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Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpartlemt0 | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 33845. (Contributed by Thierry Arnoux, 19-Sep-2017.) |
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} |
Ref | Expression |
---|---|
eulerpartlemt0 | ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnveq 5873 | . . . . . 6 ⊢ (𝑓 = 𝐴 → ◡𝑓 = ◡𝐴) | |
2 | 1 | imaeq1d 6058 | . . . . 5 ⊢ (𝑓 = 𝐴 → (◡𝑓 “ ℕ) = (◡𝐴 “ ℕ)) |
3 | 2 | sseq1d 4013 | . . . 4 ⊢ (𝑓 = 𝐴 → ((◡𝑓 “ ℕ) ⊆ 𝐽 ↔ (◡𝐴 “ ℕ) ⊆ 𝐽)) |
4 | eulerpart.t | . . . 4 ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽} | |
5 | 3, 4 | elrab2 3686 | . . 3 ⊢ (𝐴 ∈ 𝑇 ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ⊆ 𝐽)) |
6 | 2 | eleq1d 2817 | . . . 4 ⊢ (𝑓 = 𝐴 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐴 “ ℕ) ∈ Fin)) |
7 | eulerpart.r | . . . 4 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
8 | 6, 7 | elab4g 3673 | . . 3 ⊢ (𝐴 ∈ 𝑅 ↔ (𝐴 ∈ V ∧ (◡𝐴 “ ℕ) ∈ Fin)) |
9 | 5, 8 | anbi12i 626 | . 2 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ∈ 𝑅) ↔ ((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (◡𝐴 “ ℕ) ∈ Fin))) |
10 | elin 3964 | . 2 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ 𝑇 ∧ 𝐴 ∈ 𝑅)) | |
11 | elex 3492 | . . . . 5 ⊢ (𝐴 ∈ (ℕ0 ↑m ℕ) → 𝐴 ∈ V) | |
12 | 11 | pm4.71i 559 | . . . 4 ⊢ (𝐴 ∈ (ℕ0 ↑m ℕ) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ 𝐴 ∈ V)) |
13 | 12 | anbi1i 623 | . . 3 ⊢ ((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ ((◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽)) ↔ ((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ 𝐴 ∈ V) ∧ ((◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))) |
14 | 3anass 1094 | . . 3 ⊢ ((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ ((◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))) | |
15 | an42 654 | . . 3 ⊢ (((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (◡𝐴 “ ℕ) ∈ Fin)) ↔ ((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ 𝐴 ∈ V) ∧ ((◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))) | |
16 | 13, 14, 15 | 3bitr4i 303 | . 2 ⊢ ((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽) ↔ ((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (◡𝐴 “ ℕ) ∈ Fin))) |
17 | 9, 10, 16 | 3bitr4i 303 | 1 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 {cab 2708 ∀wral 3060 {crab 3431 Vcvv 3473 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 class class class wbr 5148 {copab 5210 ↦ cmpt 5231 ◡ccnv 5675 “ cima 5679 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 supp csupp 8151 ↑m cmap 8826 Fincfn 8945 1c1 11117 · cmul 11121 ≤ cle 11256 ℕcn 12219 2c2 12274 ℕ0cn0 12479 ↑cexp 14034 Σcsu 15639 ∥ cdvds 16204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 |
This theorem is referenced by: eulerpartlemf 33833 eulerpartlemt 33834 eulerpartlemmf 33838 eulerpartlemgvv 33839 eulerpartlemgu 33840 eulerpartlemgh 33841 eulerpartlemgs2 33843 eulerpartlemn 33844 |
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