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Theorem eulerpartlemf 33010
Description: Lemma for eulerpart 33022: Odd partitions are zero for even numbers. (Contributed by Thierry Arnoux, 9-Sep-2017.)
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}
Assertion
Ref Expression
eulerpartlemf ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) β†’ (π΄β€˜π‘‘) = 0)
Distinct variable groups:   𝑧,𝑑   𝑓,𝑔,π‘˜,𝑛,𝑑,𝐴   𝑓,𝐽   𝑓,𝑁   𝑃,𝑔
Allowed substitution hints:   𝐴(π‘₯,𝑦,𝑧,π‘Ÿ)   𝐷(π‘₯,𝑦,𝑧,𝑑,𝑓,𝑔,π‘˜,𝑛,π‘Ÿ)   𝑃(π‘₯,𝑦,𝑧,𝑑,𝑓,π‘˜,𝑛,π‘Ÿ)   𝑅(π‘₯,𝑦,𝑧,𝑑,𝑓,𝑔,π‘˜,𝑛,π‘Ÿ)   𝑇(π‘₯,𝑦,𝑧,𝑑,𝑓,𝑔,π‘˜,𝑛,π‘Ÿ)   𝐹(π‘₯,𝑦,𝑧,𝑑,𝑓,𝑔,π‘˜,𝑛,π‘Ÿ)   𝐻(π‘₯,𝑦,𝑧,𝑑,𝑓,𝑔,π‘˜,𝑛,π‘Ÿ)   𝐽(π‘₯,𝑦,𝑧,𝑑,𝑔,π‘˜,𝑛,π‘Ÿ)   𝑀(π‘₯,𝑦,𝑧,𝑑,𝑓,𝑔,π‘˜,𝑛,π‘Ÿ)   𝑁(π‘₯,𝑦,𝑧,𝑑,𝑔,π‘˜,𝑛,π‘Ÿ)   𝑂(π‘₯,𝑦,𝑧,𝑑,𝑓,𝑔,π‘˜,𝑛,π‘Ÿ)

Proof of Theorem eulerpartlemf
StepHypRef Expression
1 eldif 3925 . . . . . 6 (𝑑 ∈ (β„• βˆ– 𝐽) ↔ (𝑑 ∈ β„• ∧ Β¬ 𝑑 ∈ 𝐽))
2 breq2 5114 . . . . . . . . . . 11 (𝑧 = 𝑑 β†’ (2 βˆ₯ 𝑧 ↔ 2 βˆ₯ 𝑑))
32notbid 318 . . . . . . . . . 10 (𝑧 = 𝑑 β†’ (Β¬ 2 βˆ₯ 𝑧 ↔ Β¬ 2 βˆ₯ 𝑑))
4 eulerpart.j . . . . . . . . . 10 𝐽 = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}
53, 4elrab2 3653 . . . . . . . . 9 (𝑑 ∈ 𝐽 ↔ (𝑑 ∈ β„• ∧ Β¬ 2 βˆ₯ 𝑑))
65simplbi2 502 . . . . . . . 8 (𝑑 ∈ β„• β†’ (Β¬ 2 βˆ₯ 𝑑 β†’ 𝑑 ∈ 𝐽))
76con1d 145 . . . . . . 7 (𝑑 ∈ β„• β†’ (Β¬ 𝑑 ∈ 𝐽 β†’ 2 βˆ₯ 𝑑))
87imp 408 . . . . . 6 ((𝑑 ∈ β„• ∧ Β¬ 𝑑 ∈ 𝐽) β†’ 2 βˆ₯ 𝑑)
91, 8sylbi 216 . . . . 5 (𝑑 ∈ (β„• βˆ– 𝐽) β†’ 2 βˆ₯ 𝑑)
109adantl 483 . . . 4 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) β†’ 2 βˆ₯ 𝑑)
1110adantr 482 . . 3 (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) ∧ (π΄β€˜π‘‘) ∈ β„•) β†’ 2 βˆ₯ 𝑑)
12 simpll 766 . . . 4 (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) ∧ (π΄β€˜π‘‘) ∈ β„•) β†’ 𝐴 ∈ (𝑇 ∩ 𝑅))
13 eldifi 4091 . . . . . 6 (𝑑 ∈ (β„• βˆ– 𝐽) β†’ 𝑑 ∈ β„•)
14 eulerpart.p . . . . . . . . . . 11 𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}
15 eulerpart.o . . . . . . . . . . 11 𝑂 = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}
16 eulerpart.d . . . . . . . . . . 11 𝐷 = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}
17 eulerpart.f . . . . . . . . . . 11 𝐹 = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))
18 eulerpart.h . . . . . . . . . . 11 𝐻 = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}
19 eulerpart.m . . . . . . . . . . 11 𝑀 = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})
20 eulerpart.r . . . . . . . . . . 11 𝑅 = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}
21 eulerpart.t . . . . . . . . . . 11 𝑇 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ (◑𝑓 β€œ β„•) βŠ† 𝐽}
2214, 15, 16, 4, 17, 18, 19, 20, 21eulerpartlemt0 33009 . . . . . . . . . 10 (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (β„•0 ↑m β„•) ∧ (◑𝐴 β€œ β„•) ∈ Fin ∧ (◑𝐴 β€œ β„•) βŠ† 𝐽))
2322simp1bi 1146 . . . . . . . . 9 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ 𝐴 ∈ (β„•0 ↑m β„•))
24 elmapi 8794 . . . . . . . . 9 (𝐴 ∈ (β„•0 ↑m β„•) β†’ 𝐴:β„•βŸΆβ„•0)
2523, 24syl 17 . . . . . . . 8 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ 𝐴:β„•βŸΆβ„•0)
26 ffn 6673 . . . . . . . 8 (𝐴:β„•βŸΆβ„•0 β†’ 𝐴 Fn β„•)
27 elpreima 7013 . . . . . . . 8 (𝐴 Fn β„• β†’ (𝑑 ∈ (◑𝐴 β€œ β„•) ↔ (𝑑 ∈ β„• ∧ (π΄β€˜π‘‘) ∈ β„•)))
2825, 26, 273syl 18 . . . . . . 7 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ (𝑑 ∈ (◑𝐴 β€œ β„•) ↔ (𝑑 ∈ β„• ∧ (π΄β€˜π‘‘) ∈ β„•)))
2928baibd 541 . . . . . 6 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ β„•) β†’ (𝑑 ∈ (◑𝐴 β€œ β„•) ↔ (π΄β€˜π‘‘) ∈ β„•))
3013, 29sylan2 594 . . . . 5 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) β†’ (𝑑 ∈ (◑𝐴 β€œ β„•) ↔ (π΄β€˜π‘‘) ∈ β„•))
3130biimpar 479 . . . 4 (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) ∧ (π΄β€˜π‘‘) ∈ β„•) β†’ 𝑑 ∈ (◑𝐴 β€œ β„•))
3222simp3bi 1148 . . . . . 6 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ (◑𝐴 β€œ β„•) βŠ† 𝐽)
3332sselda 3949 . . . . 5 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (◑𝐴 β€œ β„•)) β†’ 𝑑 ∈ 𝐽)
345simprbi 498 . . . . 5 (𝑑 ∈ 𝐽 β†’ Β¬ 2 βˆ₯ 𝑑)
3533, 34syl 17 . . . 4 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (◑𝐴 β€œ β„•)) β†’ Β¬ 2 βˆ₯ 𝑑)
3612, 31, 35syl2anc 585 . . 3 (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) ∧ (π΄β€˜π‘‘) ∈ β„•) β†’ Β¬ 2 βˆ₯ 𝑑)
3711, 36pm2.65da 816 . 2 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) β†’ Β¬ (π΄β€˜π‘‘) ∈ β„•)
3825adantr 482 . . . 4 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) β†’ 𝐴:β„•βŸΆβ„•0)
3913adantl 483 . . . 4 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) β†’ 𝑑 ∈ β„•)
4038, 39ffvelcdmd 7041 . . 3 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) β†’ (π΄β€˜π‘‘) ∈ β„•0)
41 elnn0 12422 . . 3 ((π΄β€˜π‘‘) ∈ β„•0 ↔ ((π΄β€˜π‘‘) ∈ β„• ∨ (π΄β€˜π‘‘) = 0))
4240, 41sylib 217 . 2 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) β†’ ((π΄β€˜π‘‘) ∈ β„• ∨ (π΄β€˜π‘‘) = 0))
43 orel1 888 . 2 (Β¬ (π΄β€˜π‘‘) ∈ β„• β†’ (((π΄β€˜π‘‘) ∈ β„• ∨ (π΄β€˜π‘‘) = 0) β†’ (π΄β€˜π‘‘) = 0))
4437, 42, 43sylc 65 1 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) β†’ (π΄β€˜π‘‘) = 0)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  {cab 2714  βˆ€wral 3065  {crab 3410   βˆ– cdif 3912   ∩ cin 3914   βŠ† wss 3915  βˆ…c0 4287  π’« cpw 4565   class class class wbr 5110  {copab 5172   ↦ cmpt 5193  β—‘ccnv 5637   β€œ cima 5641   Fn wfn 6496  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364   supp csupp 8097   ↑m cmap 8772  Fincfn 8890  0cc0 11058  1c1 11059   Β· cmul 11063   ≀ cle 11197  β„•cn 12160  2c2 12215  β„•0cn0 12420  β†‘cexp 13974  Ξ£csu 15577   βˆ₯ cdvds 16143
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  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-mulcl 11120  ax-i2m1 11126
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  df-n0 12421
This theorem is referenced by:  eulerpartlemgh  33018
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