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Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpartlemd | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 30572: 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 6228 | . . . . 5 ⊢ (𝑔 = 𝐴 → (𝑔‘𝑛) = (𝐴‘𝑛)) | |
2 | 1 | breq1d 4695 | . . . 4 ⊢ (𝑔 = 𝐴 → ((𝑔‘𝑛) ≤ 1 ↔ (𝐴‘𝑛) ≤ 1)) |
3 | 2 | ralbidv 3015 | . . 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 3399 | . 2 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ 𝑃 ∧ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1)) |
6 | 2z 11447 | . . . . . . . . 9 ⊢ 2 ∈ ℤ | |
7 | fzoval 12510 | . . . . . . . . 9 ⊢ (2 ∈ ℤ → (0..^2) = (0...(2 − 1))) | |
8 | 6, 7 | ax-mp 5 | . . . . . . . 8 ⊢ (0..^2) = (0...(2 − 1)) |
9 | fzo0to2pr 12593 | . . . . . . . 8 ⊢ (0..^2) = {0, 1} | |
10 | 2m1e1 11173 | . . . . . . . . 9 ⊢ (2 − 1) = 1 | |
11 | 10 | oveq2i 6701 | . . . . . . . 8 ⊢ (0...(2 − 1)) = (0...1) |
12 | 8, 9, 11 | 3eqtr3i 2681 | . . . . . . 7 ⊢ {0, 1} = (0...1) |
13 | 12 | eleq2i 2722 | . . . . . 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 30553 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝑃 ↔ (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘) = 𝑁)) |
16 | 15 | simp1bi 1096 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑃 → 𝐴:ℕ⟶ℕ0) |
17 | 16 | ffvelrnda 6399 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ ℕ0) |
18 | 1nn0 11346 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
19 | elfz2nn0 12469 | . . . . . . . . 9 ⊢ ((𝐴‘𝑛) ∈ (0...1) ↔ ((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0 ∧ (𝐴‘𝑛) ≤ 1)) | |
20 | df-3an 1056 | . . . . . . . . 9 ⊢ (((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0 ∧ (𝐴‘𝑛) ≤ 1) ↔ (((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (𝐴‘𝑛) ≤ 1)) | |
21 | 19, 20 | bitri 264 | . . . . . . . 8 ⊢ ((𝐴‘𝑛) ∈ (0...1) ↔ (((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (𝐴‘𝑛) ≤ 1)) |
22 | 21 | baib 964 | . . . . . . 7 ⊢ (((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0) → ((𝐴‘𝑛) ∈ (0...1) ↔ (𝐴‘𝑛) ≤ 1)) |
23 | 17, 18, 22 | sylancl 695 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛) ∈ (0...1) ↔ (𝐴‘𝑛) ≤ 1)) |
24 | 13, 23 | syl5rbb 273 | . . . . 5 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛) ≤ 1 ↔ (𝐴‘𝑛) ∈ {0, 1})) |
25 | 24 | ralbidva 3014 | . . . 4 ⊢ (𝐴 ∈ 𝑃 → (∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1 ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ∈ {0, 1})) |
26 | ffun 6086 | . . . . . 6 ⊢ (𝐴:ℕ⟶ℕ0 → Fun 𝐴) | |
27 | 16, 26 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ 𝑃 → Fun 𝐴) |
28 | fdm 6089 | . . . . . 6 ⊢ (𝐴:ℕ⟶ℕ0 → dom 𝐴 = ℕ) | |
29 | eqimss2 3691 | . . . . . 6 ⊢ (dom 𝐴 = ℕ → ℕ ⊆ dom 𝐴) | |
30 | 16, 28, 29 | 3syl 18 | . . . . 5 ⊢ (𝐴 ∈ 𝑃 → ℕ ⊆ dom 𝐴) |
31 | funimass4 6286 | . . . . 5 ⊢ ((Fun 𝐴 ∧ ℕ ⊆ dom 𝐴) → ((𝐴 “ ℕ) ⊆ {0, 1} ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ∈ {0, 1})) | |
32 | 27, 30, 31 | syl2anc 694 | . . . 4 ⊢ (𝐴 ∈ 𝑃 → ((𝐴 “ ℕ) ⊆ {0, 1} ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ∈ {0, 1})) |
33 | 25, 32 | bitr4d 271 | . . 3 ⊢ (𝐴 ∈ 𝑃 → (∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1 ↔ (𝐴 “ ℕ) ⊆ {0, 1})) |
34 | 33 | pm5.32i 670 | . 2 ⊢ ((𝐴 ∈ 𝑃 ∧ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1) ↔ (𝐴 ∈ 𝑃 ∧ (𝐴 “ ℕ) ⊆ {0, 1})) |
35 | 5, 34 | bitri 264 | 1 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ 𝑃 ∧ (𝐴 “ ℕ) ⊆ {0, 1})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ∀wral 2941 {crab 2945 ⊆ wss 3607 {cpr 4212 class class class wbr 4685 ◡ccnv 5142 dom cdm 5143 “ cima 5146 Fun wfun 5920 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 ↑𝑚 cmap 7899 Fincfn 7997 0cc0 9974 1c1 9975 · cmul 9979 ≤ cle 10113 − cmin 10304 ℕcn 11058 2c2 11108 ℕ0cn0 11330 ℤcz 11415 ...cfz 12364 ..^cfzo 12504 Σcsu 14460 ∥ cdvds 15027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-fzo 12505 df-seq 12842 df-sum 14461 |
This theorem is referenced by: eulerpartlemn 30571 |
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