Theorem etransclem5 43743

Description: A change of bound variable, often used in proofs for etransc 43787. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Assertion

Ref	Expression
etransclem5	⊢ (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑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

Step	Hyp	Ref	Expression
1		oveq1 7276	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝑗) = (𝑦 − 𝑗))
2	1	oveq1d 7284	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
3	2	cbvmptv 5192	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
4		oveq2 7277	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝑦 − 𝑗) = (𝑦 − 𝑘))
5		eqeq1 2744	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑗 = 0 ↔ 𝑘 = 0))
6	5	ifbid 4488	. . . . 5 ⊢ (𝑗 = 𝑘 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = if(𝑘 = 0, (𝑃 − 1), 𝑃))
7	4, 6	oveq12d 7287	. . . 4 ⊢ (𝑗 = 𝑘 → ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))
8	7	mpteq2dv 5181	. . 3 ⊢ (𝑗 = 𝑘 → (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
9	3, 8	eqtrid 2792	. 2 ⊢ (𝑗 = 𝑘 → (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
10	9	cbvmptv 5192	1 ⊢ (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))

Syntax hints:   = wceq 1542  ifcif 4465   ↦ cmpt 5162  (class class class)co 7269  0cc0 10864  1c1 10865   − cmin 11197  ...cfz 13230  ↑cexp 13772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-iota 6389  df-fv 6439  df-ov 7272

This theorem is referenced by:  etransclem27  43765  etransclem29  43767  etransclem31  43769  etransclem32  43770  etransclem33  43771  etransclem34  43772  etransclem35  43773  etransclem38  43776  etransclem40  43778  etransclem42  43780  etransclem44  43782  etransclem45  43783  etransclem46  43784