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Mirrors > Home > ILE Home > Th. List > Mathboxes > trilpolemres | GIF version |
Description: Lemma for trilpo 14447. 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 276 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐹:ℕ⟶{0, 1}) |
3 | trilpolemgt1.a | . . . 4 ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) | |
4 | simpr 110 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 < 1) | |
5 | 2, 3, 4 | trilpolemlt1 14445 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 1) → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0) |
6 | 5 | orcd 733 | . 2 ⊢ ((𝜑 ∧ 𝐴 < 1) → (∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0 ∨ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1)) |
7 | 1 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 1) → 𝐹:ℕ⟶{0, 1}) |
8 | simpr 110 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 1) → 𝐴 = 1) | |
9 | 7, 3, 8 | trilpolemeq1 14444 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 1) → ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) |
10 | 9 | olcd 734 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 1) → (∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0 ∨ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1)) |
11 | 1, 3 | trilpolemgt1 14443 | . . . 4 ⊢ (𝜑 → ¬ 1 < 𝐴) |
12 | 11 | pm2.21d 619 | . . 3 ⊢ (𝜑 → (1 < 𝐴 → (∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0 ∨ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1))) |
13 | 12 | imp 124 | . 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 1307 | 1 ⊢ (𝜑 → (∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0 ∨ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ wo 708 ∨ w3o 977 = wceq 1353 ∀wral 2455 ∃wrex 2456 {cpr 3592 class class class wbr 4000 ⟶wf 5208 ‘cfv 5212 (class class class)co 5869 0cc0 7802 1c1 7803 · cmul 7807 < clt 7982 / cdiv 8618 ℕcn 8908 2c2 8959 ↑cexp 10505 Σcsu 11345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-frec 6386 df-1o 6411 df-2o 6412 df-oadd 6415 df-er 6529 df-map 6644 df-en 6735 df-dom 6736 df-fin 6737 df-omni 7127 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-rp 9641 df-ico 9881 df-fz 9996 df-fzo 10129 df-seqfrec 10432 df-exp 10506 df-ihash 10740 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-clim 11271 df-sumdc 11346 |
This theorem is referenced by: trilpo 14447 |
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