Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eulerpartlemgu Structured version   Visualization version   GIF version

Theorem eulerpartlemgu 34379
Description: Lemma for eulerpart 34384: Rewriting the 𝑈 set for an odd partition Note that interestingly, this proof reuses marypha2lem2 9476. (Contributed by Thierry Arnoux, 10-Aug-2018.)
Hypotheses
Ref Expression
eulerpart.p 𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
eulerpart.o 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
eulerpart.d 𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}
eulerpart.j 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
eulerpart.f 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
eulerpart.h 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
eulerpart.m 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})
eulerpart.r 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
eulerpart.t 𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
eulerpart.g 𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
eulerpartlemgh.1 𝑈 = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))
Assertion
Ref Expression
eulerpartlemgu (𝐴 ∈ (𝑇𝑅) → 𝑈 = {⟨𝑡, 𝑛⟩ ∣ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)‘𝑡))})
Distinct variable groups:   𝑧,𝑡   𝑓,𝑔,𝑘,𝑛,𝑡,𝐴   𝑓,𝐽,𝑛,𝑡   𝑓,𝑁,𝑘,𝑛,𝑡   𝑛,𝑂,𝑡   𝑃,𝑔,𝑘   𝑅,𝑓,𝑘,𝑛,𝑡   𝑇,𝑛,𝑡
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧,𝑜,𝑟)   𝐷(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝑃(𝑥,𝑦,𝑧,𝑡,𝑓,𝑛,𝑜,𝑟)   𝑅(𝑥,𝑦,𝑧,𝑔,𝑜,𝑟)   𝑇(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑜,𝑟)   𝑈(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝐺(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝐻(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝐽(𝑥,𝑦,𝑧,𝑔,𝑘,𝑜,𝑟)   𝑀(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝑁(𝑥,𝑦,𝑧,𝑔,𝑜,𝑟)   𝑂(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑜,𝑟)

Proof of Theorem eulerpartlemgu
StepHypRef Expression
1 eulerpartlemgh.1 . 2 𝑈 = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))
2 eulerpart.p . . . . . . . . . . 11 𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
3 eulerpart.o . . . . . . . . . . 11 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
4 eulerpart.d . . . . . . . . . . 11 𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}
5 eulerpart.j . . . . . . . . . . 11 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
6 eulerpart.f . . . . . . . . . . 11 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
7 eulerpart.h . . . . . . . . . . 11 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
8 eulerpart.m . . . . . . . . . . 11 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})
9 eulerpart.r . . . . . . . . . . 11 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
10 eulerpart.t . . . . . . . . . . 11 𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
112, 3, 4, 5, 6, 7, 8, 9, 10eulerpartlemt0 34371 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
1211simp1bi 1146 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0m ℕ))
13 elmapi 8889 . . . . . . . . 9 (𝐴 ∈ (ℕ0m ℕ) → 𝐴:ℕ⟶ℕ0)
1412, 13syl 17 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → 𝐴:ℕ⟶ℕ0)
1514adantr 480 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝐴:ℕ⟶ℕ0)
1615ffund 6740 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → Fun 𝐴)
17 inss1 4237 . . . . . . . . 9 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ (𝐴 “ ℕ)
18 cnvimass 6100 . . . . . . . . . 10 (𝐴 “ ℕ) ⊆ dom 𝐴
1918, 14fssdm 6755 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ ℕ)
2017, 19sstrid 3995 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → ((𝐴 “ ℕ) ∩ 𝐽) ⊆ ℕ)
2120sselda 3983 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ℕ)
2214fdmd 6746 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → dom 𝐴 = ℕ)
2322eleq2d 2827 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → (𝑡 ∈ dom 𝐴𝑡 ∈ ℕ))
2423adantr 480 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (𝑡 ∈ dom 𝐴𝑡 ∈ ℕ))
2521, 24mpbird 257 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ dom 𝐴)
26 fvco 7007 . . . . . 6 ((Fun 𝐴𝑡 ∈ dom 𝐴) → ((bits ∘ 𝐴)‘𝑡) = (bits‘(𝐴𝑡)))
2716, 25, 26syl2anc 584 . . . . 5 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → ((bits ∘ 𝐴)‘𝑡) = (bits‘(𝐴𝑡)))
2827xpeq2d 5715 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → ({𝑡} × ((bits ∘ 𝐴)‘𝑡)) = ({𝑡} × (bits‘(𝐴𝑡))))
2928iuneq2dv 5016 . . 3 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × ((bits ∘ 𝐴)‘𝑡)) = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))))
30 eqid 2737 . . . 4 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × ((bits ∘ 𝐴)‘𝑡)) = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × ((bits ∘ 𝐴)‘𝑡))
3130marypha2lem2 9476 . . 3 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × ((bits ∘ 𝐴)‘𝑡)) = {⟨𝑡, 𝑛⟩ ∣ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)‘𝑡))}
3229, 31eqtr3di 2792 . 2 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) = {⟨𝑡, 𝑛⟩ ∣ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)‘𝑡))})
331, 32eqtrid 2789 1 (𝐴 ∈ (𝑇𝑅) → 𝑈 = {⟨𝑡, 𝑛⟩ ∣ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)‘𝑡))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  {crab 3436  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   ciun 4991   class class class wbr 5143  {copab 5205  cmpt 5225   × cxp 5683  ccnv 5684  dom cdm 5685  cres 5687  cima 5688  ccom 5689  Fun wfun 6555  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433   supp csupp 8185  m cmap 8866  Fincfn 8985  1c1 11156   · cmul 11160  cle 11296  cn 12266  2c2 12321  0cn0 12526  cexp 14102  Σcsu 15722  cdvds 16290  bitscbits 16456  𝟭cind 32835
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
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-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  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-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator