Theorem etransclem5 46230

Description: A change of bound variable, often used in proofs for etransc 46274. (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 7396	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝑗) = (𝑦 − 𝑗))
2	1	oveq1d 7404	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
3	2	cbvmptv 5213	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
4		oveq2 7397	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝑦 − 𝑗) = (𝑦 − 𝑘))
5		eqeq1 2734	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑗 = 0 ↔ 𝑘 = 0))
6	5	ifbid 4514	. . . . 5 ⊢ (𝑗 = 𝑘 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = if(𝑘 = 0, (𝑃 − 1), 𝑃))
7	4, 6	oveq12d 7407	. . . 4 ⊢ (𝑗 = 𝑘 → ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))
8	7	mpteq2dv 5203	. . 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 2777	. 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 5213	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 1540  ifcif 4490   ↦ cmpt 5190  (class class class)co 7389  0cc0 11074  1c1 11075   − cmin 11411  ...cfz 13474  ↑cexp 14032

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-iota 6466  df-fv 6521  df-ov 7392

This theorem is referenced by:  etransclem27  46252  etransclem29  46254  etransclem31  46256  etransclem32  46257  etransclem33  46258  etransclem34  46259  etransclem35  46260  etransclem38  46263  etransclem40  46265  etransclem42  46267  etransclem44  46269  etransclem45  46270  etransclem46  46271