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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 (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 6663 . . . . . . 7 (𝑛 = 𝑀 → (𝑃𝑛) = (𝑃𝑀))
21fveq1d 6665 . . . . . 6 (𝑛 = 𝑀 → ((𝑃𝑛)‘𝑦) = ((𝑃𝑀)‘𝑦))
3 eqid 2758 . . . . . 6 (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) = (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))
4 fvex 6676 . . . . . 6 ((𝑃𝑀)‘𝑦) ∈ V
52, 3, 4fvmpt 6764 . . . . 5 (𝑀 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑀) = ((𝑃𝑀)‘𝑦))
65ad2antlr 726 . . . 4 (((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑀) = ((𝑃𝑀)‘𝑦))
7 nnuz 12334 . . . . 5 ℕ = (ℤ‘1)
8 simplr 768 . . . . 5 (((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑀 ∈ ℕ)
9 itg2i1fseq.5 . . . . . . 7 (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))
10 fveq2 6663 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑃𝑛)‘𝑥) = ((𝑃𝑛)‘𝑦))
1110mpteq2dv 5132 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)))
12 fveq2 6663 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1311, 12breq12d 5049 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦)))
1413rspccva 3542 . . . . . . 7 ((∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥) ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦))
159, 14sylan 583 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦))
1615adantlr 714 . . . . 5 (((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦)) ⇝ (𝐹𝑦))
17 fveq2 6663 . . . . . . . . . 10 (𝑛 = 𝑘 → (𝑃𝑛) = (𝑃𝑘))
1817fveq1d 6665 . . . . . . . . 9 (𝑛 = 𝑘 → ((𝑃𝑛)‘𝑦) = ((𝑃𝑘)‘𝑦))
19 fvex 6676 . . . . . . . . 9 ((𝑃𝑘)‘𝑦) ∈ V
2018, 3, 19fvmpt 6764 . . . . . . . 8 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑘) = ((𝑃𝑘)‘𝑦))
2120adantl 485 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑘) = ((𝑃𝑘)‘𝑦))
22 itg2i1fseq.3 . . . . . . . . . . 11 (𝜑𝑃:ℕ⟶dom ∫1)
2322ffvelrnda 6848 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝑃𝑘) ∈ dom ∫1)
24 i1ff 24389 . . . . . . . . . 10 ((𝑃𝑘) ∈ dom ∫1 → (𝑃𝑘):ℝ⟶ℝ)
2523, 24syl 17 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝑃𝑘):ℝ⟶ℝ)
2625ffvelrnda 6848 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑘)‘𝑦) ∈ ℝ)
2726an32s 651 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑃𝑘)‘𝑦) ∈ ℝ)
2821, 27eqeltrd 2852 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑘) ∈ ℝ)
2928adantllr 718 . . . . 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 488 . . . . . . . . . . . . 13 ((0𝑝r ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))) → (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))
3231ralimi 3092 . . . . . . . . . . . 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 fvoveq1 7179 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑘 + 1)))
3517, 34breq12d 5049 . . . . . . . . . . . 12 (𝑛 = 𝑘 → ((𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃𝑘) ∘r ≤ (𝑃‘(𝑘 + 1))))
3635rspccva 3542 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)) ∧ 𝑘 ∈ ℕ) → (𝑃𝑘) ∘r ≤ (𝑃‘(𝑘 + 1)))
3733, 36sylan 583 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝑃𝑘) ∘r ≤ (𝑃‘(𝑘 + 1)))
38 ffn 6503 . . . . . . . . . . . 12 ((𝑃𝑘):ℝ⟶ℝ → (𝑃𝑘) Fn ℝ)
3923, 24, 383syl 18 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (𝑃𝑘) 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)
4222, 40, 41syl2an 598 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ 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)
4542, 43, 443syl 18 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (𝑃‘(𝑘 + 1)) Fn ℝ)
46 reex 10679 . . . . . . . . . . . 12 ℝ ∈ V
4746a1i 11 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ℝ ∈ V)
48 inidm 4125 . . . . . . . . . . 11 (ℝ ∩ ℝ) = ℝ
49 eqidd 2759 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑘)‘𝑦) = ((𝑃𝑘)‘𝑦))
50 eqidd 2759 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃‘(𝑘 + 1))‘𝑦) = ((𝑃‘(𝑘 + 1))‘𝑦))
5139, 45, 47, 47, 48, 49, 50ofrfval 7420 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → ((𝑃𝑘) ∘r ≤ (𝑃‘(𝑘 + 1)) ↔ ∀𝑦 ∈ ℝ ((𝑃𝑘)‘𝑦) ≤ ((𝑃‘(𝑘 + 1))‘𝑦)))
5237, 51mpbid 235 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃𝑘)‘𝑦) ≤ ((𝑃‘(𝑘 + 1))‘𝑦))
5352r19.21bi 3137 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑘)‘𝑦) ≤ ((𝑃‘(𝑘 + 1))‘𝑦))
5453an32s 651 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑃𝑘)‘𝑦) ≤ ((𝑃‘(𝑘 + 1))‘𝑦))
55 fveq2 6663 . . . . . . . . . . 11 (𝑛 = (𝑘 + 1) → (𝑃𝑛) = (𝑃‘(𝑘 + 1)))
5655fveq1d 6665 . . . . . . . . . 10 (𝑛 = (𝑘 + 1) → ((𝑃𝑛)‘𝑦) = ((𝑃‘(𝑘 + 1))‘𝑦))
57 fvex 6676 . . . . . . . . . 10 ((𝑃‘(𝑘 + 1))‘𝑦) ∈ V
5856, 3, 57fvmpt 6764 . . . . . . . . 9 ((𝑘 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑘 + 1)) = ((𝑃‘(𝑘 + 1))‘𝑦))
5940, 58syl 17 . . . . . . . 8 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑘 + 1)) = ((𝑃‘(𝑘 + 1))‘𝑦))
6059adantl 485 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑘 + 1)) = ((𝑃‘(𝑘 + 1))‘𝑦))
6154, 21, 603brtr4d 5068 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑘 + 1)))
6261adantllr 718 . . . . 5 ((((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘(𝑘 + 1)))
637, 8, 16, 29, 62climub 15079 . . . 4 (((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑦))‘𝑀) ≤ (𝐹𝑦))
646, 63eqbrtrrd 5060 . . 3 (((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑀)‘𝑦) ≤ (𝐹𝑦))
6564ralrimiva 3113 . 2 ((𝜑𝑀 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑃𝑀)‘𝑦) ≤ (𝐹𝑦))
6622ffvelrnda 6848 . . . 4 ((𝜑𝑀 ∈ ℕ) → (𝑃𝑀) ∈ dom ∫1)
67 i1ff 24389 . . . 4 ((𝑃𝑀) ∈ dom ∫1 → (𝑃𝑀):ℝ⟶ℝ)
68 ffn 6503 . . . 4 ((𝑃𝑀):ℝ⟶ℝ → (𝑃𝑀) Fn ℝ)
6966, 67, 683syl 18 . . 3 ((𝜑𝑀 ∈ ℕ) → (𝑃𝑀) Fn ℝ)
70 itg2i1fseq.2 . . . . 5 (𝜑𝐹:ℝ⟶(0[,)+∞))
7170ffnd 6504 . . . 4 (𝜑𝐹 Fn ℝ)
7271adantr 484 . . 3 ((𝜑𝑀 ∈ ℕ) → 𝐹 Fn ℝ)
7346a1i 11 . . 3 ((𝜑𝑀 ∈ ℕ) → ℝ ∈ V)
74 eqidd 2759 . . 3 (((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑃𝑀)‘𝑦) = ((𝑃𝑀)‘𝑦))
75 eqidd 2759 . . 3 (((𝜑𝑀 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹𝑦) = (𝐹𝑦))
7669, 72, 73, 73, 48, 74, 75ofrfval 7420 . 2 ((𝜑𝑀 ∈ ℕ) → ((𝑃𝑀) ∘r𝐹 ↔ ∀𝑦 ∈ ℝ ((𝑃𝑀)‘𝑦) ≤ (𝐹𝑦)))
7765, 76mpbird 260 1 ((𝜑𝑀 ∈ ℕ) → (𝑃𝑀) ∘r𝐹)
 Colors of variables: wff setvar class 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  0𝑝c0p 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
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