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Theorem eulerpartlemgu 33671
Description: Lemma for eulerpart 33676: Rewriting the π‘ˆ set for an odd partition Note that interestingly, this proof reuses marypha2lem2 9434. (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 33663 . . . . . . . . . 10 (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (β„•0 ↑m β„•) ∧ (◑𝐴 β€œ β„•) ∈ Fin ∧ (◑𝐴 β€œ β„•) βŠ† 𝐽))
1211simp1bi 1144 . . . . . . . . 9 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ 𝐴 ∈ (β„•0 ↑m β„•))
13 elmapi 8846 . . . . . . . . 9 (𝐴 ∈ (β„•0 ↑m β„•) β†’ 𝐴:β„•βŸΆβ„•0)
1412, 13syl 17 . . . . . . . 8 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ 𝐴:β„•βŸΆβ„•0)
1514adantr 480 . . . . . . 7 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)) β†’ 𝐴:β„•βŸΆβ„•0)
1615ffund 6722 . . . . . 6 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)) β†’ Fun 𝐴)
17 inss1 4229 . . . . . . . . 9 ((◑𝐴 β€œ β„•) ∩ 𝐽) βŠ† (◑𝐴 β€œ β„•)
18 cnvimass 6081 . . . . . . . . . 10 (◑𝐴 β€œ β„•) βŠ† dom 𝐴
1918, 14fssdm 6738 . . . . . . . . 9 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ (◑𝐴 β€œ β„•) βŠ† β„•)
2017, 19sstrid 3994 . . . . . . . 8 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ ((◑𝐴 β€œ β„•) ∩ 𝐽) βŠ† β„•)
2120sselda 3983 . . . . . . 7 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)) β†’ 𝑑 ∈ β„•)
2214fdmd 6729 . . . . . . . . 9 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ dom 𝐴 = β„•)
2322eleq2d 2818 . . . . . . . 8 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ (𝑑 ∈ dom 𝐴 ↔ 𝑑 ∈ β„•))
2423adantr 480 . . . . . . 7 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)) β†’ (𝑑 ∈ dom 𝐴 ↔ 𝑑 ∈ β„•))
2521, 24mpbird 256 . . . . . 6 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)) β†’ 𝑑 ∈ dom 𝐴)
26 fvco 6990 . . . . . 6 ((Fun 𝐴 ∧ 𝑑 ∈ dom 𝐴) β†’ ((bits ∘ 𝐴)β€˜π‘‘) = (bitsβ€˜(π΄β€˜π‘‘)))
2716, 25, 26syl2anc 583 . . . . 5 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)) β†’ ((bits ∘ 𝐴)β€˜π‘‘) = (bitsβ€˜(π΄β€˜π‘‘)))
2827xpeq2d 5707 . . . 4 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)) β†’ ({𝑑} Γ— ((bits ∘ 𝐴)β€˜π‘‘)) = ({𝑑} Γ— (bitsβ€˜(π΄β€˜π‘‘))))
2928iuneq2dv 5022 . . 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 2731 . . . 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 9434 . . 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 2786 . 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 2783 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 395   = wceq 1540   ∈ wcel 2105  {cab 2708  βˆ€wral 3060  {crab 3431   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  {csn 4629  βˆͺ ciun 4998   class class class wbr 5149  {copab 5211   ↦ cmpt 5232   Γ— cxp 5675  β—‘ccnv 5676  dom cdm 5677   β†Ύ cres 5679   β€œ cima 5680   ∘ ccom 5681  Fun wfun 6538  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7412   ∈ cmpo 7414   supp csupp 8149   ↑m cmap 8823  Fincfn 8942  1c1 11114   Β· cmul 11118   ≀ cle 11254  β„•cn 12217  2c2 12272  β„•0cn0 12477  β†‘cexp 14032  Ξ£csu 15637   βˆ₯ cdvds 16202  bitscbits 16365  πŸ­cind 33303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-map 8825
This theorem is referenced by: (None)
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