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Theorem eulerpartlemf 31742
Description: Lemma for eulerpart 31754: 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 3894 . . . . . 6 (𝑡 ∈ (ℕ ∖ 𝐽) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡𝐽))
2 breq2 5037 . . . . . . . . . . 11 (𝑧 = 𝑡 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑡))
32notbid 321 . . . . . . . . . 10 (𝑧 = 𝑡 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑡))
4 eulerpart.j . . . . . . . . . 10 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
53, 4elrab2 3634 . . . . . . . . 9 (𝑡𝐽 ↔ (𝑡 ∈ ℕ ∧ ¬ 2 ∥ 𝑡))
65simplbi2 504 . . . . . . . 8 (𝑡 ∈ ℕ → (¬ 2 ∥ 𝑡𝑡𝐽))
76con1d 147 . . . . . . 7 (𝑡 ∈ ℕ → (¬ 𝑡𝐽 → 2 ∥ 𝑡))
87imp 410 . . . . . 6 ((𝑡 ∈ ℕ ∧ ¬ 𝑡𝐽) → 2 ∥ 𝑡)
91, 8sylbi 220 . . . . 5 (𝑡 ∈ (ℕ ∖ 𝐽) → 2 ∥ 𝑡)
109adantl 485 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → 2 ∥ 𝑡)
1110adantr 484 . . 3 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴𝑡) ∈ ℕ) → 2 ∥ 𝑡)
12 simpll 766 . . . 4 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴𝑡) ∈ ℕ) → 𝐴 ∈ (𝑇𝑅))
13 eldifi 4057 . . . . . 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 31741 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
2322simp1bi 1142 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0m ℕ))
24 elmapi 8415 . . . . . . . . 9 (𝐴 ∈ (ℕ0m ℕ) → 𝐴:ℕ⟶ℕ0)
2523, 24syl 17 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → 𝐴:ℕ⟶ℕ0)
26 ffn 6491 . . . . . . . 8 (𝐴:ℕ⟶ℕ0𝐴 Fn ℕ)
27 elpreima 6809 . . . . . . . 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 543 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (𝑡 ∈ (𝐴 “ ℕ) ↔ (𝐴𝑡) ∈ ℕ))
3013, 29sylan2 595 . . . . 5 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝑡 ∈ (𝐴 “ ℕ) ↔ (𝐴𝑡) ∈ ℕ))
3130biimpar 481 . . . 4 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴𝑡) ∈ ℕ) → 𝑡 ∈ (𝐴 “ ℕ))
3222simp3bi 1144 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ 𝐽)
3332sselda 3918 . . . . 5 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (𝐴 “ ℕ)) → 𝑡𝐽)
345simprbi 500 . . . . 5 (𝑡𝐽 → ¬ 2 ∥ 𝑡)
3533, 34syl 17 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (𝐴 “ ℕ)) → ¬ 2 ∥ 𝑡)
3612, 31, 35syl2anc 587 . . 3 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴𝑡) ∈ ℕ) → ¬ 2 ∥ 𝑡)
3711, 36pm2.65da 816 . 2 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → ¬ (𝐴𝑡) ∈ ℕ)
3825adantr 484 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → 𝐴:ℕ⟶ℕ0)
3913adantl 485 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → 𝑡 ∈ ℕ)
4038, 39ffvelrnd 6833 . . 3 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴𝑡) ∈ ℕ0)
41 elnn0 11891 . . 3 ((𝐴𝑡) ∈ ℕ0 ↔ ((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0))
4240, 41sylib 221 . 2 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → ((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0))
43 orel1 886 . 2 (¬ (𝐴𝑡) ∈ ℕ → (((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0) → (𝐴𝑡) = 0))
4437, 42, 43sylc 65 1 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴𝑡) = 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2112  {cab 2779  wral 3109  {crab 3113  cdif 3881  cin 3883  wss 3884  c0 4246  𝒫 cpw 4500   class class class wbr 5033  {copab 5095  cmpt 5113  ccnv 5522  cima 5526   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141   supp csupp 7817  m cmap 8393  Fincfn 8496  0cc0 10530  1c1 10531   · cmul 10535  cle 10669  cn 11629  2c2 11684  0cn0 11889  cexp 13429  Σcsu 15038  cdvds 15603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-mulcl 10592  ax-i2m1 10598
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395  df-n0 11890
This theorem is referenced by:  eulerpartlemgh  31750
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