Theorem isibl2 23533

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 1843	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑥0
4		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
5		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑥ℜ
6		nffvmpt1 6199	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
7		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑥 /
8		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑥(i↑𝑘)
9	6, 7, 8	nfov 6676	. . . . . . . . 9 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))
10	5, 9	nffv 6198	. . . . . . . 8 ⊢ Ⅎ𝑥(ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))
11	3, 4, 10	nfbr 4699	. . . . . . 7 ⊢ Ⅎ𝑥0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))
12	2, 11	nfan 1828	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))))
13	12, 10, 3	nfif 4115	. . . . 5 ⊢ Ⅎ𝑥if((𝑦 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))), 0)
14		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑦if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))), 0)
15		eleq1 2689	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
16		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
17	16	oveq1d 6665	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)) = (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘)))
18	17	fveq2d 6195	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))))
19	18	breq2d 4665	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))) ↔ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘)))))
20	15, 19	anbi12d 747	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))) ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))))))
21	20, 18	ifbieq1d 4109	. . . . 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))
22	13, 14, 21	cbvmpt 4749	. . . 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))
23		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
24		isibl2.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
25		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
26	25	fvmpt2 6291	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
27	23, 24, 26	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
28	27	oveq1d 6665	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘)) = (𝐵 / (i↑𝑘)))
29	28	fveq2d 6195	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))))
30		isibl.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))
31	29, 30	eqtr4d 2659	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))) = 𝑇)
32	31	ibllem 23531	. . . . 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))
33	32	mpteq2dv 4745	. . . 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)))
34	22, 33	syl5eq 2668	. . 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)))
35	1, 34	eqtr4d 2659	. 2 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ ℝ ↦ if((𝑦 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))), 0)))
36		eqidd 2623	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))))
37	25, 24	dmmptd 6024	. 2 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
38		eqidd 2623	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦))
39	35, 36, 37, 38	isibl 23532	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘𝐺) ∈ ℝ)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ifcif 4086   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  0cc0 9936  ici 9938   ≤ cle 10075   / cdiv 10684  3c3 11071  ...cfz 12326  ↑cexp 12860  ℜcre 13837  MblFncmbf 23383  ∫₂citg2 23385  𝐿¹cibl 23386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-ibl 23391

This theorem is referenced by:  iblitg  23535  iblcnlem1  23554  iblss  23571  iblss2  23572  itgeqa  23580  iblconst  23584  iblabsr  23596  iblmulc2  23597  iblmulc2nc  33475  iblsplit  40182