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Theorem mbfi1fseqlem2 25766
Description: Lemma for mbfi1fseq 25771. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
mbfi1fseq.1 (𝜑𝐹 ∈ MblFn)
mbfi1fseq.2 (𝜑𝐹:ℝ⟶(0[,)+∞))
mbfi1fseq.3 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))
mbfi1fseq.4 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))
Assertion
Ref Expression
mbfi1fseqlem2 (𝐴 ∈ ℕ → (𝐺𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
Distinct variable groups:   𝑥,𝑚,𝑦,𝐹   𝑥,𝐺   𝑚,𝐽   𝜑,𝑚,𝑥,𝑦   𝐴,𝑚,𝑥,𝑦
Allowed substitution hints:   𝐺(𝑦,𝑚)   𝐽(𝑥,𝑦)

Proof of Theorem mbfi1fseqlem2
StepHypRef Expression
1 negeq 11498 . . . . . 6 (𝑚 = 𝐴 → -𝑚 = -𝐴)
2 id 22 . . . . . 6 (𝑚 = 𝐴𝑚 = 𝐴)
31, 2oveq12d 7449 . . . . 5 (𝑚 = 𝐴 → (-𝑚[,]𝑚) = (-𝐴[,]𝐴))
43eleq2d 2825 . . . 4 (𝑚 = 𝐴 → (𝑥 ∈ (-𝑚[,]𝑚) ↔ 𝑥 ∈ (-𝐴[,]𝐴)))
5 oveq1 7438 . . . . . 6 (𝑚 = 𝐴 → (𝑚𝐽𝑥) = (𝐴𝐽𝑥))
65, 2breq12d 5161 . . . . 5 (𝑚 = 𝐴 → ((𝑚𝐽𝑥) ≤ 𝑚 ↔ (𝐴𝐽𝑥) ≤ 𝐴))
76, 5, 2ifbieq12d 4559 . . . 4 (𝑚 = 𝐴 → if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚) = if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴))
84, 7ifbieq1d 4555 . . 3 (𝑚 = 𝐴 → if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0) = if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))
98mpteq2dv 5250 . 2 (𝑚 = 𝐴 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
10 mbfi1fseq.4 . 2 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))
11 reex 11244 . . 3 ℝ ∈ V
1211mptex 7243 . 2 (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)) ∈ V
139, 10, 12fvmpt 7016 1 (𝐴 ∈ ℕ → (𝐺𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  ifcif 4531   class class class wbr 5148  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  cr 11152  0cc0 11153   · cmul 11158  +∞cpnf 11290  cle 11294  -cneg 11491   / cdiv 11918  cn 12264  2c2 12319  [,)cico 13386  [,]cicc 13387  cfl 13827  cexp 14099  MblFncmbf 25663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-neg 11493
This theorem is referenced by:  mbfi1fseqlem3  25767  mbfi1fseqlem4  25768  mbfi1fseqlem5  25769  mbfi1fseqlem6  25770
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