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Theorem itg2i1fseqle 23271
Description: Subject to the conditions coming from mbfi1fseq 23238, 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 (𝜑𝐹 ∈ MblFn)
itg2i1fseq.2 (𝜑𝐹:ℝ⟶(0[,)+∞))
itg2i1fseq.3 (𝜑𝑃:ℕ⟶dom ∫1)
itg2i1fseq.4 (𝜑 → ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1))))
itg2i1fseq.5 (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))
Assertion
Ref Expression
itg2i1fseqle ((𝜑𝑀 ∈ ℕ) → (𝑃𝑀) ∘𝑟𝐹)
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.
StepHypRef Expression
1 fveq2 6087 . . . . . . 7 (𝑛 = 𝑀 → (𝑃𝑛) = (𝑃𝑀))
21fveq1d 6089 . . . . . 6 (𝑛 = 𝑀 → ((𝑃𝑛)‘𝑦) = ((𝑃𝑀)‘𝑦))
3 eqid 2609 . . . . . 6 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))
4 fvex 6097 . . . . . 6 ((𝑃𝑀)‘𝑦) ∈ V
52, 3, 4fvmpt 6175 . . . . 5 (𝑀 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑀) = ((𝑃𝑀)‘𝑦))
65ad2antlr 758 . . . 4 (((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑀) = ((𝑃𝑀)‘𝑦))
7 nnuz 11557 . . . . 5 ℕ = (ℤ‘1)
8 simplr 787 . . . . 5 (((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑀 ∈ ℕ)
9 itg2i1fseq.5 . . . . . . 7 (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))
10 fveq2 6087 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑃𝑛)‘𝑥) = ((𝑃𝑛)‘𝑦))
1110mpteq2dv 4667 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)))
12 fveq2 6087 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1311, 12breq12d 4590 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦)))
1413rspccva 3280 . . . . . . 7 ((∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥) ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦))
159, 14sylan 486 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦))
1615adantlr 746 . . . . 5 (((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦))
17 fveq2 6087 . . . . . . . . . 10 (𝑛 = 𝑘 → (𝑃𝑛) = (𝑃𝑘))
1817fveq1d 6089 . . . . . . . . 9 (𝑛 = 𝑘 → ((𝑃𝑛)‘𝑦) = ((𝑃𝑘)‘𝑦))
19 fvex 6097 . . . . . . . . 9 ((𝑃𝑘)‘𝑦) ∈ V
2018, 3, 19fvmpt 6175 . . . . . . . 8 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑘) = ((𝑃𝑘)‘𝑦))
2120adantl 480 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑘) = ((𝑃𝑘)‘𝑦))
22 itg2i1fseq.3 . . . . . . . . . . 11 (𝜑𝑃:ℕ⟶dom ∫1)
2322ffvelrnda 6251 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝑃𝑘) ∈ dom ∫1)
24 i1ff 23193 . . . . . . . . . 10 ((𝑃𝑘) ∈ dom ∫1 → (𝑃𝑘):ℝ⟶ℝ)
2523, 24syl 17 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝑃𝑘):ℝ⟶ℝ)
2625ffvelrnda 6251 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑘)‘𝑦) ∈ ℝ)
2726an32s 841 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑃𝑘)‘𝑦) ∈ ℝ)
2821, 27eqeltrd 2687 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑘) ∈ ℝ)
2928adantllr 750 . . . . 5 ((((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑘) ∈ ℝ)
30 itg2i1fseq.4 . . . . . . . . . . . 12 (𝜑 → ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1))))
31 simpr 475 . . . . . . . . . . . . 13 ((0𝑝𝑟 ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1))) → (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1)))
3231ralimi 2935 . . . . . . . . . . . 12 (∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1))) → ∀𝑛 ∈ ℕ (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1)))
3330, 32syl 17 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1)))
34 oveq1 6533 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
3534fveq2d 6091 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑘 + 1)))
3617, 35breq12d 4590 . . . . . . . . . . . 12 (𝑛 = 𝑘 → ((𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃𝑘) ∘𝑟 ≤ (𝑃‘(𝑘 + 1))))
3736rspccva 3280 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1)) ∧ 𝑘 ∈ ℕ) → (𝑃𝑘) ∘𝑟 ≤ (𝑃‘(𝑘 + 1)))
3833, 37sylan 486 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝑃𝑘) ∘𝑟 ≤ (𝑃‘(𝑘 + 1)))
39 ffn 5943 . . . . . . . . . . . 12 ((𝑃𝑘):ℝ⟶ℝ → (𝑃𝑘) Fn ℝ)
4023, 24, 393syl 18 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (𝑃𝑘) Fn ℝ)
41 peano2nn 10881 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
42 ffvelrn 6249 . . . . . . . . . . . . 13 ((𝑃:ℕ⟶dom ∫1 ∧ (𝑘 + 1) ∈ ℕ) → (𝑃‘(𝑘 + 1)) ∈ dom ∫1)
4322, 41, 42syl2an 492 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝑃‘(𝑘 + 1)) ∈ dom ∫1)
44 i1ff 23193 . . . . . . . . . . . 12 ((𝑃‘(𝑘 + 1)) ∈ dom ∫1 → (𝑃‘(𝑘 + 1)):ℝ⟶ℝ)
45 ffn 5943 . . . . . . . . . . . 12 ((𝑃‘(𝑘 + 1)):ℝ⟶ℝ → (𝑃‘(𝑘 + 1)) Fn ℝ)
4643, 44, 453syl 18 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (𝑃‘(𝑘 + 1)) Fn ℝ)
47 reex 9883 . . . . . . . . . . . 12 ℝ ∈ V
4847a1i 11 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ℝ ∈ V)
49 inidm 3783 . . . . . . . . . . 11 (ℝ ∩ ℝ) = ℝ
50 eqidd 2610 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑘)‘𝑦) = ((𝑃𝑘)‘𝑦))
51 eqidd 2610 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘(𝑘 + 1))‘𝑦) = ((𝑃‘(𝑘 + 1))‘𝑦))
5240, 46, 48, 48, 49, 50, 51ofrfval 6780 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → ((𝑃𝑘) ∘𝑟 ≤ (𝑃‘(𝑘 + 1)) ↔ ∀𝑦 ∈ ℝ ((𝑃𝑘)‘𝑦) ≤ ((𝑃‘(𝑘 + 1))‘𝑦)))
5338, 52mpbid 220 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃𝑘)‘𝑦) ≤ ((𝑃‘(𝑘 + 1))‘𝑦))
5453r19.21bi 2915 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑘)‘𝑦) ≤ ((𝑃‘(𝑘 + 1))‘𝑦))
5554an32s 841 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑃𝑘)‘𝑦) ≤ ((𝑃‘(𝑘 + 1))‘𝑦))
56 fveq2 6087 . . . . . . . . . . 11 (𝑛 = (𝑘 + 1) → (𝑃𝑛) = (𝑃‘(𝑘 + 1)))
5756fveq1d 6089 . . . . . . . . . 10 (𝑛 = (𝑘 + 1) → ((𝑃𝑛)‘𝑦) = ((𝑃‘(𝑘 + 1))‘𝑦))
58 fvex 6097 . . . . . . . . . 10 ((𝑃‘(𝑘 + 1))‘𝑦) ∈ V
5957, 3, 58fvmpt 6175 . . . . . . . . 9 ((𝑘 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑘 + 1)) = ((𝑃‘(𝑘 + 1))‘𝑦))
6041, 59syl 17 . . . . . . . 8 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑘 + 1)) = ((𝑃‘(𝑘 + 1))‘𝑦))
6160adantl 480 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑘 + 1)) = ((𝑃‘(𝑘 + 1))‘𝑦))
6255, 21, 613brtr4d 4609 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑘 + 1)))
6362adantllr 750 . . . . 5 ((((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑘 + 1)))
647, 8, 16, 29, 63climub 14188 . . . 4 (((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑀) ≤ (𝐹𝑦))
656, 64eqbrtrrd 4601 . . 3 (((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑀)‘𝑦) ≤ (𝐹𝑦))
6665ralrimiva 2948 . 2 ((𝜑𝑀 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃𝑀)‘𝑦) ≤ (𝐹𝑦))
6722ffvelrnda 6251 . . . 4 ((𝜑𝑀 ∈ ℕ) → (𝑃𝑀) ∈ dom ∫1)
68 i1ff 23193 . . . 4 ((𝑃𝑀) ∈ dom ∫1 → (𝑃𝑀):ℝ⟶ℝ)
69 ffn 5943 . . . 4 ((𝑃𝑀):ℝ⟶ℝ → (𝑃𝑀) Fn ℝ)
7067, 68, 693syl 18 . . 3 ((𝜑𝑀 ∈ ℕ) → (𝑃𝑀) Fn ℝ)
71 itg2i1fseq.2 . . . . . 6 (𝜑𝐹:ℝ⟶(0[,)+∞))
72 icossicc 12089 . . . . . 6 (0[,)+∞) ⊆ (0[,]+∞)
73 fss 5954 . . . . . 6 ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℝ⟶(0[,]+∞))
7471, 72, 73sylancl 692 . . . . 5 (𝜑𝐹:ℝ⟶(0[,]+∞))
75 ffn 5943 . . . . 5 (𝐹:ℝ⟶(0[,]+∞) → 𝐹 Fn ℝ)
7674, 75syl 17 . . . 4 (𝜑𝐹 Fn ℝ)
7776adantr 479 . . 3 ((𝜑𝑀 ∈ ℕ) → 𝐹 Fn ℝ)
7847a1i 11 . . 3 ((𝜑𝑀 ∈ ℕ) → ℝ ∈ V)
79 eqidd 2610 . . 3 (((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑀)‘𝑦) = ((𝑃𝑀)‘𝑦))
80 eqidd 2610 . . 3 (((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹𝑦) = (𝐹𝑦))
8170, 77, 78, 78, 49, 79, 80ofrfval 6780 . 2 ((𝜑𝑀 ∈ ℕ) → ((𝑃𝑀) ∘𝑟𝐹 ↔ ∀𝑦 ∈ ℝ ((𝑃𝑀)‘𝑦) ≤ (𝐹𝑦)))
8266, 81mpbird 245 1 ((𝜑𝑀 ∈ ℕ) → (𝑃𝑀) ∘𝑟𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  Vcvv 3172  wss 3539   class class class wbr 4577  cmpt 4637  dom cdm 5027   Fn wfn 5784  wf 5785  cfv 5789  (class class class)co 6526  𝑟 cofr 6771  cr 9791  0cc0 9792  1c1 9793   + caddc 9795  +∞cpnf 9927  cle 9931  cn 10869  [,)cico 12006  [,]cicc 12007  cli 14011  MblFncmbf 23133  1citg1 23134  0𝑝c0p 23186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-ofr 6773  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-sup 8208  df-inf 8209  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10536  df-nn 10870  df-2 10928  df-3 10929  df-n0 11142  df-z 11213  df-uz 11522  df-rp 11667  df-ico 12010  df-icc 12011  df-fz 12155  df-fl 12412  df-seq 12621  df-exp 12680  df-cj 13635  df-re 13636  df-im 13637  df-sqrt 13771  df-abs 13772  df-clim 14015  df-rlim 14016  df-sum 14213  df-itg1 23139
This theorem is referenced by:  itg2i1fseq  23272  itg2i1fseq3  23274  itg2addlem  23275
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