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Theorem eulerpartlemf 30900
Description: Lemma for eulerpart 30912: 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 3744 . . . . . 6 (𝑡 ∈ (ℕ ∖ 𝐽) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡𝐽))
2 breq2 4815 . . . . . . . . . . 11 (𝑧 = 𝑡 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑡))
32notbid 309 . . . . . . . . . 10 (𝑧 = 𝑡 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑡))
4 eulerpart.j . . . . . . . . . 10 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
53, 4elrab2 3525 . . . . . . . . 9 (𝑡𝐽 ↔ (𝑡 ∈ ℕ ∧ ¬ 2 ∥ 𝑡))
65simplbi2 494 . . . . . . . 8 (𝑡 ∈ ℕ → (¬ 2 ∥ 𝑡𝑡𝐽))
76con1d 141 . . . . . . 7 (𝑡 ∈ ℕ → (¬ 𝑡𝐽 → 2 ∥ 𝑡))
87imp 395 . . . . . 6 ((𝑡 ∈ ℕ ∧ ¬ 𝑡𝐽) → 2 ∥ 𝑡)
91, 8sylbi 208 . . . . 5 (𝑡 ∈ (ℕ ∖ 𝐽) → 2 ∥ 𝑡)
109adantl 473 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → 2 ∥ 𝑡)
1110adantr 472 . . 3 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴𝑡) ∈ ℕ) → 2 ∥ 𝑡)
12 simpll 783 . . . 4 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴𝑡) ∈ ℕ) → 𝐴 ∈ (𝑇𝑅))
13 eldifi 3896 . . . . . 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 30899 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
2322simp1bi 1175 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0𝑚 ℕ))
24 elmapi 8086 . . . . . . . . 9 (𝐴 ∈ (ℕ0𝑚 ℕ) → 𝐴:ℕ⟶ℕ0)
2523, 24syl 17 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → 𝐴:ℕ⟶ℕ0)
26 ffn 6225 . . . . . . . 8 (𝐴:ℕ⟶ℕ0𝐴 Fn ℕ)
27 elpreima 6531 . . . . . . . 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 535 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (𝑡 ∈ (𝐴 “ ℕ) ↔ (𝐴𝑡) ∈ ℕ))
3013, 29sylan2 586 . . . . 5 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝑡 ∈ (𝐴 “ ℕ) ↔ (𝐴𝑡) ∈ ℕ))
3130biimpar 469 . . . 4 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴𝑡) ∈ ℕ) → 𝑡 ∈ (𝐴 “ ℕ))
3222simp3bi 1177 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ 𝐽)
3332sselda 3763 . . . . 5 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (𝐴 “ ℕ)) → 𝑡𝐽)
345simprbi 490 . . . . 5 (𝑡𝐽 → ¬ 2 ∥ 𝑡)
3533, 34syl 17 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (𝐴 “ ℕ)) → ¬ 2 ∥ 𝑡)
3612, 31, 35syl2anc 579 . . 3 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴𝑡) ∈ ℕ) → ¬ 2 ∥ 𝑡)
3711, 36pm2.65da 851 . 2 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → ¬ (𝐴𝑡) ∈ ℕ)
3825adantr 472 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → 𝐴:ℕ⟶ℕ0)
3913adantl 473 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → 𝑡 ∈ ℕ)
4038, 39ffvelrnd 6554 . . 3 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴𝑡) ∈ ℕ0)
41 elnn0 11544 . . 3 ((𝐴𝑡) ∈ ℕ0 ↔ ((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0))
4240, 41sylib 209 . 2 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → ((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0))
43 orel1 912 . 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 197  wa 384  wo 873   = wceq 1652  wcel 2155  {cab 2751  wral 3055  {crab 3059  cdif 3731  cin 3733  wss 3734  c0 4081  𝒫 cpw 4317   class class class wbr 4811  {copab 4873  cmpt 4890  ccnv 5278  cima 5282   Fn wfn 6065  wf 6066  cfv 6070  (class class class)co 6846  cmpt2 6848   supp csupp 7501  𝑚 cmap 8064  Fincfn 8164  0cc0 10193  1c1 10194   · cmul 10198  cle 10333  cn 11278  2c2 11331  0cn0 11542  cexp 13072  Σcsu 14715  cdvds 15279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-mulcl 10255  ax-i2m1 10261
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-1st 7370  df-2nd 7371  df-map 8066  df-n0 11543
This theorem is referenced by:  eulerpartlemgh  30908
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