Theorem isibl2 25976

Description: The predicate "𝐹 is integrable" when 𝐹 is a mapping operation. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Hypotheses

Ref	Expression
isibl.1	⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))
isibl.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))
isibl2.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
isibl2	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘𝐺) ∈ ℝ)))

Distinct variable groups:   𝑥,𝑘,𝐴   𝐵,𝑘   𝜑,𝑘,𝑥   𝑥,𝑉

Allowed substitution hints:   𝐵(𝑥)   𝑇(𝑥, 𝑘)   𝐺(𝑥, 𝑘)   𝑉(𝑘)

Proof of Theorem isibl2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isibl.1	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))
2		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑥0
4		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
5		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑥ℜ
6		nffvmpt1 6896	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
7		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑥 /
8		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑥(i↑𝑘)
9	6, 7, 8	nfov 7449	. . . . . . . . 9 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))
10	5, 9	nffv 6895	. . . . . . . 8 ⊢ Ⅎ𝑥(ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))
11	3, 4, 10	nfbr 5160	. . . . . . 7 ⊢ Ⅎ𝑥0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))
12	2, 11	nfan 1932	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))))
13	12, 10, 3	nfif 4520	. . . . 5 ⊢ Ⅎ𝑥if((𝑦 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))), 0)
14		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑦if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))), 0)
15		eleq1w 2848	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
16		fveq2 6885	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
17	16	fvoveq1d 7441	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))))
18	17	breq2d 5123	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))) ↔ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘)))))
19	15, 18	anbi12d 644	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))) ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))))))
20	19, 17	ifbieq1d 4514	. . . . 5 ⊢ (𝑦 = 𝑥 → if((𝑦 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))), 0))
21	13, 14, 20	cbvmpt 5215	. . . 4 ⊢ (𝑦 ∈ ℝ ↦ if((𝑦 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))), 0))
22		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
23		isibl2.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
24		eqid 2765	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
25	24	fvmpt2 7005	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
26	22, 23, 25	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
27	26	fvoveq1d 7441	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))))
28		isibl.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))
29	27, 28	eqtr4d 2803	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))) = 𝑇)
30	29	ibllem 25974	. . . . 5 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))
31	30	mpteq2dv 5207	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))
32	21, 31	eqtrid 2812	. . 3 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ if((𝑦 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))
33	1, 32	eqtr4d 2803	. 2 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ ℝ ↦ if((𝑦 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))), 0)))
34		eqidd 2766	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))))
35	24, 23	dmmptd 6684	. 2 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
36		eqidd 2766	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦))
37	33, 34, 35, 36	isibl 25975	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘𝐺) ∈ ℝ)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ifcif 4489   class class class wbr 5111   ↦ cmpt 5194  ‘cfv 6540  (class class class)co 7419  ℝcr 11114  0cc0 11115  ici 11117   ≤ cle 11259   / cdiv 11886  3c3 12311  ...cfz 13551  ↑cexp 14115  ℜcre 15172  MblFncmbf 25824  ∫₂citg2 25826  𝐿¹cibl 25827

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-ibl 25832

This theorem is used by:  iblitg  25978  iblcnlem1  25998  iblss  26015  iblss2  26016  itgeqa  26024  iblconst  26028  iblabsr  26040  iblmulc2  26041  iblmulc2nc  38393  iblsplit  46738