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

Theorem eulerpartlemf 29553
Description: Lemma for eulerpart 29565: Odd partitions are zero for even numbers. (Contributed by Thierry Arnoux, 9-Sep-2017.)
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}
Assertion
Ref Expression
eulerpartlemf ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴𝑡) = 0)
Distinct variable groups:   𝑧,𝑡   𝑓,𝑔,𝑘,𝑛,𝑡,𝐴   𝑓,𝐽   𝑓,𝑁   𝑃,𝑔
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧,𝑟)   𝐷(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑟)   𝑃(𝑥,𝑦,𝑧,𝑡,𝑓,𝑘,𝑛,𝑟)   𝑅(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑟)   𝑇(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑟)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑟)   𝐻(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑟)   𝐽(𝑥,𝑦,𝑧,𝑡,𝑔,𝑘,𝑛,𝑟)   𝑀(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑟)   𝑁(𝑥,𝑦,𝑧,𝑡,𝑔,𝑘,𝑛,𝑟)   𝑂(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑟)

Proof of Theorem eulerpartlemf
StepHypRef Expression
1 eldif 3550 . . . . . 6 (𝑡 ∈ (ℕ ∖ 𝐽) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡𝐽))
2 breq2 4582 . . . . . . . . . . 11 (𝑧 = 𝑡 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑡))
32notbid 307 . . . . . . . . . 10 (𝑧 = 𝑡 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑡))
4 eulerpart.j . . . . . . . . . 10 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
53, 4elrab2 3333 . . . . . . . . 9 (𝑡𝐽 ↔ (𝑡 ∈ ℕ ∧ ¬ 2 ∥ 𝑡))
65simplbi2 653 . . . . . . . 8 (𝑡 ∈ ℕ → (¬ 2 ∥ 𝑡𝑡𝐽))
76con1d 138 . . . . . . 7 (𝑡 ∈ ℕ → (¬ 𝑡𝐽 → 2 ∥ 𝑡))
87imp 444 . . . . . 6 ((𝑡 ∈ ℕ ∧ ¬ 𝑡𝐽) → 2 ∥ 𝑡)
91, 8sylbi 206 . . . . 5 (𝑡 ∈ (ℕ ∖ 𝐽) → 2 ∥ 𝑡)
109adantl 481 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → 2 ∥ 𝑡)
1110adantr 480 . . 3 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴𝑡) ∈ ℕ) → 2 ∥ 𝑡)
12 simpll 786 . . . 4 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴𝑡) ∈ ℕ) → 𝐴 ∈ (𝑇𝑅))
13 eldifi 3694 . . . . . 6 (𝑡 ∈ (ℕ ∖ 𝐽) → 𝑡 ∈ ℕ)
14 eulerpart.p . . . . . . . . . . 11 𝑃 = {𝑓 ∈ (ℕ0𝑚 ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
15 eulerpart.o . . . . . . . . . . 11 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
16 eulerpart.d . . . . . . . . . . 11 𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}
17 eulerpart.f . . . . . . . . . . 11 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
18 eulerpart.h . . . . . . . . . . 11 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
19 eulerpart.m . . . . . . . . . . 11 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})
20 eulerpart.r . . . . . . . . . . 11 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
21 eulerpart.t . . . . . . . . . . 11 𝑇 = {𝑓 ∈ (ℕ0𝑚 ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
2214, 15, 16, 4, 17, 18, 19, 20, 21eulerpartlemt0 29552 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
2322simp1bi 1069 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0𝑚 ℕ))
24 elmapi 7743 . . . . . . . . 9 (𝐴 ∈ (ℕ0𝑚 ℕ) → 𝐴:ℕ⟶ℕ0)
2523, 24syl 17 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → 𝐴:ℕ⟶ℕ0)
26 ffn 5944 . . . . . . . 8 (𝐴:ℕ⟶ℕ0𝐴 Fn ℕ)
27 elpreima 6230 . . . . . . . 8 (𝐴 Fn ℕ → (𝑡 ∈ (𝐴 “ ℕ) ↔ (𝑡 ∈ ℕ ∧ (𝐴𝑡) ∈ ℕ)))
2825, 26, 273syl 18 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → (𝑡 ∈ (𝐴 “ ℕ) ↔ (𝑡 ∈ ℕ ∧ (𝐴𝑡) ∈ ℕ)))
2928baibd 946 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (𝑡 ∈ (𝐴 “ ℕ) ↔ (𝐴𝑡) ∈ ℕ))
3013, 29sylan2 490 . . . . 5 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝑡 ∈ (𝐴 “ ℕ) ↔ (𝐴𝑡) ∈ ℕ))
3130biimpar 501 . . . 4 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴𝑡) ∈ ℕ) → 𝑡 ∈ (𝐴 “ ℕ))
3222simp3bi 1071 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ 𝐽)
3332sselda 3568 . . . . 5 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (𝐴 “ ℕ)) → 𝑡𝐽)
345simprbi 479 . . . . 5 (𝑡𝐽 → ¬ 2 ∥ 𝑡)
3533, 34syl 17 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (𝐴 “ ℕ)) → ¬ 2 ∥ 𝑡)
3612, 31, 35syl2anc 691 . . 3 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴𝑡) ∈ ℕ) → ¬ 2 ∥ 𝑡)
3711, 36pm2.65da 598 . 2 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → ¬ (𝐴𝑡) ∈ ℕ)
3825adantr 480 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → 𝐴:ℕ⟶ℕ0)
3913adantl 481 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → 𝑡 ∈ ℕ)
4038, 39ffvelrnd 6253 . . 3 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴𝑡) ∈ ℕ0)
41 elnn0 11144 . . 3 ((𝐴𝑡) ∈ ℕ0 ↔ ((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0))
4240, 41sylib 207 . 2 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → ((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0))
43 orel1 396 . 2 (¬ (𝐴𝑡) ∈ ℕ → (((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0) → (𝐴𝑡) = 0))
4437, 42, 43sylc 63 1 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴𝑡) = 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383   = wceq 1475  wcel 1977  {cab 2596  wral 2896  {crab 2900  cdif 3537  cin 3539  wss 3540  c0 3874  𝒫 cpw 4108   class class class wbr 4578  {copab 4637  cmpt 4638  ccnv 5027  cima 5031   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  cmpt2 6529   supp csupp 7160  𝑚 cmap 7722  Fincfn 7819  0cc0 9793  1c1 9794   · cmul 9798  cle 9932  cn 10870  2c2 10920  0cn0 11142  cexp 12680  Σcsu 14213  cdvds 14770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-mulcl 9855  ax-i2m1 9861
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7037  df-2nd 7038  df-map 7724  df-n0 11143
This theorem is referenced by:  eulerpartlemgh  29561
  Copyright terms: Public domain W3C validator