Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  etransclem11 Structured version   Visualization version   GIF version

Theorem etransclem11 40780
 Description: A change of bound variable, often used in proofs for etransc 40818. (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 6698 . . . . 5 (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
21oveq1d 6705 . . . 4 (𝑛 = 𝑚 → ((0...𝑛) ↑𝑚 (0...𝑀)) = ((0...𝑚) ↑𝑚 (0...𝑀)))
32rabeqdv 3225 . . 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 6229 . . . . . . . 8 (𝑗 = 𝑘 → (𝑐𝑗) = (𝑐𝑘))
54cbvsumv 14470 . . . . . . 7 Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑐𝑘)
6 fveq1 6228 . . . . . . . 8 (𝑐 = 𝑑 → (𝑐𝑘) = (𝑑𝑘))
76sumeq2ad 14478 . . . . . . 7 (𝑐 = 𝑑 → Σ𝑘 ∈ (0...𝑀)(𝑐𝑘) = Σ𝑘 ∈ (0...𝑀)(𝑑𝑘))
85, 7syl5eq 2697 . . . . . 6 (𝑐 = 𝑑 → Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑑𝑘))
98eqeq1d 2653 . . . . 5 (𝑐 = 𝑑 → (Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛 ↔ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑛))
109cbvrabv 3230 . . . 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 2662 . . . . 5 (𝑛 = 𝑚 → (Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑛 ↔ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚))
1211rabbidv 3220 . . . 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, 12syl5eq 2697 . . 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 2685 . 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 4783 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 1523  {crab 2945   ↦ cmpt 4762  ‘cfv 5926  (class class class)co 6690   ↑𝑚 cmap 7899  0cc0 9974  ℕ0cn0 11330  ...cfz 12364  Σcsu 14460 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-seq 12842  df-sum 14461 This theorem is referenced by:  etransclem32  40801  etransclem33  40802  etransclem36  40805  etransclem37  40806  etransclem38  40807  etransclem40  40809  etransclem41  40810  etransclem42  40811  etransclem44  40813  etransclem45  40814
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