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Theorem eulerpartlemgu 33017
Description: Lemma for eulerpart 33022: Rewriting the π‘ˆ set for an odd partition Note that interestingly, this proof reuses marypha2lem2 9379. (Contributed by Thierry Arnoux, 10-Aug-2018.)
Hypotheses
Ref Expression
eulerpart.p 𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}
eulerpart.o 𝑂 = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}
eulerpart.d 𝐷 = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}
eulerpart.j 𝐽 = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}
eulerpart.f 𝐹 = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))
eulerpart.h 𝐻 = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}
eulerpart.m 𝑀 = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})
eulerpart.r 𝑅 = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}
eulerpart.t 𝑇 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ (◑𝑓 β€œ β„•) βŠ† 𝐽}
eulerpart.g 𝐺 = (π‘œ ∈ (𝑇 ∩ 𝑅) ↦ ((πŸ­β€˜β„•)β€˜(𝐹 β€œ (π‘€β€˜(bits ∘ (π‘œ β†Ύ 𝐽))))))
eulerpartlemgh.1 π‘ˆ = βˆͺ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)({𝑑} Γ— (bitsβ€˜(π΄β€˜π‘‘)))
Assertion
Ref Expression
eulerpartlemgu (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ π‘ˆ = {βŸ¨π‘‘, π‘›βŸ© ∣ (𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)β€˜π‘‘))})
Distinct variable groups:   𝑧,𝑑   𝑓,𝑔,π‘˜,𝑛,𝑑,𝐴   𝑓,𝐽,𝑛,𝑑   𝑓,𝑁,π‘˜,𝑛,𝑑   𝑛,𝑂,𝑑   𝑃,𝑔,π‘˜   𝑅,𝑓,π‘˜,𝑛,𝑑   𝑇,𝑛,𝑑
Allowed substitution hints:   𝐴(π‘₯,𝑦,𝑧,π‘œ,π‘Ÿ)   𝐷(π‘₯,𝑦,𝑧,𝑑,𝑓,𝑔,π‘˜,𝑛,π‘œ,π‘Ÿ)   𝑃(π‘₯,𝑦,𝑧,𝑑,𝑓,𝑛,π‘œ,π‘Ÿ)   𝑅(π‘₯,𝑦,𝑧,𝑔,π‘œ,π‘Ÿ)   𝑇(π‘₯,𝑦,𝑧,𝑓,𝑔,π‘˜,π‘œ,π‘Ÿ)   π‘ˆ(π‘₯,𝑦,𝑧,𝑑,𝑓,𝑔,π‘˜,𝑛,π‘œ,π‘Ÿ)   𝐹(π‘₯,𝑦,𝑧,𝑑,𝑓,𝑔,π‘˜,𝑛,π‘œ,π‘Ÿ)   𝐺(π‘₯,𝑦,𝑧,𝑑,𝑓,𝑔,π‘˜,𝑛,π‘œ,π‘Ÿ)   𝐻(π‘₯,𝑦,𝑧,𝑑,𝑓,𝑔,π‘˜,𝑛,π‘œ,π‘Ÿ)   𝐽(π‘₯,𝑦,𝑧,𝑔,π‘˜,π‘œ,π‘Ÿ)   𝑀(π‘₯,𝑦,𝑧,𝑑,𝑓,𝑔,π‘˜,𝑛,π‘œ,π‘Ÿ)   𝑁(π‘₯,𝑦,𝑧,𝑔,π‘œ,π‘Ÿ)   𝑂(π‘₯,𝑦,𝑧,𝑓,𝑔,π‘˜,π‘œ,π‘Ÿ)

Proof of Theorem eulerpartlemgu
StepHypRef Expression
1 eulerpartlemgh.1 . 2 π‘ˆ = βˆͺ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)({𝑑} Γ— (bitsβ€˜(π΄β€˜π‘‘)))
2 eulerpart.p . . . . . . . . . . 11 𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}
3 eulerpart.o . . . . . . . . . . 11 𝑂 = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}
4 eulerpart.d . . . . . . . . . . 11 𝐷 = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}
5 eulerpart.j . . . . . . . . . . 11 𝐽 = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}
6 eulerpart.f . . . . . . . . . . 11 𝐹 = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))
7 eulerpart.h . . . . . . . . . . 11 𝐻 = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}
8 eulerpart.m . . . . . . . . . . 11 𝑀 = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})
9 eulerpart.r . . . . . . . . . . 11 𝑅 = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}
10 eulerpart.t . . . . . . . . . . 11 𝑇 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ (◑𝑓 β€œ β„•) βŠ† 𝐽}
112, 3, 4, 5, 6, 7, 8, 9, 10eulerpartlemt0 33009 . . . . . . . . . 10 (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (β„•0 ↑m β„•) ∧ (◑𝐴 β€œ β„•) ∈ Fin ∧ (◑𝐴 β€œ β„•) βŠ† 𝐽))
1211simp1bi 1146 . . . . . . . . 9 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ 𝐴 ∈ (β„•0 ↑m β„•))
13 elmapi 8794 . . . . . . . . 9 (𝐴 ∈ (β„•0 ↑m β„•) β†’ 𝐴:β„•βŸΆβ„•0)
1412, 13syl 17 . . . . . . . 8 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ 𝐴:β„•βŸΆβ„•0)
1514adantr 482 . . . . . . 7 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)) β†’ 𝐴:β„•βŸΆβ„•0)
1615ffund 6677 . . . . . 6 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)) β†’ Fun 𝐴)
17 inss1 4193 . . . . . . . . 9 ((◑𝐴 β€œ β„•) ∩ 𝐽) βŠ† (◑𝐴 β€œ β„•)
18 cnvimass 6038 . . . . . . . . . 10 (◑𝐴 β€œ β„•) βŠ† dom 𝐴
1918, 14fssdm 6693 . . . . . . . . 9 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ (◑𝐴 β€œ β„•) βŠ† β„•)
2017, 19sstrid 3960 . . . . . . . 8 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ ((◑𝐴 β€œ β„•) ∩ 𝐽) βŠ† β„•)
2120sselda 3949 . . . . . . 7 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)) β†’ 𝑑 ∈ β„•)
2214fdmd 6684 . . . . . . . . 9 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ dom 𝐴 = β„•)
2322eleq2d 2824 . . . . . . . 8 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ (𝑑 ∈ dom 𝐴 ↔ 𝑑 ∈ β„•))
2423adantr 482 . . . . . . 7 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)) β†’ (𝑑 ∈ dom 𝐴 ↔ 𝑑 ∈ β„•))
2521, 24mpbird 257 . . . . . 6 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)) β†’ 𝑑 ∈ dom 𝐴)
26 fvco 6944 . . . . . 6 ((Fun 𝐴 ∧ 𝑑 ∈ dom 𝐴) β†’ ((bits ∘ 𝐴)β€˜π‘‘) = (bitsβ€˜(π΄β€˜π‘‘)))
2716, 25, 26syl2anc 585 . . . . 5 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)) β†’ ((bits ∘ 𝐴)β€˜π‘‘) = (bitsβ€˜(π΄β€˜π‘‘)))
2827xpeq2d 5668 . . . 4 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)) β†’ ({𝑑} Γ— ((bits ∘ 𝐴)β€˜π‘‘)) = ({𝑑} Γ— (bitsβ€˜(π΄β€˜π‘‘))))
2928iuneq2dv 4983 . . 3 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ βˆͺ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)({𝑑} Γ— ((bits ∘ 𝐴)β€˜π‘‘)) = βˆͺ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)({𝑑} Γ— (bitsβ€˜(π΄β€˜π‘‘))))
30 eqid 2737 . . . 4 βˆͺ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)({𝑑} Γ— ((bits ∘ 𝐴)β€˜π‘‘)) = βˆͺ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)({𝑑} Γ— ((bits ∘ 𝐴)β€˜π‘‘))
3130marypha2lem2 9379 . . 3 βˆͺ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)({𝑑} Γ— ((bits ∘ 𝐴)β€˜π‘‘)) = {βŸ¨π‘‘, π‘›βŸ© ∣ (𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)β€˜π‘‘))}
3229, 31eqtr3di 2792 . 2 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ βˆͺ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)({𝑑} Γ— (bitsβ€˜(π΄β€˜π‘‘))) = {βŸ¨π‘‘, π‘›βŸ© ∣ (𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)β€˜π‘‘))})
331, 32eqtrid 2789 1 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ π‘ˆ = {βŸ¨π‘‘, π‘›βŸ© ∣ (𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)β€˜π‘‘))})
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2714  βˆ€wral 3065  {crab 3410   ∩ cin 3914   βŠ† wss 3915  βˆ…c0 4287  π’« cpw 4565  {csn 4591  βˆͺ ciun 4959   class class class wbr 5110  {copab 5172   ↦ cmpt 5193   Γ— cxp 5636  β—‘ccnv 5637  dom cdm 5638   β†Ύ cres 5640   β€œ cima 5641   ∘ ccom 5642  Fun wfun 6495  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364   supp csupp 8097   ↑m cmap 8772  Fincfn 8890  1c1 11059   Β· cmul 11063   ≀ cle 11197  β„•cn 12160  2c2 12215  β„•0cn0 12420  β†‘cexp 13974  Ξ£csu 15577   βˆ₯ cdvds 16143  bitscbits 16306  πŸ­cind 32649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-map 8774
This theorem is referenced by: (None)
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