Theorem eulerpartlem1 34348

Description: Lemma for eulerpart 34363. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)

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))})

Assertion

Ref	Expression
eulerpartlem1	⊢ 𝑀:𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)

Distinct variable groups:   𝑥,𝑟,𝑦,𝐽   𝐻,𝑟

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

Proof of Theorem eulerpartlem1

Step	Hyp	Ref	Expression
1		eulerpart.j	. . . 4 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
2		nnex 12122	. . . 4 ⊢ ℕ ∈ V
3	1, 2	rabex2 5276	. . 3 ⊢ 𝐽 ∈ V
4		nn0ex 12378	. . 3 ⊢ ℕ0 ∈ V
5		eqid 2729	. . 3 ⊢ (𝑟 ∈ (𝒫 ℕ0 ↑m 𝐽) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) = (𝑟 ∈ (𝒫 ℕ0 ↑m 𝐽) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})
6		eulerpart.h	. . 3 ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
7	3, 4, 5, 6	fpwrelmapffs 32669	. 2 ⊢ ((𝑟 ∈ (𝒫 ℕ0 ↑m 𝐽) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ↾ 𝐻):𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)
8		eulerpart.m	. . . 4 ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})
9		ssrab2 4027	. . . . . . 7 ⊢ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} ⊆ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽)
10	4	pwex 5315	. . . . . . . 8 ⊢ 𝒫 ℕ0 ∈ V
11		inss1 4184	. . . . . . . 8 ⊢ (𝒫 ℕ0 ∩ Fin) ⊆ 𝒫 ℕ0
12		mapss 8807	. . . . . . . 8 ⊢ ((𝒫 ℕ0 ∈ V ∧ (𝒫 ℕ0 ∩ Fin) ⊆ 𝒫 ℕ0) → ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ⊆ (𝒫 ℕ0 ↑m 𝐽))
13	10, 11, 12	mp2an 692	. . . . . . 7 ⊢ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ⊆ (𝒫 ℕ0 ↑m 𝐽)
14	9, 13	sstri 3941	. . . . . 6 ⊢ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} ⊆ (𝒫 ℕ0 ↑m 𝐽)
15	6, 14	eqsstri 3978	. . . . 5 ⊢ 𝐻 ⊆ (𝒫 ℕ0 ↑m 𝐽)
16		resmpt 5982	. . . . 5 ⊢ (𝐻 ⊆ (𝒫 ℕ0 ↑m 𝐽) → ((𝑟 ∈ (𝒫 ℕ0 ↑m 𝐽) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ↾ 𝐻) = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}))
17	15, 16	ax-mp 5	. . . 4 ⊢ ((𝑟 ∈ (𝒫 ℕ0 ↑m 𝐽) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ↾ 𝐻) = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})
18	8, 17	eqtr4i 2755	. . 3 ⊢ 𝑀 = ((𝑟 ∈ (𝒫 ℕ0 ↑m 𝐽) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ↾ 𝐻)
19		f1oeq1 6746	. . 3 ⊢ (𝑀 = ((𝑟 ∈ (𝒫 ℕ0 ↑m 𝐽) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ↾ 𝐻) → (𝑀:𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) ↔ ((𝑟 ∈ (𝒫 ℕ0 ↑m 𝐽) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ↾ 𝐻):𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)))
20	18, 19	ax-mp 5	. 2 ⊢ (𝑀:𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) ↔ ((𝑟 ∈ (𝒫 ℕ0 ↑m 𝐽) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ↾ 𝐻):𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin))
21	7, 20	mpbir 231	1 ⊢ 𝑀:𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3392  Vcvv 3433   ∩ cin 3898   ⊆ wss 3899  ∅c0 4280  𝒫 cpw 4547   class class class wbr 5088  {copab 5150   ↦ cmpt 5169   × cxp 5611  ^◡ccnv 5612   ↾ cres 5615   “ cima 5616  –1-1-onto→wf1o 6475  ‘cfv 6476  (class class class)co 7340   ∈ cmpo 7342   supp csupp 8084   ↑_m cmap 8744  Fincfn 8863  1c1 10998   · cmul 11002   ≤ cle 11138  ℕcn 12116  2c2 12171  ℕ₀cn0 12372  ↑cexp 13956  Σcsu 15580   ∥ cdvds 16150

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5214  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662  ax-ac2 10345  ax-cnex 11053  ax-1cn 11055  ax-addcl 11057

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4895  df-iun 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-tr 5196  df-id 5508  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5566  df-se 5567  df-we 5568  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-isom 6485  df-riota 7297  df-ov 7343  df-oprab 7344  df-mpo 7345  df-om 7791  df-1st 7915  df-2nd 7916  df-supp 8085  df-frecs 8205  df-wrecs 8236  df-recs 8285  df-rdg 8323  df-1o 8379  df-er 8616  df-map 8746  df-en 8864  df-dom 8865  df-fin 8867  df-card 9823  df-acn 9826  df-ac 9998  df-nn 12117  df-n0 12373

This theorem is referenced by:  eulerpartgbij  34353  eulerpartlemgvv  34357  eulerpartlemgf  34360