Theorem isibl2 25131

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 6853	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
7		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑥 /
8		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑥(i↑𝑘)
9	6, 7, 8	nfov 7387	. . . . . . . . 9 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))
10	5, 9	nffv 6852	. . . . . . . 8 ⊢ Ⅎ𝑥(ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘)))
11	3, 4, 10	nfbr 5152	. . . . . . 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 4516	. . . . 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 2820	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
16		fveq2 6842	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
17	16	fvoveq1d 7379	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))))
18	17	breq2d 5117	. . . . . . 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 4510	. . . . 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 5216	. . . 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 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
25	24	fvmpt2 6959	. . . . . . . . 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 7379	. . . . . . 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 2779	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) / (i↑𝑘))) = 𝑇)
30	29	ibllem 25129	. . . . 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 2788	. . 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 2779	. 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 2737	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) / (i↑𝑘))))
35	24, 23	dmmptd 6646	. 2 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
36		eqidd 2737	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦))
37	33, 34, 35, 36	isibl 25130	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 1541   ∈ wcel 2106  ∀wral 3064  ifcif 4486   class class class wbr 5105   ↦ cmpt 5188  ‘cfv 6496  (class class class)co 7357  ℝcr 11050  0cc0 11051  ici 11053   ≤ cle 11190   / cdiv 11812  3c3 12209  ...cfz 13424  ↑cexp 13967  ℜcre 14982  MblFncmbf 24978  ∫₂citg2 24980  𝐿¹cibl 24981

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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-ibl 24986

This theorem is referenced by:  iblitg  25133  iblcnlem1  25152  iblss  25169  iblss2  25170  itgeqa  25178  iblconst  25182  iblabsr  25194  iblmulc2  25195  iblmulc2nc  36143  iblsplit  44197