Theorem etransclem5 47248

Description: A change of bound variable, often used in proofs for etransc 47292. (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 7427	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝑗) = (𝑦 − 𝑗))
2	1	oveq1d 7435	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
3	2	cbvmptv 5209	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
4		oveq2 7428	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝑦 − 𝑗) = (𝑦 − 𝑘))
5		eqeq1 2765	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑗 = 0 ↔ 𝑘 = 0))
6	5	ifbid 4506	. . . . 5 ⊢ (𝑗 = 𝑘 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = if(𝑘 = 0, (𝑃 − 1), 𝑃))
7	4, 6	oveq12d 7438	. . . 4 ⊢ (𝑗 = 𝑘 → ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))
8	7	mpteq2dv 5199	. . 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 2808	. 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 5209	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))))

This proof depends on syntax axioms:   = wceq 1570  ifcif 4482   ↦ cmpt 5186  (class class class)co 7420  0cc0 11200  1c1 11201   − cmin 11541  ...cfz 13639  ↑cexp 14204

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-iota 6494  df-fv 6546  df-ov 7423

This theorem is used by:  etransclem27  47270  etransclem29  47272  etransclem31  47274  etransclem32  47275  etransclem33  47276  etransclem34  47277  etransclem35  47278  etransclem38  47281  etransclem40  47283  etransclem42  47285  etransclem44  47287  etransclem45  47288  etransclem46  47289