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Theorem eulerpartlemf 33667
Description: Lemma for eulerpart 33679: 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 3957 . . . . . 6 (𝑑 ∈ (β„• βˆ– 𝐽) ↔ (𝑑 ∈ β„• ∧ Β¬ 𝑑 ∈ 𝐽))
2 breq2 5151 . . . . . . . . . . 11 (𝑧 = 𝑑 β†’ (2 βˆ₯ 𝑧 ↔ 2 βˆ₯ 𝑑))
32notbid 317 . . . . . . . . . 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 3685 . . . . . . . . 9 (𝑑 ∈ 𝐽 ↔ (𝑑 ∈ β„• ∧ Β¬ 2 βˆ₯ 𝑑))
65simplbi2 499 . . . . . . . 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 405 . . . . . 6 ((𝑑 ∈ β„• ∧ Β¬ 𝑑 ∈ 𝐽) β†’ 2 βˆ₯ 𝑑)
91, 8sylbi 216 . . . . 5 (𝑑 ∈ (β„• βˆ– 𝐽) β†’ 2 βˆ₯ 𝑑)
109adantl 480 . . . 4 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) β†’ 2 βˆ₯ 𝑑)
1110adantr 479 . . 3 (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) ∧ (π΄β€˜π‘‘) ∈ β„•) β†’ 2 βˆ₯ 𝑑)
12 simpll 763 . . . 4 (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) ∧ (π΄β€˜π‘‘) ∈ β„•) β†’ 𝐴 ∈ (𝑇 ∩ 𝑅))
13 eldifi 4125 . . . . . 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 33666 . . . . . . . . . 10 (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (β„•0 ↑m β„•) ∧ (◑𝐴 β€œ β„•) ∈ Fin ∧ (◑𝐴 β€œ β„•) βŠ† 𝐽))
2322simp1bi 1143 . . . . . . . . 9 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ 𝐴 ∈ (β„•0 ↑m β„•))
24 elmapi 8845 . . . . . . . . 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 6716 . . . . . . . 8 (𝐴:β„•βŸΆβ„•0 β†’ 𝐴 Fn β„•)
27 elpreima 7058 . . . . . . . 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 538 . . . . . 6 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ β„•) β†’ (𝑑 ∈ (◑𝐴 β€œ β„•) ↔ (π΄β€˜π‘‘) ∈ β„•))
3013, 29sylan2 591 . . . . 5 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) β†’ (𝑑 ∈ (◑𝐴 β€œ β„•) ↔ (π΄β€˜π‘‘) ∈ β„•))
3130biimpar 476 . . . 4 (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) ∧ (π΄β€˜π‘‘) ∈ β„•) β†’ 𝑑 ∈ (◑𝐴 β€œ β„•))
3222simp3bi 1145 . . . . . 6 (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ (◑𝐴 β€œ β„•) βŠ† 𝐽)
3332sselda 3981 . . . . 5 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (◑𝐴 β€œ β„•)) β†’ 𝑑 ∈ 𝐽)
345simprbi 495 . . . . 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 582 . . 3 (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) ∧ (π΄β€˜π‘‘) ∈ β„•) β†’ Β¬ 2 βˆ₯ 𝑑)
3711, 36pm2.65da 813 . 2 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) β†’ Β¬ (π΄β€˜π‘‘) ∈ β„•)
3825adantr 479 . . . 4 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) β†’ 𝐴:β„•βŸΆβ„•0)
3913adantl 480 . . . 4 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) β†’ 𝑑 ∈ β„•)
4038, 39ffvelcdmd 7086 . . 3 ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) β†’ (π΄β€˜π‘‘) ∈ β„•0)
41 elnn0 12478 . . 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 885 . 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 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104  {cab 2707  βˆ€wral 3059  {crab 3430   βˆ– cdif 3944   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601   class class class wbr 5147  {copab 5209   ↦ cmpt 5230  β—‘ccnv 5674   β€œ cima 5678   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413   supp csupp 8148   ↑m cmap 8822  Fincfn 8941  0cc0 11112  1c1 11113   Β· cmul 11117   ≀ cle 11253  β„•cn 12216  2c2 12271  β„•0cn0 12476  β†‘cexp 14031  Ξ£csu 15636   βˆ₯ cdvds 16201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-mulcl 11174  ax-i2m1 11180
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824  df-n0 12477
This theorem is referenced by:  eulerpartlemgh  33675
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