Theorem eulerpartlemf 32337

Description: Lemma for eulerpart 32349: Odd partitions are zero for even numbers. (Contributed by Thierry Arnoux, 9-Sep-2017.)

Hypotheses

Ref	Expression
eulerpart.p	⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
eulerpart.o	⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
eulerpart.d	⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}
eulerpart.j	⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
eulerpart.f	⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
eulerpart.h	⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
eulerpart.m	⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})
eulerpart.r	⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
eulerpart.t	⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}

Assertion

Ref	Expression
eulerpartlemf	⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴‘𝑡) = 0)

Distinct variable groups:   𝑧,𝑡   𝑓,𝑔,𝑘,𝑛,𝑡,𝐴   𝑓,𝐽   𝑓,𝑁   𝑃,𝑔

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧,𝑟)   𝐷(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑟)   𝑃(𝑥,𝑦,𝑧,𝑡,𝑓,𝑘,𝑛,𝑟)   𝑅(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑟)   𝑇(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑟)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑟)   𝐻(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑟)   𝐽(𝑥,𝑦,𝑧,𝑡,𝑔,𝑘,𝑛,𝑟)   𝑀(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑟)   𝑁(𝑥,𝑦,𝑧,𝑡,𝑔,𝑘,𝑛,𝑟)   𝑂(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑟)

Proof of Theorem eulerpartlemf

Step	Hyp	Ref	Expression
1		eldif 3897	. . . . . 6 ⊢ (𝑡 ∈ (ℕ ∖ 𝐽) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ 𝐽))
2		breq2 5078	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑡 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑡))
3	2	notbid 318	. . . . . . . . . 10 ⊢ (𝑧 = 𝑡 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑡))
4		eulerpart.j	. . . . . . . . . 10 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
5	3, 4	elrab2 3627	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝐽 ↔ (𝑡 ∈ ℕ ∧ ¬ 2 ∥ 𝑡))
6	5	simplbi2 501	. . . . . . . 8 ⊢ (𝑡 ∈ ℕ → (¬ 2 ∥ 𝑡 → 𝑡 ∈ 𝐽))
7	6	con1d 145	. . . . . . 7 ⊢ (𝑡 ∈ ℕ → (¬ 𝑡 ∈ 𝐽 → 2 ∥ 𝑡))
8	7	imp 407	. . . . . 6 ⊢ ((𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ 𝐽) → 2 ∥ 𝑡)
9	1, 8	sylbi 216	. . . . 5 ⊢ (𝑡 ∈ (ℕ ∖ 𝐽) → 2 ∥ 𝑡)
10	9	adantl 482	. . . 4 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → 2 ∥ 𝑡)
11	10	adantr 481	. . 3 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴‘𝑡) ∈ ℕ) → 2 ∥ 𝑡)
12		simpll 764	. . . 4 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴‘𝑡) ∈ ℕ) → 𝐴 ∈ (𝑇 ∩ 𝑅))
13		eldifi 4061	. . . . . 6 ⊢ (𝑡 ∈ (ℕ ∖ 𝐽) → 𝑡 ∈ ℕ)
14		eulerpart.p	. . . . . . . . . . 11 ⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
15		eulerpart.o	. . . . . . . . . . 11 ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
16		eulerpart.d	. . . . . . . . . . 11 ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}
17		eulerpart.f	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
18		eulerpart.h	. . . . . . . . . . 11 ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
19		eulerpart.m	. . . . . . . . . . 11 ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})
20		eulerpart.r	. . . . . . . . . . 11 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
21		eulerpart.t	. . . . . . . . . . 11 ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}
22	14, 15, 16, 4, 17, 18, 19, 20, 21	eulerpartlemt0 32336	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))
23	22	simp1bi 1144	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴 ∈ (ℕ0 ↑m ℕ))
24		elmapi 8637	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℕ0 ↑m ℕ) → 𝐴:ℕ⟶ℕ0)
25	23, 24	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
26		ffn 6600	. . . . . . . 8 ⊢ (𝐴:ℕ⟶ℕ0 → 𝐴 Fn ℕ)
27		elpreima 6935	. . . . . . . 8 ⊢ (𝐴 Fn ℕ → (𝑡 ∈ (◡𝐴 “ ℕ) ↔ (𝑡 ∈ ℕ ∧ (𝐴‘𝑡) ∈ ℕ)))
28	25, 26, 27	3syl 18	. . . . . . 7 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑡 ∈ (◡𝐴 “ ℕ) ↔ (𝑡 ∈ ℕ ∧ (𝐴‘𝑡) ∈ ℕ)))
29	28	baibd 540	. . . . . 6 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝑡 ∈ (◡𝐴 “ ℕ) ↔ (𝐴‘𝑡) ∈ ℕ))
30	13, 29	sylan2 593	. . . . 5 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝑡 ∈ (◡𝐴 “ ℕ) ↔ (𝐴‘𝑡) ∈ ℕ))
31	30	biimpar 478	. . . 4 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴‘𝑡) ∈ ℕ) → 𝑡 ∈ (◡𝐴 “ ℕ))
32	22	simp3bi 1146	. . . . . 6 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡𝐴 “ ℕ) ⊆ 𝐽)
33	32	sselda 3921	. . . . 5 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (◡𝐴 “ ℕ)) → 𝑡 ∈ 𝐽)
34	5	simprbi 497	. . . . 5 ⊢ (𝑡 ∈ 𝐽 → ¬ 2 ∥ 𝑡)
35	33, 34	syl 17	. . . 4 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (◡𝐴 “ ℕ)) → ¬ 2 ∥ 𝑡)
36	12, 31, 35	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴‘𝑡) ∈ ℕ) → ¬ 2 ∥ 𝑡)
37	11, 36	pm2.65da 814	. 2 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → ¬ (𝐴‘𝑡) ∈ ℕ)
38	25	adantr 481	. . . 4 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → 𝐴:ℕ⟶ℕ0)
39	13	adantl 482	. . . 4 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → 𝑡 ∈ ℕ)
40	38, 39	ffvelrnd 6962	. . 3 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴‘𝑡) ∈ ℕ0)
41		elnn0 12235	. . 3 ⊢ ((𝐴‘𝑡) ∈ ℕ0 ↔ ((𝐴‘𝑡) ∈ ℕ ∨ (𝐴‘𝑡) = 0))
42	40, 41	sylib 217	. 2 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → ((𝐴‘𝑡) ∈ ℕ ∨ (𝐴‘𝑡) = 0))
43		orel1 886	. 2 ⊢ (¬ (𝐴‘𝑡) ∈ ℕ → (((𝐴‘𝑡) ∈ ℕ ∨ (𝐴‘𝑡) = 0) → (𝐴‘𝑡) = 0))
44	37, 42, 43	sylc 65	1 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴‘𝑡) = 0)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  {crab 3068   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533   class class class wbr 5074  {copab 5136   ↦ cmpt 5157  ^◡ccnv 5588   “ cima 5592   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277   supp csupp 7977   ↑_m cmap 8615  Fincfn 8733  0cc0 10871  1c1 10872   · cmul 10876   ≤ cle 11010  ℕcn 11973  2c2 12028  ℕ₀cn0 12233  ↑cexp 13782  Σcsu 15397   ∥ cdvds 15963

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-mulcl 10933  ax-i2m1 10939

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617  df-n0 12234

This theorem is referenced by:  eulerpartlemgh  32345