Theorem etransclem5 46483

Description: A change of bound variable, often used in proofs for etransc 46527. (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 7365	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝑗) = (𝑦 − 𝑗))
2	1	oveq1d 7373	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
3	2	cbvmptv 5202	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))
4		oveq2 7366	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝑦 − 𝑗) = (𝑦 − 𝑘))
5		eqeq1 2740	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑗 = 0 ↔ 𝑘 = 0))
6	5	ifbid 4503	. . . . 5 ⊢ (𝑗 = 𝑘 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = if(𝑘 = 0, (𝑃 − 1), 𝑃))
7	4, 6	oveq12d 7376	. . . 4 ⊢ (𝑗 = 𝑘 → ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))
8	7	mpteq2dv 5192	. . 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 2783	. 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 5202	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 1541  ifcif 4479   ↦ cmpt 5179  (class class class)co 7358  0cc0 11026  1c1 11027   − cmin 11364  ...cfz 13423  ↑cexp 13984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-iota 6448  df-fv 6500  df-ov 7361

This theorem is referenced by:  etransclem27  46505  etransclem29  46507  etransclem31  46509  etransclem32  46510  etransclem33  46511  etransclem34  46512  etransclem35  46513  etransclem38  46516  etransclem40  46518  etransclem42  46520  etransclem44  46522  etransclem45  46523  etransclem46  46524