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Theorem etransclem5 44116
Description: A change of bound variable, often used in proofs for etransc 44160. (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 7344 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑗) = (𝑦𝑗))
21oveq1d 7352 . . . 4 (𝑥 = 𝑦 → ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
32cbvmptv 5205 . . 3 (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
4 oveq2 7345 . . . . 5 (𝑗 = 𝑘 → (𝑦𝑗) = (𝑦𝑘))
5 eqeq1 2740 . . . . . 6 (𝑗 = 𝑘 → (𝑗 = 0 ↔ 𝑘 = 0))
65ifbid 4496 . . . . 5 (𝑗 = 𝑘 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = if(𝑘 = 0, (𝑃 − 1), 𝑃))
74, 6oveq12d 7355 . . . 4 (𝑗 = 𝑘 → ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))
87mpteq2dv 5194 . . 3 (𝑗 = 𝑘 → (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
93, 8eqtrid 2788 . 2 (𝑗 = 𝑘 → (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
109cbvmptv 5205 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 1540  ifcif 4473  cmpt 5175  (class class class)co 7337  0cc0 10972  1c1 10973  cmin 11306  ...cfz 13340  cexp 13883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-iota 6431  df-fv 6487  df-ov 7340
This theorem is referenced by:  etransclem27  44138  etransclem29  44140  etransclem31  44142  etransclem32  44143  etransclem33  44144  etransclem34  44145  etransclem35  44146  etransclem38  44149  etransclem40  44151  etransclem42  44153  etransclem44  44155  etransclem45  44156  etransclem46  44157
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