Theorem eulerpartlemgu 34359

Description: Lemma for eulerpart 34364: Rewriting the 𝑈 set for an odd partition Note that interestingly, this proof reuses marypha2lem2 9474. (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 34351	. . . . . . . . . 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 8888	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℕ0 ↑m ℕ) → 𝐴:ℕ⟶ℕ0)
14	12, 13	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
15	14	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝐴:ℕ⟶ℕ0)
16	15	ffund 6741	. . . . . 6 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → Fun 𝐴)
17		inss1 4245	. . . . . . . . 9 ⊢ ((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ (◡𝐴 “ ℕ)
18		cnvimass 6102	. . . . . . . . . 10 ⊢ (◡𝐴 “ ℕ) ⊆ dom 𝐴
19	18, 14	fssdm 6756	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡𝐴 “ ℕ) ⊆ ℕ)
20	17, 19	sstrid 4007	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ ℕ)
21	20	sselda 3995	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ℕ)
22	14	fdmd 6747	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → dom 𝐴 = ℕ)
23	22	eleq2d 2825	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑡 ∈ dom 𝐴 ↔ 𝑡 ∈ ℕ))
24	23	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → (𝑡 ∈ dom 𝐴 ↔ 𝑡 ∈ ℕ))
25	21, 24	mpbird 257	. . . . . 6 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ dom 𝐴)
26		fvco 7007	. . . . . 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 5719	. . . 4 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → ({𝑡} × ((bits ∘ 𝐴)‘𝑡)) = ({𝑡} × (bits‘(𝐴‘𝑡))))
29	28	iuneq2dv 5021	. . 3 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × ((bits ∘ 𝐴)‘𝑡)) = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))))
30		eqid 2735	. . . 4 ⊢ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × ((bits ∘ 𝐴)‘𝑡)) = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × ((bits ∘ 𝐴)‘𝑡))
31	30	marypha2lem2 9474	. . 3 ⊢ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × ((bits ∘ 𝐴)‘𝑡)) = {⟨𝑡, 𝑛⟩ ∣ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)‘𝑡))}
32	29, 31	eqtr3di 2790	. 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 2787	1 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝑈 = {⟨𝑡, 𝑛⟩ ∣ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)‘𝑡))})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {cab 2712  ∀wral 3059  {crab 3433   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  𝒫 cpw 4605  {csn 4631  ∪ ciun 4996   class class class wbr 5148  {copab 5210   ↦ cmpt 5231   × cxp 5687  ^◡ccnv 5688  dom cdm 5689   ↾ cres 5691   “ cima 5692   ∘ ccom 5693  Fun wfun 6557  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433   supp csupp 8184   ↑_m cmap 8865  Fincfn 8984  1c1 11154   · cmul 11158   ≤ cle 11294  ℕcn 12264  2c2 12319  ℕ₀cn0 12524  ↑cexp 14099  Σcsu 15719   ∥ cdvds 16287  bitscbits 16453  𝟭cind 33991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867

This theorem is referenced by: (None)