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Theorem etransclem5 42401
Description: A change of bound variable, often used in proofs for etransc 42445. (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 7152 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑗) = (𝑦𝑗))
21oveq1d 7160 . . . 4 (𝑥 = 𝑦 → ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
32cbvmptv 5160 . . 3 (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
4 oveq2 7153 . . . . 5 (𝑗 = 𝑘 → (𝑦𝑗) = (𝑦𝑘))
5 eqeq1 2822 . . . . . 6 (𝑗 = 𝑘 → (𝑗 = 0 ↔ 𝑘 = 0))
65ifbid 4485 . . . . 5 (𝑗 = 𝑘 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = if(𝑘 = 0, (𝑃 − 1), 𝑃))
74, 6oveq12d 7163 . . . 4 (𝑗 = 𝑘 → ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))
87mpteq2dv 5153 . . 3 (𝑗 = 𝑘 → (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
93, 8syl5eq 2865 . 2 (𝑗 = 𝑘 → (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
109cbvmptv 5160 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 1528  ifcif 4463  cmpt 5137  (class class class)co 7145  0cc0 10525  1c1 10526  cmin 10858  ...cfz 12880  cexp 13417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-iota 6307  df-fv 6356  df-ov 7148
This theorem is referenced by:  etransclem27  42423  etransclem29  42425  etransclem31  42427  etransclem32  42428  etransclem33  42429  etransclem34  42430  etransclem35  42431  etransclem38  42434  etransclem40  42436  etransclem42  42438  etransclem44  42440  etransclem45  42441  etransclem46  42442
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