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Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpartlemo | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 33212: 𝑂 is the set of odd partitions of 𝑁. (Contributed by Thierry Arnoux, 10-Aug-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} |
Ref | Expression |
---|---|
eulerpartlemo | ⊢ (𝐴 ∈ 𝑂 ↔ (𝐴 ∈ 𝑃 ∧ ∀𝑛 ∈ (◡𝐴 “ ℕ) ¬ 2 ∥ 𝑛)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnveq 5865 | . . . 4 ⊢ (𝑔 = 𝐴 → ◡𝑔 = ◡𝐴) | |
2 | 1 | imaeq1d 6048 | . . 3 ⊢ (𝑔 = 𝐴 → (◡𝑔 “ ℕ) = (◡𝐴 “ ℕ)) |
3 | 2 | raleqdv 3324 | . 2 ⊢ (𝑔 = 𝐴 → (∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛 ↔ ∀𝑛 ∈ (◡𝐴 “ ℕ) ¬ 2 ∥ 𝑛)) |
4 | eulerpart.o | . 2 ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} | |
5 | 3, 4 | elrab2 3682 | 1 ⊢ (𝐴 ∈ 𝑂 ↔ (𝐴 ∈ 𝑃 ∧ ∀𝑛 ∈ (◡𝐴 “ ℕ) ¬ 2 ∥ 𝑛)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3060 {crab 3431 class class class wbr 5141 ◡ccnv 5668 “ cima 5672 ‘cfv 6532 (class class class)co 7393 ↑m cmap 8803 Fincfn 8922 1c1 11093 · cmul 11097 ≤ cle 11231 ℕcn 12194 2c2 12249 ℕ0cn0 12454 Σcsu 15614 ∥ cdvds 16179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 |
This theorem is referenced by: eulerpartlemr 33204 |
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