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Theorem eulerpartlemr 31249
Description: Lemma for eulerpart 31257. (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 4090 . . . 4 ( ∈ (𝑇𝑅) ↔ (𝑇𝑅))
21anbi1i 623 . . 3 (( ∈ (𝑇𝑅) ∧ 𝑃) ↔ ((𝑇𝑅) ∧ 𝑃))
3 elin 4090 . . 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 31240 . . . 4 (𝑂 ↔ (𝑃 ∧ ∀𝑛 ∈ ( “ ℕ) ¬ 2 ∥ 𝑛))
8 cnveq 5630 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑓 = )
98imaeq1d 5805 . . . . . . . . . . . . . . . 16 (𝑓 = → (𝑓 “ ℕ) = ( “ ℕ))
109eleq1d 2867 . . . . . . . . . . . . . . 15 (𝑓 = → ((𝑓 “ ℕ) ∈ Fin ↔ ( “ ℕ) ∈ Fin))
11 fveq1 6537 . . . . . . . . . . . . . . . . . 18 (𝑓 = → (𝑓𝑘) = (𝑘))
1211oveq1d 7031 . . . . . . . . . . . . . . . . 17 (𝑓 = → ((𝑓𝑘) · 𝑘) = ((𝑘) · 𝑘))
1312sumeq2sdv 14894 . . . . . . . . . . . . . . . 16 (𝑓 = → Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑘) · 𝑘))
1413eqeq1d 2797 . . . . . . . . . . . . . . 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 3621 . . . . . . . . . . . . 13 (𝑃 ↔ ( ∈ (ℕ0𝑚 ℕ) ∧ (( “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑘) · 𝑘) = 𝑁)))
1716simplbi 498 . . . . . . . . . . . 12 (𝑃 ∈ (ℕ0𝑚 ℕ))
18 cnvimass 5825 . . . . . . . . . . . . 13 ( “ ℕ) ⊆ dom
19 nn0ex 11751 . . . . . . . . . . . . . . 15 0 ∈ V
20 nnex 11492 . . . . . . . . . . . . . . 15 ℕ ∈ V
2119, 20elmap 8285 . . . . . . . . . . . . . 14 ( ∈ (ℕ0𝑚 ℕ) ↔ :ℕ⟶ℕ0)
22 fdm 6390 . . . . . . . . . . . . . 14 (:ℕ⟶ℕ0 → dom = ℕ)
2321, 22sylbi 218 . . . . . . . . . . . . 13 ( ∈ (ℕ0𝑚 ℕ) → dom = ℕ)
2418, 23sseqtrid 3940 . . . . . . . . . . . 12 ( ∈ (ℕ0𝑚 ℕ) → ( “ ℕ) ⊆ ℕ)
2517, 24syl 17 . . . . . . . . . . 11 (𝑃 → ( “ ℕ) ⊆ ℕ)
2625sselda 3889 . . . . . . . . . 10 ((𝑃𝑛 ∈ ( “ ℕ)) → 𝑛 ∈ ℕ)
2726ralrimiva 3149 . . . . . . . . 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 3878 . . . . . . . . . 10 (( “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ ( “ ℕ)𝑛𝐽)
35 breq2 4966 . . . . . . . . . . . . 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 3621 . . . . . . . . . . 11 (𝑛𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
3938ralbii 3132 . . . . . . . . . 10 (∀𝑛 ∈ ( “ ℕ)𝑛𝐽 ↔ ∀𝑛 ∈ ( “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
40 r19.26 3137 . . . . . . . . . 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 3919 . . . . . . . 8 (𝑓 = → ((𝑓 “ ℕ) ⊆ 𝐽 ↔ ( “ ℕ) ⊆ 𝐽))
46 eulerpart.t . . . . . . . 8 𝑇 = {𝑓 ∈ (ℕ0𝑚 ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
4745, 46elrab2 3621 . . . . . . 7 (𝑇 ↔ ( ∈ (ℕ0𝑚 ℕ) ∧ ( “ ℕ) ⊆ 𝐽))
48 vex 3440 . . . . . . . 8 ∈ V
49 eulerpart.r . . . . . . . 8 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
5048, 10, 49elab2 3608 . . . . . . 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 2792 1 𝑂 = ((𝑇𝑅) ∩ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1522  wcel 2081  {cab 2775  wral 3105  {crab 3109  cin 3858  wss 3859  c0 4211  𝒫 cpw 4453   class class class wbr 4962  {copab 5024  cmpt 5041  ccnv 5442  dom cdm 5443  cres 5445  cima 5446  ccom 5447  wf 6221  cfv 6225  (class class class)co 7016  cmpo 7018   supp csupp 7681  𝑚 cmap 8256  Fincfn 8357  1c1 10384   · cmul 10388  cle 10522  cn 11486  2c2 11540  0cn0 11745  cexp 13279  Σcsu 14876  cdvds 15440  bitscbits 15601  𝟭cind 30886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-er 8139  df-map 8258  df-en 8358  df-dom 8359  df-sdom 8360  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-n0 11746  df-z 11830  df-uz 12094  df-fz 12743  df-seq 13220  df-sum 14877
This theorem is referenced by: (None)
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