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Theorem eulerpartlemr 30883
Description: Lemma for eulerpart 30891. (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 3958 . . . 4 ( ∈ (𝑇𝑅) ↔ (𝑇𝑅))
21anbi1i 617 . . 3 (( ∈ (𝑇𝑅) ∧ 𝑃) ↔ ((𝑇𝑅) ∧ 𝑃))
3 elin 3958 . . 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 30874 . . . 4 (𝑂 ↔ (𝑃 ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛))
8 cnveq 5464 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑓 = )
98imaeq1d 5647 . . . . . . . . . . . . . . . 16 (𝑓 = → (𝑓 “ ℕ) = ( “ ℕ))
109eleq1d 2829 . . . . . . . . . . . . . . 15 (𝑓 = → ((𝑓 “ ℕ) ∈ Fin ↔ ( “ ℕ) ∈ Fin))
11 fveq1 6374 . . . . . . . . . . . . . . . . . 18 (𝑓 = → (𝑓𝑘) = (𝑘))
1211oveq1d 6857 . . . . . . . . . . . . . . . . 17 (𝑓 = → ((𝑓𝑘) · 𝑘) = ((𝑘) · 𝑘))
1312sumeq2sdv 14720 . . . . . . . . . . . . . . . 16 (𝑓 = → Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑘) · 𝑘))
1413eqeq1d 2767 . . . . . . . . . . . . . . 15 (𝑓 = → (Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁))
1510, 14anbi12d 624 . . . . . . . . . . . . . 14 (𝑓 = → (((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁) ↔ (( “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁)))
1615, 4elrab2 3523 . . . . . . . . . . . . 13 (𝑃 ↔ ( ∈ (ℕ0𝑚 ℕ) ∧ (( “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁)))
1716simplbi 491 . . . . . . . . . . . 12 (𝑃 ∈ (ℕ0𝑚 ℕ))
18 cnvimass 5667 . . . . . . . . . . . . 13 ( “ ℕ) ⊆ dom
19 nn0ex 11545 . . . . . . . . . . . . . . 15 0 ∈ V
20 nnex 11281 . . . . . . . . . . . . . . 15 ℕ ∈ V
2119, 20elmap 8089 . . . . . . . . . . . . . 14 ( ∈ (ℕ0𝑚 ℕ) ↔ :ℕ⟶ℕ0)
22 fdm 6231 . . . . . . . . . . . . . 14 (:ℕ⟶ℕ0 → dom = ℕ)
2321, 22sylbi 208 . . . . . . . . . . . . 13 ( ∈ (ℕ0𝑚 ℕ) → dom = ℕ)
2418, 23syl5sseq 3813 . . . . . . . . . . . 12 ( ∈ (ℕ0𝑚 ℕ) → ( “ ℕ) ⊆ ℕ)
2517, 24syl 17 . . . . . . . . . . 11 (𝑃 → ( “ ℕ) ⊆ ℕ)
2625sselda 3761 . . . . . . . . . 10 ((𝑃𝑛 ∈ ( “ ℕ)) → 𝑛 ∈ ℕ)
2726ralrimiva 3113 . . . . . . . . 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 490 . . . . . . . . . 10 (𝑃 → (( “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁))
3130simpld 488 . . . . . . . . 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 296 . . . . . . 7 (𝑃 → (∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛 ↔ (( ∈ (ℕ0𝑚 ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)) ∧ ( “ ℕ) ∈ Fin)))
34 dfss3 3750 . . . . . . . . . 10 (( “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ ( “ ℕ)𝑛𝐽)
35 breq2 4813 . . . . . . . . . . . . 13 (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
3635notbid 309 . . . . . . . . . . . 12 (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
37 eulerpart.j . . . . . . . . . . . 12 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
3836, 37elrab2 3523 . . . . . . . . . . 11 (𝑛𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
3938ralbii 3127 . . . . . . . . . 10 (∀𝑛 ∈ ( “ ℕ)𝑛𝐽 ↔ ∀𝑛 ∈ ( “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
40 r19.26 3211 . . . . . . . . . 10 (∀𝑛 ∈ ( “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛) ↔ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛))
4134, 39, 403bitri 288 . . . . . . . . 9 (( “ ℕ) ⊆ 𝐽 ↔ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛))
4241anbi2i 616 . . . . . . . 8 (( ∈ (ℕ0𝑚 ℕ) ∧ ( “ ℕ) ⊆ 𝐽) ↔ ( ∈ (ℕ0𝑚 ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)))
4342anbi1i 617 . . . . . . 7 ((( ∈ (ℕ0𝑚 ℕ) ∧ ( “ ℕ) ⊆ 𝐽) ∧ ( “ ℕ) ∈ Fin) ↔ (( ∈ (ℕ0𝑚 ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)) ∧ ( “ ℕ) ∈ Fin))
4433, 43syl6bbr 280 . . . . . 6 (𝑃 → (∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛 ↔ (( ∈ (ℕ0𝑚 ℕ) ∧ ( “ ℕ) ⊆ 𝐽) ∧ ( “ ℕ) ∈ Fin)))
459sseq1d 3792 . . . . . . . 8 (𝑓 = → ((𝑓 “ ℕ) ⊆ 𝐽 ↔ ( “ ℕ) ⊆ 𝐽))
46 eulerpart.t . . . . . . . 8 𝑇 = {𝑓 ∈ (ℕ0𝑚 ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
4745, 46elrab2 3523 . . . . . . 7 (𝑇 ↔ ( ∈ (ℕ0𝑚 ℕ) ∧ ( “ ℕ) ⊆ 𝐽))
48 vex 3353 . . . . . . . 8 ∈ V
49 eulerpart.r . . . . . . . 8 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
5048, 10, 49elab2 3509 . . . . . . 7 (𝑅 ↔ ( “ ℕ) ∈ Fin)
5147, 50anbi12i 620 . . . . . 6 ((𝑇𝑅) ↔ (( ∈ (ℕ0𝑚 ℕ) ∧ ( “ ℕ) ⊆ 𝐽) ∧ ( “ ℕ) ∈ Fin))
5244, 51syl6bbr 280 . . . . 5 (𝑃 → (∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛 ↔ (𝑇𝑅)))
5352pm5.32i 570 . . . 4 ((𝑃 ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛) ↔ (𝑃 ∧ (𝑇𝑅)))
54 ancom 452 . . . 4 ((𝑃 ∧ (𝑇𝑅)) ↔ ((𝑇𝑅) ∧ 𝑃))
557, 53, 543bitri 288 . . 3 (𝑂 ↔ ((𝑇𝑅) ∧ 𝑃))
562, 3, 553bitr4ri 295 . 2 (𝑂 ∈ ((𝑇𝑅) ∩ 𝑃))
5756eqriv 2762 1 𝑂 = ((𝑇𝑅) ∩ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1652  wcel 2155  {cab 2751  wral 3055  {crab 3059  cin 3731  wss 3732  c0 4079  𝒫 cpw 4315   class class class wbr 4809  {copab 4871  cmpt 4888  ccnv 5276  dom cdm 5277  cres 5279  cima 5280  ccom 5281  wf 6064  cfv 6068  (class class class)co 6842  cmpt2 6844   supp csupp 7497  𝑚 cmap 8060  Fincfn 8160  1c1 10190   · cmul 10194  cle 10329  cn 11274  2c2 11327  0cn0 11538  cexp 13067  Σcsu 14701  cdvds 15265  bitscbits 15422  𝟭cind 30519
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 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  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-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-seq 13009  df-sum 14702
This theorem is referenced by: (None)
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