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Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpartlemd | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 34339: 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 6918 | . . . . 5 ⊢ (𝑔 = 𝐴 → (𝑔‘𝑛) = (𝐴‘𝑛)) | |
2 | 1 | breq1d 5179 | . . . 4 ⊢ (𝑔 = 𝐴 → ((𝑔‘𝑛) ≤ 1 ↔ (𝐴‘𝑛) ≤ 1)) |
3 | 2 | ralbidv 3180 | . . 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 3706 | . 2 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ 𝑃 ∧ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1)) |
6 | 2z 12671 | . . . . . . . . 9 ⊢ 2 ∈ ℤ | |
7 | fzoval 13713 | . . . . . . . . 9 ⊢ (2 ∈ ℤ → (0..^2) = (0...(2 − 1))) | |
8 | 6, 7 | ax-mp 5 | . . . . . . . 8 ⊢ (0..^2) = (0...(2 − 1)) |
9 | fzo0to2pr 13797 | . . . . . . . 8 ⊢ (0..^2) = {0, 1} | |
10 | 2m1e1 12415 | . . . . . . . . 9 ⊢ (2 − 1) = 1 | |
11 | 10 | oveq2i 7456 | . . . . . . . 8 ⊢ (0...(2 − 1)) = (0...1) |
12 | 8, 9, 11 | 3eqtr3i 2770 | . . . . . . 7 ⊢ {0, 1} = (0...1) |
13 | 12 | eleq2i 2830 | . . . . . 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 34320 | . . . . . . . . 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 7116 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ ℕ0) |
18 | 1nn0 12565 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
19 | elfz2nn0 13671 | . . . . . . . . 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 3178 | . . . 4 ⊢ (𝐴 ∈ 𝑃 → (∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1 ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ∈ {0, 1})) |
26 | 16 | ffund 6750 | . . . . 5 ⊢ (𝐴 ∈ 𝑃 → Fun 𝐴) |
27 | fdm 6755 | . . . . . 6 ⊢ (𝐴:ℕ⟶ℕ0 → dom 𝐴 = ℕ) | |
28 | eqimss2 4062 | . . . . . 6 ⊢ (dom 𝐴 = ℕ → ℕ ⊆ dom 𝐴) | |
29 | 16, 27, 28 | 3syl 18 | . . . . 5 ⊢ (𝐴 ∈ 𝑃 → ℕ ⊆ dom 𝐴) |
30 | funimass4 6985 | . . . . 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 2103 ∀wral 3063 {crab 3438 ⊆ wss 3970 {cpr 4650 class class class wbr 5169 ◡ccnv 5698 dom cdm 5699 “ cima 5702 Fun wfun 6566 ⟶wf 6568 ‘cfv 6572 (class class class)co 7445 ↑m cmap 8880 Fincfn 8999 0cc0 11180 1c1 11181 · cmul 11185 ≤ cle 11321 − cmin 11516 ℕcn 12289 2c2 12344 ℕ0cn0 12549 ℤcz 12635 ...cfz 13563 ..^cfzo 13707 Σcsu 15730 ∥ cdvds 16296 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-er 8759 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-n0 12550 df-z 12636 df-uz 12900 df-fz 13564 df-fzo 13708 df-seq 14049 df-sum 15731 |
This theorem is referenced by: eulerpartlemn 34338 |
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