐴, 1, 0))
38		1ex 11206	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ V
39		c0ex 11204	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
40	38, 39	ifex 4577	. . . . . . . . . . . . . . 15 ⊢ if(𝑦 ∈ 𝐴, 1, 0) ∈ V
41	37, 12, 40	fvmpt 6995	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → (𝐹‘𝑦) = if(𝑦 ∈ 𝐴, 1, 0))
42	35, 41	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = if(𝑦 ∈ 𝐴, 1, 0))
43		iftrue 4533	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐴 → if(𝑦 ∈ 𝐴, 1, 0) = 1)
44	43	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → if(𝑦 ∈ 𝐴, 1, 0) = 1)
45	42, 44	eqtrd 2772	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = 1)
46		ffn 6714	. . . . . . . . . . . . . 14 ⊢ (𝐹:ℝ⟶{0, 1} → 𝐹 Fn ℝ)
47	25, 46	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 𝐹 Fn ℝ
48		fnfvelrn 7079	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn ℝ ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ran 𝐹)
49	47, 35, 48	sylancr 587	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ran 𝐹)
50	45, 49	eqeltrrd 2834	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 1 ∈ ran 𝐹)
51		ax-1ne0 11175	. . . . . . . . . . 11 ⊢ 1 ≠ 0
52		eldifsn 4789	. . . . . . . . . . 11 ⊢ (1 ∈ (ran 𝐹 ∖ {0}) ↔ (1 ∈ ran 𝐹 ∧ 1 ≠ 0))
53	50, 51, 52	sylanblrc 590	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 1 ∈ (ran 𝐹 ∖ {0}))
54	53	snssd 4811	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → {1} ⊆ (ran 𝐹 ∖ {0}))
55	32, 54	eqssd 3998	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (ran 𝐹 ∖ {0}) = {1})
56	55	sumeq1d 15643	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → Σ𝑧 ∈ (ran 𝐹 ∖ {0})(𝑧 · (vol‘(^◡𝐹 “ {𝑧}))) = Σ𝑧 ∈ {1} (𝑧 · (vol‘(^◡𝐹 “ {𝑧}))))
57		1re 11210	. . . . . . . . 9 ⊢ 1 ∈ ℝ
58	24	simpri 486	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom vol → (^◡𝐹 “ {1}) = 𝐴)
59	58	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (^◡𝐹 “ {1}) = 𝐴)
60	59	fveq2d 6892	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (vol‘(^◡𝐹 “ {1})) = (vol‘𝐴))
61	60	oveq2d 7421	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (1 · (vol‘(^◡𝐹 “ {1}))) = (1 · (vol‘𝐴)))
62		simplr 767	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (vol‘𝐴) ∈ ℝ)
63	62	recnd 11238	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (vol‘𝐴) ∈ ℂ)
64	63	mullidd 11228	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (1 · (vol‘𝐴)) = (vol‘𝐴))
65	61, 64	eqtrd 2772	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (1 · (vol‘(^◡𝐹 “ {1}))) = (vol‘𝐴))
66	65, 63	eqeltrd 2833	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (1 · (vol‘(^◡𝐹 “ {1}))) ∈ ℂ)
67		id 22	. . . . . . . . . . 11 ⊢ (𝑧 = 1 → 𝑧 = 1)
68		sneq 4637	. . . . . . . . . . . . 13 ⊢ (𝑧 = 1 → {𝑧} = {1})
69	68	imaeq2d 6057	. . . . . . . . . . . 12 ⊢ (𝑧 = 1 → (^◡𝐹 “ {𝑧}) = (^◡𝐹 “ {1}))
70	69	fveq2d 6892	. . . . . . . . . . 11 ⊢ (𝑧 = 1 → (vol‘(^◡𝐹 “ {𝑧})) = (vol‘(^◡𝐹 “ {1})))
71	67, 70	oveq12d 7423	. . . . . . . . . 10 ⊢ (𝑧 = 1 → (𝑧 · (vol‘(^◡𝐹 “ {𝑧}))) = (1 · (vol‘(^◡𝐹 “ {1}))))
72	71	sumsn 15688	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ (1 · (vol‘(^◡𝐹 “ {1}))) ∈ ℂ) → Σ𝑧 ∈ {1} (𝑧 · (vol‘(^◡𝐹 “ {𝑧}))) = (1 · (vol‘(^◡𝐹 “ {1}))))
73	57, 66, 72	sylancr 587	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → Σ𝑧 ∈ {1} (𝑧 · (vol‘(^◡𝐹 “ {𝑧}))) = (1 · (vol‘(^◡𝐹 “ {1}))))
74	73, 65	eqtrd 2772	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → Σ𝑧 ∈ {1} (𝑧 · (vol‘(^◡𝐹 “ {𝑧}))) = (vol‘𝐴))
75	56, 74	eqtrd 2772	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → Σ𝑧 ∈ (ran 𝐹 ∖ {0})(𝑧 · (vol‘(^◡𝐹 “ {𝑧}))) = (vol‘𝐴))
76	23, 75	eqtrd 2772	. . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (∫₁‘𝐹) = (vol‘𝐴))
77	76	ex 413	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (𝑦 ∈ 𝐴 → (∫₁‘𝐹) = (vol‘𝐴)))
78	77	exlimdv 1936	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (∃𝑦 𝑦 ∈ 𝐴 → (∫₁‘𝐹) = (vol‘𝐴)))
79	19, 78	biimtrid 241	. 2 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (𝐴 ≠ ∅ → (∫₁‘𝐹) = (vol‘𝐴)))
80	18, 79	pm2.61dne 3028	1 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (∫₁‘𝐹) = (vol‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4321 ifcif 4527 {csn 4627 {cpr 4629 ↦ cmpt 5230 × cxp 5673 ^◡ccnv 5674 dom cdm 5675 ran crn 5676 “ cima 5678 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 Σcsu 15628 vol*covol 24970 volcvol 24971 ∫₁citg1 25123

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xadd 13089 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-xmet 20929 df-met 20930 df-ovol 24972 df-vol 24973 df-mbf 25127 df-itg1 25128

This theorem is referenced by: itg2const 25249 itg2addnclem 36527

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