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Theorem etransclem5 46195
Description: A change of bound variable, often used in proofs for etransc 46239. (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 7438 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑗) = (𝑦𝑗))
21oveq1d 7446 . . . 4 (𝑥 = 𝑦 → ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
32cbvmptv 5261 . . 3 (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
4 oveq2 7439 . . . . 5 (𝑗 = 𝑘 → (𝑦𝑗) = (𝑦𝑘))
5 eqeq1 2739 . . . . . 6 (𝑗 = 𝑘 → (𝑗 = 0 ↔ 𝑘 = 0))
65ifbid 4554 . . . . 5 (𝑗 = 𝑘 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = if(𝑘 = 0, (𝑃 − 1), 𝑃))
74, 6oveq12d 7449 . . . 4 (𝑗 = 𝑘 → ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))
87mpteq2dv 5250 . . 3 (𝑗 = 𝑘 → (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
93, 8eqtrid 2787 . 2 (𝑗 = 𝑘 → (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
109cbvmptv 5261 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 1537  ifcif 4531  cmpt 5231  (class class class)co 7431  0cc0 11153  1c1 11154  cmin 11490  ...cfz 13544  cexp 14099
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-ext 2706
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-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by:  etransclem27  46217  etransclem29  46219  etransclem31  46221  etransclem32  46222  etransclem33  46223  etransclem34  46224  etransclem35  46225  etransclem38  46228  etransclem40  46230  etransclem42  46232  etransclem44  46234  etransclem45  46235  etransclem46  46236
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