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Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpartlemd | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 34347: 𝐷 is the set of distinct part. of 𝑁. (Contributed by Thierry Arnoux, 11-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 |
---|---|
eulerpartlemd | ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ 𝑃 ∧ (𝐴 “ ℕ) ⊆ {0, 1})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6919 | . . . . 5 ⊢ (𝑔 = 𝐴 → (𝑔‘𝑛) = (𝐴‘𝑛)) | |
2 | 1 | breq1d 5176 | . . . 4 ⊢ (𝑔 = 𝐴 → ((𝑔‘𝑛) ≤ 1 ↔ (𝐴‘𝑛) ≤ 1)) |
3 | 2 | ralbidv 3184 | . . 3 ⊢ (𝑔 = 𝐴 → (∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1 ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1)) |
4 | eulerpart.d | . . 3 ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} | |
5 | 3, 4 | elrab2 3711 | . 2 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ 𝑃 ∧ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1)) |
6 | 2z 12675 | . . . . . . . . 9 ⊢ 2 ∈ ℤ | |
7 | fzoval 13717 | . . . . . . . . 9 ⊢ (2 ∈ ℤ → (0..^2) = (0...(2 − 1))) | |
8 | 6, 7 | ax-mp 5 | . . . . . . . 8 ⊢ (0..^2) = (0...(2 − 1)) |
9 | fzo0to2pr 13801 | . . . . . . . 8 ⊢ (0..^2) = {0, 1} | |
10 | 2m1e1 12419 | . . . . . . . . 9 ⊢ (2 − 1) = 1 | |
11 | 10 | oveq2i 7459 | . . . . . . . 8 ⊢ (0...(2 − 1)) = (0...1) |
12 | 8, 9, 11 | 3eqtr3i 2776 | . . . . . . 7 ⊢ {0, 1} = (0...1) |
13 | 12 | eleq2i 2836 | . . . . . 6 ⊢ ((𝐴‘𝑛) ∈ {0, 1} ↔ (𝐴‘𝑛) ∈ (0...1)) |
14 | eulerpart.p | . . . . . . . . . 10 ⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} | |
15 | 14 | eulerpartleme 34328 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝑃 ↔ (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘) = 𝑁)) |
16 | 15 | simp1bi 1145 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑃 → 𝐴:ℕ⟶ℕ0) |
17 | 16 | ffvelcdmda 7118 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ ℕ0) |
18 | 1nn0 12569 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
19 | elfz2nn0 13675 | . . . . . . . . 9 ⊢ ((𝐴‘𝑛) ∈ (0...1) ↔ ((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0 ∧ (𝐴‘𝑛) ≤ 1)) | |
20 | df-3an 1089 | . . . . . . . . 9 ⊢ (((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0 ∧ (𝐴‘𝑛) ≤ 1) ↔ (((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (𝐴‘𝑛) ≤ 1)) | |
21 | 19, 20 | bitri 275 | . . . . . . . 8 ⊢ ((𝐴‘𝑛) ∈ (0...1) ↔ (((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (𝐴‘𝑛) ≤ 1)) |
22 | 21 | baib 535 | . . . . . . 7 ⊢ (((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0) → ((𝐴‘𝑛) ∈ (0...1) ↔ (𝐴‘𝑛) ≤ 1)) |
23 | 17, 18, 22 | sylancl 585 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛) ∈ (0...1) ↔ (𝐴‘𝑛) ≤ 1)) |
24 | 13, 23 | bitr2id 284 | . . . . 5 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛) ≤ 1 ↔ (𝐴‘𝑛) ∈ {0, 1})) |
25 | 24 | ralbidva 3182 | . . . 4 ⊢ (𝐴 ∈ 𝑃 → (∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1 ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ∈ {0, 1})) |
26 | 16 | ffund 6751 | . . . . 5 ⊢ (𝐴 ∈ 𝑃 → Fun 𝐴) |
27 | fdm 6756 | . . . . . 6 ⊢ (𝐴:ℕ⟶ℕ0 → dom 𝐴 = ℕ) | |
28 | eqimss2 4068 | . . . . . 6 ⊢ (dom 𝐴 = ℕ → ℕ ⊆ dom 𝐴) | |
29 | 16, 27, 28 | 3syl 18 | . . . . 5 ⊢ (𝐴 ∈ 𝑃 → ℕ ⊆ dom 𝐴) |
30 | funimass4 6986 | . . . . 5 ⊢ ((Fun 𝐴 ∧ ℕ ⊆ dom 𝐴) → ((𝐴 “ ℕ) ⊆ {0, 1} ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ∈ {0, 1})) | |
31 | 26, 29, 30 | syl2anc 583 | . . . 4 ⊢ (𝐴 ∈ 𝑃 → ((𝐴 “ ℕ) ⊆ {0, 1} ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ∈ {0, 1})) |
32 | 25, 31 | bitr4d 282 | . . 3 ⊢ (𝐴 ∈ 𝑃 → (∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1 ↔ (𝐴 “ ℕ) ⊆ {0, 1})) |
33 | 32 | pm5.32i 574 | . 2 ⊢ ((𝐴 ∈ 𝑃 ∧ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1) ↔ (𝐴 ∈ 𝑃 ∧ (𝐴 “ ℕ) ⊆ {0, 1})) |
34 | 5, 33 | bitri 275 | 1 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ 𝑃 ∧ (𝐴 “ ℕ) ⊆ {0, 1})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 {crab 3443 ⊆ wss 3976 {cpr 4650 class class class wbr 5166 ◡ccnv 5699 dom cdm 5700 “ cima 5703 Fun wfun 6567 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ↑m cmap 8884 Fincfn 9003 0cc0 11184 1c1 11185 · cmul 11189 ≤ cle 11325 − cmin 11520 ℕcn 12293 2c2 12348 ℕ0cn0 12553 ℤcz 12639 ...cfz 13567 ..^cfzo 13711 Σcsu 15734 ∥ cdvds 16302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-sum 15735 |
This theorem is referenced by: eulerpartlemn 34346 |
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