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Theorem etransclem11 46260
Description: A change of bound variable, often used in proofs for etransc 46298. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Assertion
Ref Expression
etransclem11 (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛}) = (𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 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
StepHypRef Expression
1 oveq2 7439 . . . . 5 (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
21oveq1d 7446 . . . 4 (𝑛 = 𝑚 → ((0...𝑛) ↑m (0...𝑀)) = ((0...𝑚) ↑m (0...𝑀)))
32rabeqdv 3452 . . 3 (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})
4 fveq2 6906 . . . . . . . 8 (𝑗 = 𝑘 → (𝑐𝑗) = (𝑐𝑘))
54cbvsumv 15732 . . . . . . 7 Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑐𝑘)
6 fveq1 6905 . . . . . . . 8 (𝑐 = 𝑑 → (𝑐𝑘) = (𝑑𝑘))
76sumeq2sdv 15739 . . . . . . 7 (𝑐 = 𝑑 → Σ𝑘 ∈ (0...𝑀)(𝑐𝑘) = Σ𝑘 ∈ (0...𝑀)(𝑑𝑘))
85, 7eqtrid 2789 . . . . . 6 (𝑐 = 𝑑 → Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑑𝑘))
98eqeq1d 2739 . . . . 5 (𝑐 = 𝑑 → (Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛 ↔ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑛))
109cbvrabv 3447 . . . 4 {𝑐 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛} = {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑛}
11 eqeq2 2749 . . . . 5 (𝑛 = 𝑚 → (Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑛 ↔ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚))
1211rabbidv 3444 . . . 4 (𝑛 = 𝑚 → {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑛} = {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})
1310, 12eqtrid 2789 . . 3 (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛} = {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})
143, 13eqtrd 2777 . 2 (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛} = {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})
1514cbvmptv 5255 1 (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛}) = (𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  {crab 3436  cmpt 5225  cfv 6561  (class class class)co 7431  m cmap 8866  0cc0 11155  0cn0 12526  ...cfz 13547  Σcsu 15722
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seq 14043  df-sum 15723
This theorem is referenced by:  etransclem32  46281  etransclem33  46282  etransclem36  46285  etransclem37  46286  etransclem38  46287  etransclem40  46289  etransclem41  46290  etransclem42  46291  etransclem44  46293  etransclem45  46294
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