Theorem eulerpartlemf 34769

Description: Lemma for eulerpart 34781: 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 3914	. . . . . 6 ⊢ (𝑡 ∈ (ℕ ∖ 𝐽) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ 𝐽))
2		breq2 5112	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑡 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑡))
3	2	notbid 321	. . . . . . . . . 10 ⊢ (𝑧 = 𝑡 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑡))
4		eulerpart.j	. . . . . . . . . 10 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
5	3, 4	elrab2 3653	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝐽 ↔ (𝑡 ∈ ℕ ∧ ¬ 2 ∥ 𝑡))
6	5	simplbi2 505	. . . . . . . 8 ⊢ (𝑡 ∈ ℕ → (¬ 2 ∥ 𝑡 → 𝑡 ∈ 𝐽))
7	6	con1d 146	. . . . . . 7 ⊢ (𝑡 ∈ ℕ → (¬ 𝑡 ∈ 𝐽 → 2 ∥ 𝑡))
8	7	imp 411	. . . . . 6 ⊢ ((𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ 𝐽) → 2 ∥ 𝑡)
9	1, 8	sylbi 220	. . . . 5 ⊢ (𝑡 ∈ (ℕ ∖ 𝐽) → 2 ∥ 𝑡)
10	9	adantl 486	. . . 4 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → 2 ∥ 𝑡)
11	10	adantr 485	. . 3 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴‘𝑡) ∈ ℕ) → 2 ∥ 𝑡)
12		simpll 778	. . . 4 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴‘𝑡) ∈ ℕ) → 𝐴 ∈ (𝑇 ∩ 𝑅))
13		eldifi 4084	. . . . . 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 34768	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))
23	22	simp1bi 1162	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴 ∈ (ℕ0 ↑m ℕ))
24		elmapi 8844	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℕ0 ↑m ℕ) → 𝐴:ℕ⟶ℕ0)
25	23, 24	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
26		ffn 6705	. . . . . . . 8 ⊢ (𝐴:ℕ⟶ℕ0 → 𝐴 Fn ℕ)
27		elpreima 7053	. . . . . . . 8 ⊢ (𝐴 Fn ℕ → (𝑡 ∈ (◡𝐴 “ ℕ) ↔ (𝑡 ∈ ℕ ∧ (𝐴‘𝑡) ∈ ℕ)))
28	25, 26, 27	3syl 19	. . . . . . 7 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑡 ∈ (◡𝐴 “ ℕ) ↔ (𝑡 ∈ ℕ ∧ (𝐴‘𝑡) ∈ ℕ)))
29	28	baibd 548	. . . . . 6 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝑡 ∈ (◡𝐴 “ ℕ) ↔ (𝐴‘𝑡) ∈ ℕ))
30	13, 29	sylan2 604	. . . . 5 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝑡 ∈ (◡𝐴 “ ℕ) ↔ (𝐴‘𝑡) ∈ ℕ))
31	30	biimpar 482	. . . 4 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴‘𝑡) ∈ ℕ) → 𝑡 ∈ (◡𝐴 “ ℕ))
32	22	simp3bi 1164	. . . . . 6 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡𝐴 “ ℕ) ⊆ 𝐽)
33	32	sselda 3936	. . . . 5 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (◡𝐴 “ ℕ)) → 𝑡 ∈ 𝐽)
34	5	simprbi 502	. . . . 5 ⊢ (𝑡 ∈ 𝐽 → ¬ 2 ∥ 𝑡)
35	33, 34	syl 18	. . . 4 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (◡𝐴 “ ℕ)) → ¬ 2 ∥ 𝑡)
36	12, 31, 35	syl2anc 595	. . 3 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) ∧ (𝐴‘𝑡) ∈ ℕ) → ¬ 2 ∥ 𝑡)
37	11, 36	pm2.65da 828	. 2 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → ¬ (𝐴‘𝑡) ∈ ℕ)
38	25	adantr 485	. . . 4 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → 𝐴:ℕ⟶ℕ0)
39	13	adantl 486	. . . 4 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → 𝑡 ∈ ℕ)
40	38, 39	ffvelcdmd 7080	. . 3 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴‘𝑡) ∈ ℕ0)
41		elnn0 12512	. . 3 ⊢ ((𝐴‘𝑡) ∈ ℕ0 ↔ ((𝐴‘𝑡) ∈ ℕ ∨ (𝐴‘𝑡) = 0))
42	40, 41	sylib 221	. 2 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → ((𝐴‘𝑡) ∈ ℕ ∨ (𝐴‘𝑡) = 0))
43		orel1 901	. 2 ⊢ (¬ (𝐴‘𝑡) ∈ ℕ → (((𝐴‘𝑡) ∈ ℕ ∨ (𝐴‘𝑡) = 0) → (𝐴‘𝑡) = 0))
44	37, 42, 43	sylc 66	1 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴‘𝑡) = 0)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1569   ∈ wcel 2142  {cab 2740  ∀wral 3078  {crab 3415   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4561   class class class wbr 5108  {copab 5172   ↦ cmpt 5191  ^◡ccnv 5659   “ cima 5663   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7412   ∈ cmpo 7414   supp csupp 8154   ↑_m cmap 8822  Fincfn 8941  0cc0 11106  1c1 11107   · cmul 11111   ≤ cle 11250  ℕcn 12239  2c2 12301  ℕ₀cn0 12510  ↑cexp 14104  Σcsu 15744   ∥ cdvds 16316

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-mulcl 11168  ax-i2m1 11174

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-map 8824  df-n0 12511

This theorem is used by:  eulerpartlemgh  34777