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Theorem mbfi1fseqlem2 25570
Description: Lemma for mbfi1fseq 25575. (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 11450 . . . . . 6 (𝑚 = 𝐴 → -𝑚 = -𝐴)
2 id 22 . . . . . 6 (𝑚 = 𝐴𝑚 = 𝐴)
31, 2oveq12d 7420 . . . . 5 (𝑚 = 𝐴 → (-𝑚[,]𝑚) = (-𝐴[,]𝐴))
43eleq2d 2811 . . . 4 (𝑚 = 𝐴 → (𝑥 ∈ (-𝑚[,]𝑚) ↔ 𝑥 ∈ (-𝐴[,]𝐴)))
5 oveq1 7409 . . . . . 6 (𝑚 = 𝐴 → (𝑚𝐽𝑥) = (𝐴𝐽𝑥))
65, 2breq12d 5152 . . . . 5 (𝑚 = 𝐴 → ((𝑚𝐽𝑥) ≤ 𝑚 ↔ (𝐴𝐽𝑥) ≤ 𝐴))
76, 5, 2ifbieq12d 4549 . . . 4 (𝑚 = 𝐴 → if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚) = if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴))
84, 7ifbieq1d 4545 . . 3 (𝑚 = 𝐴 → if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0) = if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))
98mpteq2dv 5241 . 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 11198 . . 3 ℝ ∈ V
1211mptex 7217 . 2 (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)) ∈ V
139, 10, 12fvmpt 6989 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 1533  wcel 2098  ifcif 4521   class class class wbr 5139  cmpt 5222  wf 6530  cfv 6534  (class class class)co 7402  cmpo 7404  cr 11106  0cc0 11107   · cmul 11112  +∞cpnf 11243  cle 11247  -cneg 11443   / cdiv 11869  cn 12210  2c2 12265  [,)cico 13324  [,]cicc 13325  cfl 13753  cexp 14025  MblFncmbf 25467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-cnex 11163  ax-resscn 11164
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-neg 11445
This theorem is referenced by:  mbfi1fseqlem3  25571  mbfi1fseqlem4  25572  mbfi1fseqlem5  25573  mbfi1fseqlem6  25574
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