Theorem isibl2 25674

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 1914	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑥0
4		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
5		nfcv 2892	. . . . . . . . 9 ⊢ Ⅎ𝑥ℜ
6		nffvmpt1 6872	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
7		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑥 /
8		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑥(i↑𝑘)
9	6, 7, 8	nfov 7420	. . . . . . . . 9 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))
10	5, 9	nffv 6871	. . . . . . . 8 ⊢ Ⅎ𝑥(ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))
11	3, 4, 10	nfbr 5157	. . . . . . 7 ⊢ Ⅎ𝑥0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))
12	2, 11	nfan 1899	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))))
13	12, 10, 3	nfif 4522	. . . . 5 ⊢ Ⅎ𝑥if((𝑦 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))), 0)
14		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑦if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))), 0)
15		eleq1w 2812	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
16		fveq2 6861	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
17	16	fvoveq1d 7412	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))))
18	17	breq2d 5122	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))) ↔ 0 ≤ (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘)))))
19	15, 18	anbi12d 632	. . . . . 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 4516	. . . . 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 5212	. . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
23		isibl2.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
24		eqid 2730	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
25	24	fvmpt2 6982	. . . . . . . . 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 7412	. . . . . . 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 2768	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))) = 𝑇)
30	29	ibllem 25672	. . . . 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 5204	. . . 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 2777	. . 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 2768	. 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 2731	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))))
35	24, 23	dmmptd 6666	. 2 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
36		eqidd 2731	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦))
37	33, 34, 35, 36	isibl 25673	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘𝐺) ∈ ℝ)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ifcif 4491   class class class wbr 5110   ↦ cmpt 5191  ‘cfv 6514  (class class class)co 7390  ℝcr 11074  0cc0 11075  ici 11077   ≤ cle 11216   / cdiv 11842  3c3 12249  ...cfz 13475  ↑cexp 14033  ℜcre 15070  MblFncmbf 25522  ∫₂citg2 25524  𝐿¹cibl 25525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-ibl 25530

This theorem is referenced by:  iblitg  25676  iblcnlem1  25696  iblss  25713  iblss2  25714  itgeqa  25722  iblconst  25726  iblabsr  25738  iblmulc2  25739  iblmulc2nc  37686  iblsplit  45971