Theorem etransclem11 46819

Description: A change of bound variable, often used in proofs for etransc 46857. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Assertion

Ref	Expression
etransclem11	⊢ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛}) = (𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})

Distinct variable groups:   𝑀,𝑐,𝑑,𝑗,𝑘   𝑚,𝑀,𝑐,𝑑,𝑗   𝑛,𝑀,𝑐,𝑑,𝑘   𝑚,𝑛

Proof of Theorem etransclem11

Step	Hyp	Ref	Expression
1		oveq2 7404	. . . . 5 ⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
2	1	oveq1d 7411	. . . 4 ⊢ (𝑛 = 𝑚 → ((0...𝑛) ↑m (0...𝑀)) = ((0...𝑚) ↑m (0...𝑀)))
3	2	rabeqdv 3429	. . 3 ⊢ (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})
4		fveq2 6867	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑐‘𝑗) = (𝑐‘𝑘))
5	4	cbvsumv 15723	. . . . . . 7 ⊢ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘)
6		fveq1 6866	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑘) = (𝑑‘𝑘))
7	6	sumeq2sdv 15730	. . . . . . 7 ⊢ (𝑐 = 𝑑 → Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘))
8	5, 7	eqtrid 2809	. . . . . 6 ⊢ (𝑐 = 𝑑 → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘))
9	8	eqeq1d 2764	. . . . 5 ⊢ (𝑐 = 𝑑 → (Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛 ↔ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑛))
10	9	cbvrabv 3424	. . . 4 ⊢ {𝑐 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛} = {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑛}
11		eqeq2 2774	. . . . 5 ⊢ (𝑛 = 𝑚 → (Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑛 ↔ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚))
12	11	rabbidv 3421	. . . 4 ⊢ (𝑛 = 𝑚 → {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑛} = {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})
13	10, 12	eqtrid 2809	. . 3 ⊢ (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛} = {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})
14	3, 13	eqtrd 2797	. 2 ⊢ (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛} = {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})
15	14	cbvmptv 5204	1 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛}) = (𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})

Syntax hints:   = wceq 1560  {crab 3414   ↦ cmpt 5181  ‘cfv 6521  (class class class)co 7396   ↑_m cmap 8808  0cc0 11073  ℕ₀cn0 12481  ...cfz 13512  Σcsu 15713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-xp 5653  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-iota 6477  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-seq 14015  df-sum 15714

This theorem is referenced by:  etransclem32  46840  etransclem33  46841  etransclem36  46844  etransclem37  46845  etransclem38  46846  etransclem40  46848  etransclem41  46849  etransclem42  46850  etransclem44  46852  etransclem45  46853