Theorem itg2i1fseqle 24467

Description: Subject to the conditions coming from mbfi1fseq 24434, the sequence of simple functions are all less than the target function 𝐹. (Contributed by Mario Carneiro, 17-Aug-2014.)

Hypotheses

Ref	Expression
itg2i1fseq.1	⊢ (𝜑 → 𝐹 ∈ MblFn)
itg2i1fseq.2	⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
itg2i1fseq.3	⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1)
itg2i1fseq.4	⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))
itg2i1fseq.5	⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))

Assertion

Ref	Expression
itg2i1fseqle	⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (𝑃‘𝑀) ∘r ≤ 𝐹)

Distinct variable groups:   𝑥,𝑛,𝐹   𝑛,𝑀   𝑃,𝑛,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑛)   𝑀(𝑥)

Proof of Theorem itg2i1fseqle

Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6663	. . . . . . 7 ⊢ (𝑛 = 𝑀 → (𝑃‘𝑛) = (𝑃‘𝑀))
2	1	fveq1d 6665	. . . . . 6 ⊢ (𝑛 = 𝑀 → ((𝑃‘𝑛)‘𝑦) = ((𝑃‘𝑀)‘𝑦))
3		eqid 2758	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) = (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))
4		fvex 6676	. . . . . 6 ⊢ ((𝑃‘𝑀)‘𝑦) ∈ V
5	2, 3, 4	fvmpt 6764	. . . . 5 ⊢ (𝑀 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑀) = ((𝑃‘𝑀)‘𝑦))
6	5	ad2antlr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑀) = ((𝑃‘𝑀)‘𝑦))
7		nnuz 12334	. . . . 5 ⊢ ℕ = (ℤ≥‘1)
8		simplr 768	. . . . 5 ⊢ (((𝜑 ∧ 𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑀 ∈ ℕ)
9		itg2i1fseq.5	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))
10		fveq2 6663	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑃‘𝑛)‘𝑥) = ((𝑃‘𝑛)‘𝑦))
11	10	mpteq2dv 5132	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)))
12		fveq2 6663	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
13	11, 12	breq12d 5049	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦)))
14	13	rspccva 3542	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥) ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦))
15	9, 14	sylan 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦))
16	15	adantlr 714	. . . . 5 ⊢ (((𝜑 ∧ 𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦)) ⇝ (𝐹‘𝑦))
17		fveq2 6663	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (𝑃‘𝑛) = (𝑃‘𝑘))
18	17	fveq1d 6665	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → ((𝑃‘𝑛)‘𝑦) = ((𝑃‘𝑘)‘𝑦))
19		fvex 6676	. . . . . . . . 9 ⊢ ((𝑃‘𝑘)‘𝑦) ∈ V
20	18, 3, 19	fvmpt 6764	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑘) = ((𝑃‘𝑘)‘𝑦))
21	20	adantl 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑘) = ((𝑃‘𝑘)‘𝑦))
22		itg2i1fseq.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1)
23	22	ffvelrnda 6848	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∈ dom ∫1)
24		i1ff 24389	. . . . . . . . . 10 ⊢ ((𝑃‘𝑘) ∈ dom ∫1 → (𝑃‘𝑘):ℝ⟶ℝ)
25	23, 24	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘):ℝ⟶ℝ)
26	25	ffvelrnda 6848	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑘)‘𝑦) ∈ ℝ)
27	26	an32s 651	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑃‘𝑘)‘𝑦) ∈ ℝ)
28	21, 27	eqeltrd 2852	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑘) ∈ ℝ)
29	28	adantllr 718	. . . . 5 ⊢ ((((𝜑 ∧ 𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑘) ∈ ℝ)
30		itg2i1fseq.4	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))
31		simpr 488	. . . . . . . . . . . . 13 ⊢ ((0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) → (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))
32	31	ralimi 3092	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) → ∀𝑛 ∈ ℕ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))
33	30, 32	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))
34		fvoveq1 7179	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑘 + 1)))
35	17, 34	breq12d 5049	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃‘𝑘) ∘r ≤ (𝑃‘(𝑘 + 1))))
36	35	rspccva 3542	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)) ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∘r ≤ (𝑃‘(𝑘 + 1)))
37	33, 36	sylan 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∘r ≤ (𝑃‘(𝑘 + 1)))
38		ffn 6503	. . . . . . . . . . . 12 ⊢ ((𝑃‘𝑘):ℝ⟶ℝ → (𝑃‘𝑘) Fn ℝ)
39	23, 24, 38	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) Fn ℝ)
40		peano2nn 11699	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
41		ffvelrn 6846	. . . . . . . . . . . . 13 ⊢ ((𝑃:ℕ⟶dom ∫1 ∧ (𝑘 + 1) ∈ ℕ) → (𝑃‘(𝑘 + 1)) ∈ dom ∫1)
42	22, 40, 41	syl2an 598	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘(𝑘 + 1)) ∈ dom ∫1)
43		i1ff 24389	. . . . . . . . . . . 12 ⊢ ((𝑃‘(𝑘 + 1)) ∈ dom ∫1 → (𝑃‘(𝑘 + 1)):ℝ⟶ℝ)
44		ffn 6503	. . . . . . . . . . . 12 ⊢ ((𝑃‘(𝑘 + 1)):ℝ⟶ℝ → (𝑃‘(𝑘 + 1)) Fn ℝ)
45	42, 43, 44	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘(𝑘 + 1)) Fn ℝ)
46		reex 10679	. . . . . . . . . . . 12 ⊢ ℝ ∈ V
47	46	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ℝ ∈ V)
48		inidm 4125	. . . . . . . . . . 11 ⊢ (ℝ ∩ ℝ) = ℝ
49		eqidd 2759	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑘)‘𝑦) = ((𝑃‘𝑘)‘𝑦))
50		eqidd 2759	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘(𝑘 + 1))‘𝑦) = ((𝑃‘(𝑘 + 1))‘𝑦))
51	39, 45, 47, 47, 48, 49, 50	ofrfval 7420	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑃‘𝑘) ∘r ≤ (𝑃‘(𝑘 + 1)) ↔ ∀𝑦 ∈ ℝ ((𝑃‘𝑘)‘𝑦) ≤ ((𝑃‘(𝑘 + 1))‘𝑦)))
52	37, 51	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃‘𝑘)‘𝑦) ≤ ((𝑃‘(𝑘 + 1))‘𝑦))
53	52	r19.21bi 3137	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑘)‘𝑦) ≤ ((𝑃‘(𝑘 + 1))‘𝑦))
54	53	an32s 651	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑃‘𝑘)‘𝑦) ≤ ((𝑃‘(𝑘 + 1))‘𝑦))
55		fveq2 6663	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑘 + 1) → (𝑃‘𝑛) = (𝑃‘(𝑘 + 1)))
56	55	fveq1d 6665	. . . . . . . . . 10 ⊢ (𝑛 = (𝑘 + 1) → ((𝑃‘𝑛)‘𝑦) = ((𝑃‘(𝑘 + 1))‘𝑦))
57		fvex 6676	. . . . . . . . . 10 ⊢ ((𝑃‘(𝑘 + 1))‘𝑦) ∈ V
58	56, 3, 57	fvmpt 6764	. . . . . . . . 9 ⊢ ((𝑘 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑘 + 1)) = ((𝑃‘(𝑘 + 1))‘𝑦))
59	40, 58	syl 17	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑘 + 1)) = ((𝑃‘(𝑘 + 1))‘𝑦))
60	59	adantl 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑘 + 1)) = ((𝑃‘(𝑘 + 1))‘𝑦))
61	54, 21, 60	3brtr4d 5068	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑘 + 1)))
62	61	adantllr 718	. . . . 5 ⊢ ((((𝜑 ∧ 𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘(𝑘 + 1)))
63	7, 8, 16, 29, 62	climub 15079	. . . 4 ⊢ (((𝜑 ∧ 𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑦))‘𝑀) ≤ (𝐹‘𝑦))
64	6, 63	eqbrtrrd 5060	. . 3 ⊢ (((𝜑 ∧ 𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑀)‘𝑦) ≤ (𝐹‘𝑦))
65	64	ralrimiva 3113	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃‘𝑀)‘𝑦) ≤ (𝐹‘𝑦))
66	22	ffvelrnda 6848	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (𝑃‘𝑀) ∈ dom ∫1)
67		i1ff 24389	. . . 4 ⊢ ((𝑃‘𝑀) ∈ dom ∫1 → (𝑃‘𝑀):ℝ⟶ℝ)
68		ffn 6503	. . . 4 ⊢ ((𝑃‘𝑀):ℝ⟶ℝ → (𝑃‘𝑀) Fn ℝ)
69	66, 67, 68	3syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (𝑃‘𝑀) Fn ℝ)
70		itg2i1fseq.2	. . . . 5 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
71	70	ffnd 6504	. . . 4 ⊢ (𝜑 → 𝐹 Fn ℝ)
72	71	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → 𝐹 Fn ℝ)
73	46	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → ℝ ∈ V)
74		eqidd 2759	. . 3 ⊢ (((𝜑 ∧ 𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘𝑀)‘𝑦) = ((𝑃‘𝑀)‘𝑦))
75		eqidd 2759	. . 3 ⊢ (((𝜑 ∧ 𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) = (𝐹‘𝑦))
76	69, 72, 73, 73, 48, 74, 75	ofrfval 7420	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → ((𝑃‘𝑀) ∘r ≤ 𝐹 ↔ ∀𝑦 ∈ ℝ ((𝑃‘𝑀)‘𝑦) ≤ (𝐹‘𝑦)))
77	65, 76	mpbird 260	1 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (𝑃‘𝑀) ∘r ≤ 𝐹)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409   class class class wbr 5036   ↦ cmpt 5116  dom cdm 5528   Fn wfn 6335  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156   ∘_r cofr 7410  ℝcr 10587  0cc0 10588  1c1 10589   + caddc 10591  +∞cpnf 10723   ≤ cle 10727  ℕcn 11687  [,)cico 12794   ⇝ cli 14902  MblFncmbf 24327  ∫₁citg1 24328  0_𝑝c0p 24382

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665  ax-pre-sup 10666

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-ofr 7412  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-er 8305  df-pm 8425  df-en 8541  df-dom 8542  df-sdom 8543  df-sup 8952  df-inf 8953  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-div 11349  df-nn 11688  df-2 11750  df-3 11751  df-n0 11948  df-z 12034  df-uz 12296  df-rp 12444  df-fz 12953  df-fl 13224  df-seq 13432  df-exp 13493  df-cj 14519  df-re 14520  df-im 14521  df-sqrt 14655  df-abs 14656  df-clim 14906  df-rlim 14907  df-sum 15104  df-itg1 24333

This theorem is referenced by:  itg2i1fseq  24468  itg2i1fseq3  24470  itg2addlem  24471