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Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpartlemd | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 31318: 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 6496 | . . . . 5 ⊢ (𝑔 = 𝐴 → (𝑔‘𝑛) = (𝐴‘𝑛)) | |
2 | 1 | breq1d 4936 | . . . 4 ⊢ (𝑔 = 𝐴 → ((𝑔‘𝑛) ≤ 1 ↔ (𝐴‘𝑛) ≤ 1)) |
3 | 2 | ralbidv 3142 | . . 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 3594 | . 2 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ 𝑃 ∧ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1)) |
6 | 2z 11826 | . . . . . . . . 9 ⊢ 2 ∈ ℤ | |
7 | fzoval 12854 | . . . . . . . . 9 ⊢ (2 ∈ ℤ → (0..^2) = (0...(2 − 1))) | |
8 | 6, 7 | ax-mp 5 | . . . . . . . 8 ⊢ (0..^2) = (0...(2 − 1)) |
9 | fzo0to2pr 12936 | . . . . . . . 8 ⊢ (0..^2) = {0, 1} | |
10 | 2m1e1 11572 | . . . . . . . . 9 ⊢ (2 − 1) = 1 | |
11 | 10 | oveq2i 6986 | . . . . . . . 8 ⊢ (0...(2 − 1)) = (0...1) |
12 | 8, 9, 11 | 3eqtr3i 2805 | . . . . . . 7 ⊢ {0, 1} = (0...1) |
13 | 12 | eleq2i 2852 | . . . . . 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 31299 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝑃 ↔ (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘) = 𝑁)) |
16 | 15 | simp1bi 1126 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑃 → 𝐴:ℕ⟶ℕ0) |
17 | 16 | ffvelrnda 6675 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ ℕ0) |
18 | 1nn0 11724 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
19 | elfz2nn0 12813 | . . . . . . . . 9 ⊢ ((𝐴‘𝑛) ∈ (0...1) ↔ ((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0 ∧ (𝐴‘𝑛) ≤ 1)) | |
20 | df-3an 1071 | . . . . . . . . 9 ⊢ (((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0 ∧ (𝐴‘𝑛) ≤ 1) ↔ (((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (𝐴‘𝑛) ≤ 1)) | |
21 | 19, 20 | bitri 267 | . . . . . . . 8 ⊢ ((𝐴‘𝑛) ∈ (0...1) ↔ (((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (𝐴‘𝑛) ≤ 1)) |
22 | 21 | baib 528 | . . . . . . 7 ⊢ (((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0) → ((𝐴‘𝑛) ∈ (0...1) ↔ (𝐴‘𝑛) ≤ 1)) |
23 | 17, 18, 22 | sylancl 578 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛) ∈ (0...1) ↔ (𝐴‘𝑛) ≤ 1)) |
24 | 13, 23 | syl5rbb 276 | . . . . 5 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛) ≤ 1 ↔ (𝐴‘𝑛) ∈ {0, 1})) |
25 | 24 | ralbidva 3141 | . . . 4 ⊢ (𝐴 ∈ 𝑃 → (∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1 ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ∈ {0, 1})) |
26 | 16 | ffund 6346 | . . . . 5 ⊢ (𝐴 ∈ 𝑃 → Fun 𝐴) |
27 | fdm 6350 | . . . . . 6 ⊢ (𝐴:ℕ⟶ℕ0 → dom 𝐴 = ℕ) | |
28 | eqimss2 3909 | . . . . . 6 ⊢ (dom 𝐴 = ℕ → ℕ ⊆ dom 𝐴) | |
29 | 16, 27, 28 | 3syl 18 | . . . . 5 ⊢ (𝐴 ∈ 𝑃 → ℕ ⊆ dom 𝐴) |
30 | funimass4 6558 | . . . . 5 ⊢ ((Fun 𝐴 ∧ ℕ ⊆ dom 𝐴) → ((𝐴 “ ℕ) ⊆ {0, 1} ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ∈ {0, 1})) | |
31 | 26, 29, 30 | syl2anc 576 | . . . 4 ⊢ (𝐴 ∈ 𝑃 → ((𝐴 “ ℕ) ⊆ {0, 1} ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ∈ {0, 1})) |
32 | 25, 31 | bitr4d 274 | . . 3 ⊢ (𝐴 ∈ 𝑃 → (∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1 ↔ (𝐴 “ ℕ) ⊆ {0, 1})) |
33 | 32 | pm5.32i 567 | . 2 ⊢ ((𝐴 ∈ 𝑃 ∧ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1) ↔ (𝐴 ∈ 𝑃 ∧ (𝐴 “ ℕ) ⊆ {0, 1})) |
34 | 5, 33 | bitri 267 | 1 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ 𝑃 ∧ (𝐴 “ ℕ) ⊆ {0, 1})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ∀wral 3083 {crab 3087 ⊆ wss 3824 {cpr 4438 class class class wbr 4926 ◡ccnv 5403 dom cdm 5404 “ cima 5407 Fun wfun 6180 ⟶wf 6182 ‘cfv 6186 (class class class)co 6975 ↑𝑚 cmap 8205 Fincfn 8305 0cc0 10334 1c1 10335 · cmul 10339 ≤ cle 10474 − cmin 10669 ℕcn 11438 2c2 11494 ℕ0cn0 11706 ℤcz 11792 ...cfz 12707 ..^cfzo 12848 Σcsu 14902 ∥ cdvds 15466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-map 8207 df-en 8306 df-dom 8307 df-sdom 8308 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-n0 11707 df-z 11793 df-uz 12058 df-fz 12708 df-fzo 12849 df-seq 13184 df-sum 14903 |
This theorem is referenced by: eulerpartlemn 31317 |
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