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Theorem eulerpartlemf 34372
Description: Lemma for eulerpart 34384: 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 3961 . . . . . 6 (𝑡 ∈ (ℕ ∖ 𝐽) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡𝐽))
2 breq2 5147 . . . . . . . . . . 11 (𝑧 = 𝑡 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑡))
32notbid 318 . . . . . . . . . 10 (𝑧 = 𝑡 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑡))
4 eulerpart.j . . . . . . . . . 10 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
53, 4elrab2 3695 . . . . . . . . 9 (𝑡𝐽 ↔ (𝑡 ∈ ℕ ∧ ¬ 2 ∥ 𝑡))
65simplbi2 500 . . . . . . . 8 (𝑡 ∈ ℕ → (¬ 2 ∥ 𝑡𝑡𝐽))
76con1d 145 . . . . . . 7 (𝑡 ∈ ℕ → (¬ 𝑡𝐽 → 2 ∥ 𝑡))
87imp 406 . . . . . 6 ((𝑡 ∈ ℕ ∧ ¬ 𝑡𝐽) → 2 ∥ 𝑡)
91, 8sylbi 217 . . . . 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 767 . . . 4 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴𝑡) ∈ ℕ) → 𝐴 ∈ (𝑇𝑅))
13 eldifi 4131 . . . . . 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 34371 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
2322simp1bi 1146 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0m ℕ))
24 elmapi 8889 . . . . . . . . 9 (𝐴 ∈ (ℕ0m ℕ) → 𝐴:ℕ⟶ℕ0)
2523, 24syl 17 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → 𝐴:ℕ⟶ℕ0)
26 ffn 6736 . . . . . . . 8 (𝐴:ℕ⟶ℕ0𝐴 Fn ℕ)
27 elpreima 7078 . . . . . . . 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 539 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (𝑡 ∈ (𝐴 “ ℕ) ↔ (𝐴𝑡) ∈ ℕ))
3013, 29sylan2 593 . . . . 5 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝑡 ∈ (𝐴 “ ℕ) ↔ (𝐴𝑡) ∈ ℕ))
3130biimpar 477 . . . 4 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴𝑡) ∈ ℕ) → 𝑡 ∈ (𝐴 “ ℕ))
3222simp3bi 1148 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ 𝐽)
3332sselda 3983 . . . . 5 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (𝐴 “ ℕ)) → 𝑡𝐽)
345simprbi 496 . . . . 5 (𝑡𝐽 → ¬ 2 ∥ 𝑡)
3533, 34syl 17 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (𝐴 “ ℕ)) → ¬ 2 ∥ 𝑡)
3612, 31, 35syl2anc 584 . . 3 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴𝑡) ∈ ℕ) → ¬ 2 ∥ 𝑡)
3711, 36pm2.65da 817 . 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, 39ffvelcdmd 7105 . . 3 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴𝑡) ∈ ℕ0)
41 elnn0 12528 . . 3 ((𝐴𝑡) ∈ ℕ0 ↔ ((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0))
4240, 41sylib 218 . 2 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → ((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0))
43 orel1 889 . 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 206  wa 395  wo 848   = wceq 1540  wcel 2108  {cab 2714  wral 3061  {crab 3436  cdif 3948  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600   class class class wbr 5143  {copab 5205  cmpt 5225  ccnv 5684  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433   supp csupp 8185  m cmap 8866  Fincfn 8985  0cc0 11155  1c1 11156   · cmul 11160  cle 11296  cn 12266  2c2 12321  0cn0 12526  cexp 14102  Σcsu 15722  cdvds 16290
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  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-mulcl 11217  ax-i2m1 11223
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  df-n0 12527
This theorem is referenced by:  eulerpartlemgh  34380
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