Theorem etransclem11 47254

Description: A change of bound variable, often used in proofs for etransc 47292. (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 7428	. . . . 5 ⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
2	1	oveq1d 7435	. . . 4 ⊢ (𝑛 = 𝑚 → ((0...𝑛) ↑m (0...𝑀)) = ((0...𝑚) ↑m (0...𝑀)))
3	2	rabeqdv 3428	. . 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 6885	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑐‘𝑗) = (𝑐‘𝑘))
5	4	cbvsumv 15863	. . . . . . 7 ⊢ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘)
6		fveq1 6884	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑘) = (𝑑‘𝑘))
7	6	sumeq2sdv 15870	. . . . . . 7 ⊢ (𝑐 = 𝑑 → Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘))
8	5, 7	eqtrid 2808	. . . . . 6 ⊢ (𝑐 = 𝑑 → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘))
9	8	eqeq1d 2763	. . . . 5 ⊢ (𝑐 = 𝑑 → (Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛 ↔ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑛))
10	9	cbvrabv 3423	. . . 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 2773	. . . . 5 ⊢ (𝑛 = 𝑚 → (Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑛 ↔ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚))
12	11	rabbidv 3420	. . . 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 2808	. . 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 2796	. 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 5209	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})

This proof depends on syntax axioms:   = wceq 1570  {crab 3413   ↦ cmpt 5186  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8847  0cc0 11200  ℕ₀cn0 12606  ...cfz 13639  Σcsu 15853

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-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  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-xp 5657  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-iota 6494  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-seq 14145  df-sum 15854

This theorem is used by:  etransclem32  47275  etransclem33  47276  etransclem36  47279  etransclem37  47280  etransclem38  47281  etransclem40  47283  etransclem41  47284  etransclem42  47285  etransclem44  47287  etransclem45  47288