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Theorem eulerpartlemr 31746
Description: Lemma for eulerpart 31754. (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 3900 . . . 4 ( ∈ (𝑇𝑅) ↔ (𝑇𝑅))
21anbi1i 626 . . 3 (( ∈ (𝑇𝑅) ∧ 𝑃) ↔ ((𝑇𝑅) ∧ 𝑃))
3 elin 3900 . . 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 31737 . . . 4 (𝑂 ↔ (𝑃 ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛))
8 cnveq 5712 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑓 = )
98imaeq1d 5899 . . . . . . . . . . . . . . . 16 (𝑓 = → (𝑓 “ ℕ) = ( “ ℕ))
109eleq1d 2877 . . . . . . . . . . . . . . 15 (𝑓 = → ((𝑓 “ ℕ) ∈ Fin ↔ ( “ ℕ) ∈ Fin))
11 fveq1 6648 . . . . . . . . . . . . . . . . . 18 (𝑓 = → (𝑓𝑘) = (𝑘))
1211oveq1d 7154 . . . . . . . . . . . . . . . . 17 (𝑓 = → ((𝑓𝑘) · 𝑘) = ((𝑘) · 𝑘))
1312sumeq2sdv 15057 . . . . . . . . . . . . . . . 16 (𝑓 = → Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑘) · 𝑘))
1413eqeq1d 2803 . . . . . . . . . . . . . . 15 (𝑓 = → (Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁))
1510, 14anbi12d 633 . . . . . . . . . . . . . 14 (𝑓 = → (((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁) ↔ (( “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁)))
1615, 4elrab2 3634 . . . . . . . . . . . . 13 (𝑃 ↔ ( ∈ (ℕ0m ℕ) ∧ (( “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁)))
1716simplbi 501 . . . . . . . . . . . 12 (𝑃 ∈ (ℕ0m ℕ))
18 cnvimass 5920 . . . . . . . . . . . . 13 ( “ ℕ) ⊆ dom
19 nn0ex 11895 . . . . . . . . . . . . . . 15 0 ∈ V
20 nnex 11635 . . . . . . . . . . . . . . 15 ℕ ∈ V
2119, 20elmap 8422 . . . . . . . . . . . . . 14 ( ∈ (ℕ0m ℕ) ↔ :ℕ⟶ℕ0)
22 fdm 6499 . . . . . . . . . . . . . 14 (:ℕ⟶ℕ0 → dom = ℕ)
2321, 22sylbi 220 . . . . . . . . . . . . 13 ( ∈ (ℕ0m ℕ) → dom = ℕ)
2418, 23sseqtrid 3970 . . . . . . . . . . . 12 ( ∈ (ℕ0m ℕ) → ( “ ℕ) ⊆ ℕ)
2517, 24syl 17 . . . . . . . . . . 11 (𝑃 → ( “ ℕ) ⊆ ℕ)
2625sselda 3918 . . . . . . . . . 10 ((𝑃𝑛 ∈ ( “ ℕ)) → 𝑛 ∈ ℕ)
2726ralrimiva 3152 . . . . . . . . 9 (𝑃 → ∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ)
2827biantrurd 536 . . . . . . . 8 (𝑃 → (∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛 ↔ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)))
2917biantrurd 536 . . . . . . . 8 (𝑃 → ((∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛) ↔ ( ∈ (ℕ0m ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛))))
3016simprbi 500 . . . . . . . . . 10 (𝑃 → (( “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁))
3130simpld 498 . . . . . . . . 9 (𝑃 → ( “ ℕ) ∈ Fin)
3231biantrud 535 . . . . . . . 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 308 . . . . . . 7 (𝑃 → (∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛 ↔ (( ∈ (ℕ0m ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)) ∧ ( “ ℕ) ∈ Fin)))
34 dfss3 3906 . . . . . . . . . 10 (( “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ ( “ ℕ)𝑛𝐽)
35 breq2 5037 . . . . . . . . . . . . 13 (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
3635notbid 321 . . . . . . . . . . . 12 (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
37 eulerpart.j . . . . . . . . . . . 12 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
3836, 37elrab2 3634 . . . . . . . . . . 11 (𝑛𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
3938ralbii 3136 . . . . . . . . . 10 (∀𝑛 ∈ ( “ ℕ)𝑛𝐽 ↔ ∀𝑛 ∈ ( “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
40 r19.26 3140 . . . . . . . . . 10 (∀𝑛 ∈ ( “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛) ↔ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛))
4134, 39, 403bitri 300 . . . . . . . . 9 (( “ ℕ) ⊆ 𝐽 ↔ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛))
4241anbi2i 625 . . . . . . . 8 (( ∈ (ℕ0m ℕ) ∧ ( “ ℕ) ⊆ 𝐽) ↔ ( ∈ (ℕ0m ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)))
4342anbi1i 626 . . . . . . 7 ((( ∈ (ℕ0m ℕ) ∧ ( “ ℕ) ⊆ 𝐽) ∧ ( “ ℕ) ∈ Fin) ↔ (( ∈ (ℕ0m ℕ) ∧ (∀𝑛 ∈ ( “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛)) ∧ ( “ ℕ) ∈ Fin))
4433, 43syl6bbr 292 . . . . . 6 (𝑃 → (∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛 ↔ (( ∈ (ℕ0m ℕ) ∧ ( “ ℕ) ⊆ 𝐽) ∧ ( “ ℕ) ∈ Fin)))
459sseq1d 3949 . . . . . . . 8 (𝑓 = → ((𝑓 “ ℕ) ⊆ 𝐽 ↔ ( “ ℕ) ⊆ 𝐽))
46 eulerpart.t . . . . . . . 8 𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
4745, 46elrab2 3634 . . . . . . 7 (𝑇 ↔ ( ∈ (ℕ0m ℕ) ∧ ( “ ℕ) ⊆ 𝐽))
48 vex 3447 . . . . . . . 8 ∈ V
49 eulerpart.r . . . . . . . 8 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
5048, 10, 49elab2 3621 . . . . . . 7 (𝑅 ↔ ( “ ℕ) ∈ Fin)
5147, 50anbi12i 629 . . . . . 6 ((𝑇𝑅) ↔ (( ∈ (ℕ0m ℕ) ∧ ( “ ℕ) ⊆ 𝐽) ∧ ( “ ℕ) ∈ Fin))
5244, 51syl6bbr 292 . . . . 5 (𝑃 → (∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛 ↔ (𝑇𝑅)))
5352pm5.32i 578 . . . 4 ((𝑃 ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛) ↔ (𝑃 ∧ (𝑇𝑅)))
54 ancom 464 . . . 4 ((𝑃 ∧ (𝑇𝑅)) ↔ ((𝑇𝑅) ∧ 𝑃))
557, 53, 543bitri 300 . . 3 (𝑂 ↔ ((𝑇𝑅) ∧ 𝑃))
562, 3, 553bitr4ri 307 . 2 (𝑂 ∈ ((𝑇𝑅) ∩ 𝑃))
5756eqriv 2798 1 𝑂 = ((𝑇𝑅) ∩ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399   = wceq 1538  wcel 2112  {cab 2779  wral 3109  {crab 3113  cin 3883  wss 3884  c0 4246  𝒫 cpw 4500   class class class wbr 5033  {copab 5095  cmpt 5113  ccnv 5522  dom cdm 5523  cres 5525  cima 5526  ccom 5527  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141   supp csupp 7817  m cmap 8393  Fincfn 8496  1c1 10531   · cmul 10535  cle 10669  cn 11629  2c2 11684  0cn0 11889  cexp 13429  Σcsu 15038  cdvds 15603  bitscbits 15762  𝟭cind 31383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-seq 13369  df-sum 15039
This theorem is referenced by: (None)
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