Theorem eulerpartlemgu 32344

Description: Lemma for eulerpart 32349: Rewriting the 𝑈 set for an odd partition Note that interestingly, this proof reuses marypha2lem2 9195. (Contributed by Thierry Arnoux, 10-Aug-2018.)

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}
eulerpart.g	⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
eulerpartlemgh.1	⊢ 𝑈 = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))

Assertion

Ref	Expression
eulerpartlemgu	⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝑈 = {⟨𝑡, 𝑛⟩ ∣ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)‘𝑡))})

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

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

Proof of Theorem eulerpartlemgu

Step	Hyp	Ref	Expression
1		eulerpartlemgh.1	. 2 ⊢ 𝑈 = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))
2		eulerpart.p	. . . . . . . . . . 11 ⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
3		eulerpart.o	. . . . . . . . . . 11 ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
4		eulerpart.d	. . . . . . . . . . 11 ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}
5		eulerpart.j	. . . . . . . . . . 11 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
6		eulerpart.f	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
7		eulerpart.h	. . . . . . . . . . 11 ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
8		eulerpart.m	. . . . . . . . . . 11 ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})
9		eulerpart.r	. . . . . . . . . . 11 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
10		eulerpart.t	. . . . . . . . . . 11 ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}
11	2, 3, 4, 5, 6, 7, 8, 9, 10	eulerpartlemt0 32336	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))
12	11	simp1bi 1144	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴 ∈ (ℕ0 ↑m ℕ))
13		elmapi 8637	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℕ0 ↑m ℕ) → 𝐴:ℕ⟶ℕ0)
14	12, 13	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
15	14	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝐴:ℕ⟶ℕ0)
16	15	ffund 6604	. . . . . 6 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → Fun 𝐴)
17		inss1 4162	. . . . . . . . 9 ⊢ ((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ (◡𝐴 “ ℕ)
18		cnvimass 5989	. . . . . . . . . 10 ⊢ (◡𝐴 “ ℕ) ⊆ dom 𝐴
19	18, 14	fssdm 6620	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡𝐴 “ ℕ) ⊆ ℕ)
20	17, 19	sstrid 3932	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ ℕ)
21	20	sselda 3921	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ℕ)
22	14	fdmd 6611	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → dom 𝐴 = ℕ)
23	22	eleq2d 2824	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑡 ∈ dom 𝐴 ↔ 𝑡 ∈ ℕ))
24	23	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → (𝑡 ∈ dom 𝐴 ↔ 𝑡 ∈ ℕ))
25	21, 24	mpbird 256	. . . . . 6 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ dom 𝐴)
26		fvco 6866	. . . . . 6 ⊢ ((Fun 𝐴 ∧ 𝑡 ∈ dom 𝐴) → ((bits ∘ 𝐴)‘𝑡) = (bits‘(𝐴‘𝑡)))
27	16, 25, 26	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → ((bits ∘ 𝐴)‘𝑡) = (bits‘(𝐴‘𝑡)))
28	27	xpeq2d 5619	. . . 4 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → ({𝑡} × ((bits ∘ 𝐴)‘𝑡)) = ({𝑡} × (bits‘(𝐴‘𝑡))))
29	28	iuneq2dv 4948	. . 3 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × ((bits ∘ 𝐴)‘𝑡)) = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))))
30		eqid 2738	. . . 4 ⊢ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × ((bits ∘ 𝐴)‘𝑡)) = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × ((bits ∘ 𝐴)‘𝑡))
31	30	marypha2lem2 9195	. . 3 ⊢ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × ((bits ∘ 𝐴)‘𝑡)) = {⟨𝑡, 𝑛⟩ ∣ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)‘𝑡))}
32	29, 31	eqtr3di 2793	. 2 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) = {⟨𝑡, 𝑛⟩ ∣ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)‘𝑡))})
33	1, 32	eqtrid 2790	1 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝑈 = {⟨𝑡, 𝑛⟩ ∣ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)‘𝑡))})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  {crab 3068   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561  ∪ ciun 4924   class class class wbr 5074  {copab 5136   ↦ cmpt 5157   × cxp 5587  ^◡ccnv 5588  dom cdm 5589   ↾ cres 5591   “ cima 5592   ∘ ccom 5593  Fun wfun 6427  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277   supp csupp 7977   ↑_m cmap 8615  Fincfn 8733  1c1 10872   · cmul 10876   ≤ cle 11010  ℕcn 11973  2c2 12028  ℕ₀cn0 12233  ↑cexp 13782  Σcsu 15397   ∥ cdvds 15963  bitscbits 16126  𝟭cind 31978

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

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

This theorem is referenced by: (None)