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Theorem eulerpartlemr 31531
Description: Lemma for eulerpart 31539. (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 4166 . . . 4 ( ∈ (𝑇𝑅) ↔ (𝑇𝑅))
21anbi1i 623 . . 3 (( ∈ (𝑇𝑅) ∧ 𝑃) ↔ ((𝑇𝑅) ∧ 𝑃))
3 elin 4166 . . 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 31522 . . . 4 (𝑂 ↔ (𝑃 ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛))
8 cnveq 5737 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑓 = )
98imaeq1d 5921 . . . . . . . . . . . . . . . 16 (𝑓 = → (𝑓 “ ℕ) = ( “ ℕ))
109eleq1d 2894 . . . . . . . . . . . . . . 15 (𝑓 = → ((𝑓 “ ℕ) ∈ Fin ↔ ( “ ℕ) ∈ Fin))
11 fveq1 6662 . . . . . . . . . . . . . . . . . 18 (𝑓 = → (𝑓𝑘) = (𝑘))
1211oveq1d 7160 . . . . . . . . . . . . . . . . 17 (𝑓 = → ((𝑓𝑘) · 𝑘) = ((𝑘) · 𝑘))
1312sumeq2sdv 15049 . . . . . . . . . . . . . . . 16 (𝑓 = → Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑘) · 𝑘))
1413eqeq1d 2820 . . . . . . . . . . . . . . 15 (𝑓 = → (Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁))
1510, 14anbi12d 630 . . . . . . . . . . . . . 14 (𝑓 = → (((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁) ↔ (( “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁)))
1615, 4elrab2 3680 . . . . . . . . . . . . 13 (𝑃 ↔ ( ∈ (ℕ0m ℕ) ∧ (( “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁)))
1716simplbi 498 . . . . . . . . . . . 12 (𝑃 ∈ (ℕ0m ℕ))
18 cnvimass 5942 . . . . . . . . . . . . 13 ( “ ℕ) ⊆ dom
19 nn0ex 11891 . . . . . . . . . . . . . . 15 0 ∈ V
20 nnex 11632 . . . . . . . . . . . . . . 15 ℕ ∈ V
2119, 20elmap 8424 . . . . . . . . . . . . . 14 ( ∈ (ℕ0m ℕ) ↔ :ℕ⟶ℕ0)
22 fdm 6515 . . . . . . . . . . . . . 14 (:ℕ⟶ℕ0 → dom = ℕ)
2321, 22sylbi 218 . . . . . . . . . . . . 13 ( ∈ (ℕ0m ℕ) → dom = ℕ)
2418, 23sseqtrid 4016 . . . . . . . . . . . 12 ( ∈ (ℕ0m ℕ) → ( “ ℕ) ⊆ ℕ)
2517, 24syl 17 . . . . . . . . . . 11 (𝑃 → ( “ ℕ) ⊆ ℕ)
2625sselda 3964 . . . . . . . . . 10 ((𝑃𝑛 ∈ ( “ ℕ)) → 𝑛 ∈ ℕ)
2726ralrimiva 3179 . . . . . . . . 9 (𝑃 → ∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ)
2827biantrurd 533 . . . . . . . 8 (𝑃 → (∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛 ↔ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)))
2917biantrurd 533 . . . . . . . 8 (𝑃 → ((∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛) ↔ ( ∈ (ℕ0m ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛))))
3016simprbi 497 . . . . . . . . . 10 (𝑃 → (( “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁))
3130simpld 495 . . . . . . . . 9 (𝑃 → ( “ ℕ) ∈ Fin)
3231biantrud 532 . . . . . . . 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 306 . . . . . . 7 (𝑃 → (∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛 ↔ (( ∈ (ℕ0m ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)) ∧ ( “ ℕ) ∈ Fin)))
34 dfss3 3953 . . . . . . . . . 10 (( “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ ( “ ℕ)𝑛𝐽)
35 breq2 5061 . . . . . . . . . . . . 13 (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
3635notbid 319 . . . . . . . . . . . 12 (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
37 eulerpart.j . . . . . . . . . . . 12 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
3836, 37elrab2 3680 . . . . . . . . . . 11 (𝑛𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
3938ralbii 3162 . . . . . . . . . 10 (∀𝑛 ∈ ( “ ℕ)𝑛𝐽 ↔ ∀𝑛 ∈ ( “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
40 r19.26 3167 . . . . . . . . . 10 (∀𝑛 ∈ ( “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛) ↔ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛))
4134, 39, 403bitri 298 . . . . . . . . 9 (( “ ℕ) ⊆ 𝐽 ↔ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛))
4241anbi2i 622 . . . . . . . 8 (( ∈ (ℕ0m ℕ) ∧ ( “ ℕ) ⊆ 𝐽) ↔ ( ∈ (ℕ0m ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)))
4342anbi1i 623 . . . . . . 7 ((( ∈ (ℕ0m ℕ) ∧ ( “ ℕ) ⊆ 𝐽) ∧ ( “ ℕ) ∈ Fin) ↔ (( ∈ (ℕ0m ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)) ∧ ( “ ℕ) ∈ Fin))
4433, 43syl6bbr 290 . . . . . 6 (𝑃 → (∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛 ↔ (( ∈ (ℕ0m ℕ) ∧ ( “ ℕ) ⊆ 𝐽) ∧ ( “ ℕ) ∈ Fin)))
459sseq1d 3995 . . . . . . . 8 (𝑓 = → ((𝑓 “ ℕ) ⊆ 𝐽 ↔ ( “ ℕ) ⊆ 𝐽))
46 eulerpart.t . . . . . . . 8 𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
4745, 46elrab2 3680 . . . . . . 7 (𝑇 ↔ ( ∈ (ℕ0m ℕ) ∧ ( “ ℕ) ⊆ 𝐽))
48 vex 3495 . . . . . . . 8 ∈ V
49 eulerpart.r . . . . . . . 8 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
5048, 10, 49elab2 3667 . . . . . . 7 (𝑅 ↔ ( “ ℕ) ∈ Fin)
5147, 50anbi12i 626 . . . . . 6 ((𝑇𝑅) ↔ (( ∈ (ℕ0m ℕ) ∧ ( “ ℕ) ⊆ 𝐽) ∧ ( “ ℕ) ∈ Fin))
5244, 51syl6bbr 290 . . . . 5 (𝑃 → (∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛 ↔ (𝑇𝑅)))
5352pm5.32i 575 . . . 4 ((𝑃 ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛) ↔ (𝑃 ∧ (𝑇𝑅)))
54 ancom 461 . . . 4 ((𝑃 ∧ (𝑇𝑅)) ↔ ((𝑇𝑅) ∧ 𝑃))
557, 53, 543bitri 298 . . 3 (𝑂 ↔ ((𝑇𝑅) ∧ 𝑃))
562, 3, 553bitr4ri 305 . 2 (𝑂 ∈ ((𝑇𝑅) ∩ 𝑃))
5756eqriv 2815 1 𝑂 = ((𝑇𝑅) ∩ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1528  wcel 2105  {cab 2796  wral 3135  {crab 3139  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535   class class class wbr 5057  {copab 5119  cmpt 5137  ccnv 5547  dom cdm 5548  cres 5550  cima 5551  ccom 5552  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147   supp csupp 7819  m cmap 8395  Fincfn 8497  1c1 10526   · cmul 10530  cle 10664  cn 11626  2c2 11680  0cn0 11885  cexp 13417  Σcsu 15030  cdvds 15595  bitscbits 15756  𝟭cind 31168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-seq 13358  df-sum 15031
This theorem is referenced by: (None)
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