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Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpartlemd | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 31642: 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 6671 | . . . . 5 ⊢ (𝑔 = 𝐴 → (𝑔‘𝑛) = (𝐴‘𝑛)) | |
2 | 1 | breq1d 5078 | . . . 4 ⊢ (𝑔 = 𝐴 → ((𝑔‘𝑛) ≤ 1 ↔ (𝐴‘𝑛) ≤ 1)) |
3 | 2 | ralbidv 3199 | . . 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 3685 | . 2 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ 𝑃 ∧ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1)) |
6 | 2z 12017 | . . . . . . . . 9 ⊢ 2 ∈ ℤ | |
7 | fzoval 13042 | . . . . . . . . 9 ⊢ (2 ∈ ℤ → (0..^2) = (0...(2 − 1))) | |
8 | 6, 7 | ax-mp 5 | . . . . . . . 8 ⊢ (0..^2) = (0...(2 − 1)) |
9 | fzo0to2pr 13125 | . . . . . . . 8 ⊢ (0..^2) = {0, 1} | |
10 | 2m1e1 11766 | . . . . . . . . 9 ⊢ (2 − 1) = 1 | |
11 | 10 | oveq2i 7169 | . . . . . . . 8 ⊢ (0...(2 − 1)) = (0...1) |
12 | 8, 9, 11 | 3eqtr3i 2854 | . . . . . . 7 ⊢ {0, 1} = (0...1) |
13 | 12 | eleq2i 2906 | . . . . . 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 31623 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝑃 ↔ (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘) = 𝑁)) |
16 | 15 | simp1bi 1141 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑃 → 𝐴:ℕ⟶ℕ0) |
17 | 16 | ffvelrnda 6853 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ ℕ0) |
18 | 1nn0 11916 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
19 | elfz2nn0 13001 | . . . . . . . . 9 ⊢ ((𝐴‘𝑛) ∈ (0...1) ↔ ((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0 ∧ (𝐴‘𝑛) ≤ 1)) | |
20 | df-3an 1085 | . . . . . . . . 9 ⊢ (((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0 ∧ (𝐴‘𝑛) ≤ 1) ↔ (((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (𝐴‘𝑛) ≤ 1)) | |
21 | 19, 20 | bitri 277 | . . . . . . . 8 ⊢ ((𝐴‘𝑛) ∈ (0...1) ↔ (((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (𝐴‘𝑛) ≤ 1)) |
22 | 21 | baib 538 | . . . . . . 7 ⊢ (((𝐴‘𝑛) ∈ ℕ0 ∧ 1 ∈ ℕ0) → ((𝐴‘𝑛) ∈ (0...1) ↔ (𝐴‘𝑛) ≤ 1)) |
23 | 17, 18, 22 | sylancl 588 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛) ∈ (0...1) ↔ (𝐴‘𝑛) ≤ 1)) |
24 | 13, 23 | syl5rbb 286 | . . . . 5 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑛) ≤ 1 ↔ (𝐴‘𝑛) ∈ {0, 1})) |
25 | 24 | ralbidva 3198 | . . . 4 ⊢ (𝐴 ∈ 𝑃 → (∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1 ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ∈ {0, 1})) |
26 | 16 | ffund 6520 | . . . . 5 ⊢ (𝐴 ∈ 𝑃 → Fun 𝐴) |
27 | fdm 6524 | . . . . . 6 ⊢ (𝐴:ℕ⟶ℕ0 → dom 𝐴 = ℕ) | |
28 | eqimss2 4026 | . . . . . 6 ⊢ (dom 𝐴 = ℕ → ℕ ⊆ dom 𝐴) | |
29 | 16, 27, 28 | 3syl 18 | . . . . 5 ⊢ (𝐴 ∈ 𝑃 → ℕ ⊆ dom 𝐴) |
30 | funimass4 6732 | . . . . 5 ⊢ ((Fun 𝐴 ∧ ℕ ⊆ dom 𝐴) → ((𝐴 “ ℕ) ⊆ {0, 1} ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ∈ {0, 1})) | |
31 | 26, 29, 30 | syl2anc 586 | . . . 4 ⊢ (𝐴 ∈ 𝑃 → ((𝐴 “ ℕ) ⊆ {0, 1} ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ∈ {0, 1})) |
32 | 25, 31 | bitr4d 284 | . . 3 ⊢ (𝐴 ∈ 𝑃 → (∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1 ↔ (𝐴 “ ℕ) ⊆ {0, 1})) |
33 | 32 | pm5.32i 577 | . 2 ⊢ ((𝐴 ∈ 𝑃 ∧ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ≤ 1) ↔ (𝐴 ∈ 𝑃 ∧ (𝐴 “ ℕ) ⊆ {0, 1})) |
34 | 5, 33 | bitri 277 | 1 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ 𝑃 ∧ (𝐴 “ ℕ) ⊆ {0, 1})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 ⊆ wss 3938 {cpr 4571 class class class wbr 5068 ◡ccnv 5556 dom cdm 5557 “ cima 5560 Fun wfun 6351 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 Fincfn 8511 0cc0 10539 1c1 10540 · cmul 10544 ≤ cle 10678 − cmin 10872 ℕcn 11640 2c2 11695 ℕ0cn0 11900 ℤcz 11984 ...cfz 12895 ..^cfzo 13036 Σcsu 15044 ∥ cdvds 15609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-sum 15045 |
This theorem is referenced by: eulerpartlemn 31641 |
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