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Mirrors > Home > ILE Home > Th. List > Mathboxes > trilpolemres | GIF version |
Description: Lemma for trilpo 13239. The result. (Contributed by Jim Kingdon, 23-Aug-2023.) |
Ref | Expression |
---|---|
trilpolemgt1.f | ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1}) |
trilpolemgt1.a | ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) |
trilpolemres.a | ⊢ (𝜑 → (𝐴 < 1 ∨ 𝐴 = 1 ∨ 1 < 𝐴)) |
Ref | Expression |
---|---|
trilpolemres | ⊢ (𝜑 → (∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0 ∨ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trilpolemgt1.f | . . . . 5 ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1}) | |
2 | 1 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐹:ℕ⟶{0, 1}) |
3 | trilpolemgt1.a | . . . 4 ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) | |
4 | simpr 109 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 < 1) | |
5 | 2, 3, 4 | trilpolemlt1 13237 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 1) → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0) |
6 | 5 | orcd 722 | . 2 ⊢ ((𝜑 ∧ 𝐴 < 1) → (∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0 ∨ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1)) |
7 | 1 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 1) → 𝐹:ℕ⟶{0, 1}) |
8 | simpr 109 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 1) → 𝐴 = 1) | |
9 | 7, 3, 8 | trilpolemeq1 13236 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 1) → ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) |
10 | 9 | olcd 723 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 1) → (∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0 ∨ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1)) |
11 | 1, 3 | trilpolemgt1 13235 | . . . 4 ⊢ (𝜑 → ¬ 1 < 𝐴) |
12 | 11 | pm2.21d 608 | . . 3 ⊢ (𝜑 → (1 < 𝐴 → (∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0 ∨ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1))) |
13 | 12 | imp 123 | . 2 ⊢ ((𝜑 ∧ 1 < 𝐴) → (∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0 ∨ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1)) |
14 | trilpolemres.a | . 2 ⊢ (𝜑 → (𝐴 < 1 ∨ 𝐴 = 1 ∨ 1 < 𝐴)) | |
15 | 6, 10, 13, 14 | mpjao3dan 1285 | 1 ⊢ (𝜑 → (∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0 ∨ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 697 ∨ w3o 961 = wceq 1331 ∀wral 2416 ∃wrex 2417 {cpr 3528 class class class wbr 3929 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 0cc0 7623 1c1 7624 · cmul 7628 < clt 7803 / cdiv 8435 ℕcn 8723 2c2 8774 ↑cexp 10295 Σcsu 11125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 ax-pre-mulext 7741 ax-arch 7742 ax-caucvg 7743 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-2o 6314 df-oadd 6317 df-er 6429 df-map 6544 df-en 6635 df-dom 6636 df-fin 6637 df-omni 7006 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-reap 8340 df-ap 8347 df-div 8436 df-inn 8724 df-2 8782 df-3 8783 df-4 8784 df-n0 8981 df-z 9058 df-uz 9330 df-q 9415 df-rp 9445 df-ico 9680 df-fz 9794 df-fzo 9923 df-seqfrec 10222 df-exp 10296 df-ihash 10525 df-cj 10617 df-re 10618 df-im 10619 df-rsqrt 10773 df-abs 10774 df-clim 11051 df-sumdc 11126 |
This theorem is referenced by: trilpo 13239 |
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