Theorem isibl2 24931

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 1917	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥0
4		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
5		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑥ℜ
6		nffvmpt1 6785	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
7		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑥 /
8		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑥(i↑𝑘)
9	6, 7, 8	nfov 7305	. . . . . . . . 9 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))
10	5, 9	nffv 6784	. . . . . . . 8 ⊢ Ⅎ𝑥(ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))
11	3, 4, 10	nfbr 5121	. . . . . . 7 ⊢ Ⅎ𝑥0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))
12	2, 11	nfan 1902	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))))
13	12, 10, 3	nfif 4489	. . . . 5 ⊢ Ⅎ𝑥if((𝑦 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))), 0)
14		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑦if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))), 0)
15		eleq1w 2821	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
16		fveq2 6774	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
17	16	fvoveq1d 7297	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))))
18	17	breq2d 5086	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))) ↔ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘)))))
19	15, 18	anbi12d 631	. . . . . 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 4483	. . . . 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 5185	. . . 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 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
23		isibl2.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
24		eqid 2738	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
25	24	fvmpt2 6886	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
26	22, 23, 25	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
27	26	fvoveq1d 7297	. . . . . . 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 2781	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))) = 𝑇)
30	29	ibllem 24929	. . . . 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 5176	. . . 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 2790	. . 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 2781	. 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 2739	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))))
35	24, 23	dmmptd 6578	. 2 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
36		eqidd 2739	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦))
37	33, 34, 35, 36	isibl 24930	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘𝐺) ∈ ℝ)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ifcif 4459   class class class wbr 5074   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275  ℝcr 10870  0cc0 10871  ici 10873   ≤ cle 11010   / cdiv 11632  3c3 12029  ...cfz 13239  ↑cexp 13782  ℜcre 14808  MblFncmbf 24778  ∫₂citg2 24780  𝐿¹cibl 24781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-ibl 24786

This theorem is referenced by:  iblitg  24933  iblcnlem1  24952  iblss  24969  iblss2  24970  itgeqa  24978  iblconst  24982  iblabsr  24994  iblmulc2  24995  iblmulc2nc  35842  iblsplit  43507