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Theorem eulerpartlemr 30564
Description: Lemma for eulerpart 30572. (Contributed by Thierry Arnoux, 13-Nov-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}
eulerpart.g 𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
Assertion
Ref Expression
eulerpartlemr 𝑂 = ((𝑇𝑅) ∩ 𝑃)
Distinct variable groups:   𝑓,𝑘,𝑛,𝑧   𝑓,𝐽,𝑛   𝑓,𝑁   𝑔,𝑛,𝑃
Allowed substitution hints:   𝐷(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑘,𝑜,𝑟)   𝑅(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝑇(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝐹(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝐺(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝐽(𝑥,𝑦,𝑧,𝑔,𝑘,𝑜,𝑟)   𝑀(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝑁(𝑥,𝑦,𝑧,𝑔,𝑘,𝑛,𝑜,𝑟)   𝑂(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)

Proof of Theorem eulerpartlemr
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elin 3829 . . . 4 ( ∈ (𝑇𝑅) ↔ (𝑇𝑅))
21anbi1i 731 . . 3 (( ∈ (𝑇𝑅) ∧ 𝑃) ↔ ((𝑇𝑅) ∧ 𝑃))
3 elin 3829 . . 3 ( ∈ ((𝑇𝑅) ∩ 𝑃) ↔ ( ∈ (𝑇𝑅) ∧ 𝑃))
4 eulerpart.p . . . . 5 𝑃 = {𝑓 ∈ (ℕ0𝑚 ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
5 eulerpart.o . . . . 5 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
6 eulerpart.d . . . . 5 𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}
74, 5, 6eulerpartlemo 30555 . . . 4 (𝑂 ↔ (𝑃 ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛))
8 cnveq 5328 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑓 = )
98imaeq1d 5500 . . . . . . . . . . . . . . . 16 (𝑓 = → (𝑓 “ ℕ) = ( “ ℕ))
109eleq1d 2715 . . . . . . . . . . . . . . 15 (𝑓 = → ((𝑓 “ ℕ) ∈ Fin ↔ ( “ ℕ) ∈ Fin))
11 fveq1 6228 . . . . . . . . . . . . . . . . . 18 (𝑓 = → (𝑓𝑘) = (𝑘))
1211oveq1d 6705 . . . . . . . . . . . . . . . . 17 (𝑓 = → ((𝑓𝑘) · 𝑘) = ((𝑘) · 𝑘))
1312sumeq2sdv 14479 . . . . . . . . . . . . . . . 16 (𝑓 = → Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑘) · 𝑘))
1413eqeq1d 2653 . . . . . . . . . . . . . . 15 (𝑓 = → (Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁))
1510, 14anbi12d 747 . . . . . . . . . . . . . 14 (𝑓 = → (((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁) ↔ (( “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁)))
1615, 4elrab2 3399 . . . . . . . . . . . . 13 (𝑃 ↔ ( ∈ (ℕ0𝑚 ℕ) ∧ (( “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁)))
1716simplbi 475 . . . . . . . . . . . 12 (𝑃 ∈ (ℕ0𝑚 ℕ))
18 cnvimass 5520 . . . . . . . . . . . . 13 ( “ ℕ) ⊆ dom
19 nn0ex 11336 . . . . . . . . . . . . . . 15 0 ∈ V
20 nnex 11064 . . . . . . . . . . . . . . 15 ℕ ∈ V
2119, 20elmap 7928 . . . . . . . . . . . . . 14 ( ∈ (ℕ0𝑚 ℕ) ↔ :ℕ⟶ℕ0)
22 fdm 6089 . . . . . . . . . . . . . 14 (:ℕ⟶ℕ0 → dom = ℕ)
2321, 22sylbi 207 . . . . . . . . . . . . 13 ( ∈ (ℕ0𝑚 ℕ) → dom = ℕ)
2418, 23syl5sseq 3686 . . . . . . . . . . . 12 ( ∈ (ℕ0𝑚 ℕ) → ( “ ℕ) ⊆ ℕ)
2517, 24syl 17 . . . . . . . . . . 11 (𝑃 → ( “ ℕ) ⊆ ℕ)
2625sselda 3636 . . . . . . . . . 10 ((𝑃𝑛 ∈ ( “ ℕ)) → 𝑛 ∈ ℕ)
2726ralrimiva 2995 . . . . . . . . 9 (𝑃 → ∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ)
2827biantrurd 528 . . . . . . . 8 (𝑃 → (∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛 ↔ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)))
2917biantrurd 528 . . . . . . . 8 (𝑃 → ((∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛) ↔ ( ∈ (ℕ0𝑚 ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛))))
3016simprbi 479 . . . . . . . . . 10 (𝑃 → (( “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁))
3130simpld 474 . . . . . . . . 9 (𝑃 → ( “ ℕ) ∈ Fin)
3231biantrud 527 . . . . . . . 8 (𝑃 → (( ∈ (ℕ0𝑚 ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)) ↔ (( ∈ (ℕ0𝑚 ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)) ∧ ( “ ℕ) ∈ Fin)))
3328, 29, 323bitrd 294 . . . . . . 7 (𝑃 → (∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛 ↔ (( ∈ (ℕ0𝑚 ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)) ∧ ( “ ℕ) ∈ Fin)))
34 dfss3 3625 . . . . . . . . . 10 (( “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ ( “ ℕ)𝑛𝐽)
35 breq2 4689 . . . . . . . . . . . . 13 (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
3635notbid 307 . . . . . . . . . . . 12 (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
37 eulerpart.j . . . . . . . . . . . 12 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
3836, 37elrab2 3399 . . . . . . . . . . 11 (𝑛𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
3938ralbii 3009 . . . . . . . . . 10 (∀𝑛 ∈ ( “ ℕ)𝑛𝐽 ↔ ∀𝑛 ∈ ( “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
40 r19.26 3093 . . . . . . . . . 10 (∀𝑛 ∈ ( “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛) ↔ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛))
4134, 39, 403bitri 286 . . . . . . . . 9 (( “ ℕ) ⊆ 𝐽 ↔ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛))
4241anbi2i 730 . . . . . . . 8 (( ∈ (ℕ0𝑚 ℕ) ∧ ( “ ℕ) ⊆ 𝐽) ↔ ( ∈ (ℕ0𝑚 ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)))
4342anbi1i 731 . . . . . . 7 ((( ∈ (ℕ0𝑚 ℕ) ∧ ( “ ℕ) ⊆ 𝐽) ∧ ( “ ℕ) ∈ Fin) ↔ (( ∈ (ℕ0𝑚 ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)) ∧ ( “ ℕ) ∈ Fin))
4433, 43syl6bbr 278 . . . . . 6 (𝑃 → (∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛 ↔ (( ∈ (ℕ0𝑚 ℕ) ∧ ( “ ℕ) ⊆ 𝐽) ∧ ( “ ℕ) ∈ Fin)))
459sseq1d 3665 . . . . . . . 8 (𝑓 = → ((𝑓 “ ℕ) ⊆ 𝐽 ↔ ( “ ℕ) ⊆ 𝐽))
46 eulerpart.t . . . . . . . 8 𝑇 = {𝑓 ∈ (ℕ0𝑚 ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
4745, 46elrab2 3399 . . . . . . 7 (𝑇 ↔ ( ∈ (ℕ0𝑚 ℕ) ∧ ( “ ℕ) ⊆ 𝐽))
48 vex 3234 . . . . . . . 8 ∈ V
49 eulerpart.r . . . . . . . 8 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
5048, 10, 49elab2 3386 . . . . . . 7 (𝑅 ↔ ( “ ℕ) ∈ Fin)
5147, 50anbi12i 733 . . . . . 6 ((𝑇𝑅) ↔ (( ∈ (ℕ0𝑚 ℕ) ∧ ( “ ℕ) ⊆ 𝐽) ∧ ( “ ℕ) ∈ Fin))
5244, 51syl6bbr 278 . . . . 5 (𝑃 → (∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛 ↔ (𝑇𝑅)))
5352pm5.32i 670 . . . 4 ((𝑃 ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛) ↔ (𝑃 ∧ (𝑇𝑅)))
54 ancom 465 . . . 4 ((𝑃 ∧ (𝑇𝑅)) ↔ ((𝑇𝑅) ∧ 𝑃))
557, 53, 543bitri 286 . . 3 (𝑂 ↔ ((𝑇𝑅) ∧ 𝑃))
562, 3, 553bitr4ri 293 . 2 (𝑂 ∈ ((𝑇𝑅) ∩ 𝑃))
5756eqriv 2648 1 𝑂 = ((𝑇𝑅) ∩ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1523  wcel 2030  {cab 2637  wral 2941  {crab 2945  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191   class class class wbr 4685  {copab 4745  cmpt 4762  ccnv 5142  dom cdm 5143  cres 5145  cima 5146  ccom 5147  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692   supp csupp 7340  𝑚 cmap 7899  Fincfn 7997  1c1 9975   · cmul 9979  cle 10113  cn 11058  2c2 11108  0cn0 11330  cexp 12900  Σcsu 14460  cdvds 15027  bitscbits 15188  𝟭cind 30200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-seq 12842  df-sum 14461
This theorem is referenced by: (None)
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