Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  etransclem5 Structured version   Visualization version   GIF version

Theorem etransclem5 44955
Description: A change of bound variable, often used in proofs for etransc 44999. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Assertion
Ref Expression
etransclem5 (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
Distinct variable groups:   𝑗,𝑀,𝑘   𝑃,𝑗,𝑘,𝑥,𝑦   𝑗,𝑋,𝑘,𝑥,𝑦
Allowed substitution hints:   𝑀(𝑥,𝑦)

Proof of Theorem etransclem5
StepHypRef Expression
1 oveq1 7416 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑗) = (𝑦𝑗))
21oveq1d 7424 . . . 4 (𝑥 = 𝑦 → ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
32cbvmptv 5262 . . 3 (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
4 oveq2 7417 . . . . 5 (𝑗 = 𝑘 → (𝑦𝑗) = (𝑦𝑘))
5 eqeq1 2737 . . . . . 6 (𝑗 = 𝑘 → (𝑗 = 0 ↔ 𝑘 = 0))
65ifbid 4552 . . . . 5 (𝑗 = 𝑘 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = if(𝑘 = 0, (𝑃 − 1), 𝑃))
74, 6oveq12d 7427 . . . 4 (𝑗 = 𝑘 → ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))
87mpteq2dv 5251 . . 3 (𝑗 = 𝑘 → (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
93, 8eqtrid 2785 . 2 (𝑗 = 𝑘 → (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
109cbvmptv 5262 1 (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  ifcif 4529  cmpt 5232  (class class class)co 7409  0cc0 11110  1c1 11111  cmin 11444  ...cfz 13484  cexp 14027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-iota 6496  df-fv 6552  df-ov 7412
This theorem is referenced by:  etransclem27  44977  etransclem29  44979  etransclem31  44981  etransclem32  44982  etransclem33  44983  etransclem34  44984  etransclem35  44985  etransclem38  44988  etransclem40  44990  etransclem42  44992  etransclem44  44994  etransclem45  44995  etransclem46  44996
  Copyright terms: Public domain W3C validator