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Theorem etransclem5 45253
Description: A change of bound variable, often used in proofs for etransc 45297. (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 7418 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑗) = (𝑦𝑗))
21oveq1d 7426 . . . 4 (𝑥 = 𝑦 → ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
32cbvmptv 5260 . . 3 (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
4 oveq2 7419 . . . . 5 (𝑗 = 𝑘 → (𝑦𝑗) = (𝑦𝑘))
5 eqeq1 2734 . . . . . 6 (𝑗 = 𝑘 → (𝑗 = 0 ↔ 𝑘 = 0))
65ifbid 4550 . . . . 5 (𝑗 = 𝑘 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = if(𝑘 = 0, (𝑃 − 1), 𝑃))
74, 6oveq12d 7429 . . . 4 (𝑗 = 𝑘 → ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))
87mpteq2dv 5249 . . 3 (𝑗 = 𝑘 → (𝑦𝑋 ↦ ((𝑦𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
93, 8eqtrid 2782 . 2 (𝑗 = 𝑘 → (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
109cbvmptv 5260 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 1539  ifcif 4527  cmpt 5230  (class class class)co 7411  0cc0 11112  1c1 11113  cmin 11448  ...cfz 13488  cexp 14031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-iota 6494  df-fv 6550  df-ov 7414
This theorem is referenced by:  etransclem27  45275  etransclem29  45277  etransclem31  45279  etransclem32  45280  etransclem33  45281  etransclem34  45282  etransclem35  45283  etransclem38  45286  etransclem40  45288  etransclem42  45290  etransclem44  45292  etransclem45  45293  etransclem46  45294
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