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Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpartlemt0 | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 29771. (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 5218 | . . . . . 6 ⊢ (𝑓 = 𝐴 → ◡𝑓 = ◡𝐴) | |
2 | 1 | imaeq1d 5384 | . . . . 5 ⊢ (𝑓 = 𝐴 → (◡𝑓 “ ℕ) = (◡𝐴 “ ℕ)) |
3 | 2 | sseq1d 3595 | . . . 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 3333 | . . 3 ⊢ (𝐴 ∈ 𝑇 ↔ (𝐴 ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡𝐴 “ ℕ) ⊆ 𝐽)) |
6 | 2 | eleq1d 2672 | . . . 4 ⊢ (𝑓 = 𝐴 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐴 “ ℕ) ∈ Fin)) |
7 | eulerpart.r | . . . 4 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
8 | 6, 7 | elab4g 3324 | . . 3 ⊢ (𝐴 ∈ 𝑅 ↔ (𝐴 ∈ V ∧ (◡𝐴 “ ℕ) ∈ Fin)) |
9 | 5, 8 | anbi12i 729 | . 2 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ∈ 𝑅) ↔ ((𝐴 ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (◡𝐴 “ ℕ) ∈ Fin))) |
10 | elin 3758 | . 2 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ 𝑇 ∧ 𝐴 ∈ 𝑅)) | |
11 | elex 3185 | . . . . 5 ⊢ (𝐴 ∈ (ℕ0 ↑𝑚 ℕ) → 𝐴 ∈ V) | |
12 | 11 | pm4.71i 662 | . . . 4 ⊢ (𝐴 ∈ (ℕ0 ↑𝑚 ℕ) ↔ (𝐴 ∈ (ℕ0 ↑𝑚 ℕ) ∧ 𝐴 ∈ V)) |
13 | 12 | anbi1i 727 | . . 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 1035 | . . 3 ⊢ ((𝐴 ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽) ↔ (𝐴 ∈ (ℕ0 ↑𝑚 ℕ) ∧ ((◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))) | |
15 | an42 862 | . . 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 291 | . 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 291 | 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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {cab 2596 ∀wral 2896 {crab 2900 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 class class class wbr 4583 {copab 4642 ↦ cmpt 4643 ◡ccnv 5037 “ cima 5041 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 supp csupp 7182 ↑𝑚 cmap 7744 Fincfn 7841 1c1 9816 · cmul 9820 ≤ cle 9954 ℕcn 10897 2c2 10947 ℕ0cn0 11169 ↑cexp 12722 Σcsu 14264 ∥ cdvds 14821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: eulerpartlemf 29759 eulerpartlemt 29760 eulerpartlemmf 29764 eulerpartlemgvv 29765 eulerpartlemgu 29766 eulerpartlemgh 29767 eulerpartlemgs2 29769 eulerpartlemn 29770 |
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