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Mirrors > Home > ILE Home > Th. List > Mathboxes > redcwlpolemeq1 | GIF version |
Description: Lemma for redcwlpo 13567. A biconditionalized version of trilpolemeq1 13552. (Contributed by Jim Kingdon, 21-Jun-2024.) |
Ref | Expression |
---|---|
redcwlpolemeq1.f | ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1}) |
redcwlpolemeq1.a | ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) |
Ref | Expression |
---|---|
redcwlpolemeq1 | ⊢ (𝜑 → (𝐴 = 1 ↔ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | redcwlpolemeq1.f | . . . 4 ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1}) | |
2 | 1 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 1) → 𝐹:ℕ⟶{0, 1}) |
3 | redcwlpolemeq1.a | . . 3 ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) | |
4 | simpr 109 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 1) → 𝐴 = 1) | |
5 | 2, 3, 4 | trilpolemeq1 13552 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 1) → ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) |
6 | fveqeq2 5470 | . . . . . . 7 ⊢ (𝑥 = 𝑖 → ((𝐹‘𝑥) = 1 ↔ (𝐹‘𝑖) = 1)) | |
7 | simplr 520 | . . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) | |
8 | simpr 109 | . . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ) | |
9 | 6, 7, 8 | rspcdva 2818 | . . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) = 1) |
10 | 9 | oveq2d 5830 | . . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((1 / (2↑𝑖)) · 1)) |
11 | 2nn 8973 | . . . . . . . . . 10 ⊢ 2 ∈ ℕ | |
12 | 11 | a1i 9 | . . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → 2 ∈ ℕ) |
13 | 8 | nnnn0d 9122 | . . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ0) |
14 | 12, 13 | nnexpcld 10550 | . . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈ ℕ) |
15 | 14 | nncnd 8826 | . . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈ ℂ) |
16 | 14 | nnap0d 8858 | . . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → (2↑𝑖) # 0) |
17 | 15, 16 | recclapd 8633 | . . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℂ) |
18 | 17 | mulid1d 7874 | . . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · 1) = (1 / (2↑𝑖))) |
19 | 10, 18 | eqtrd 2187 | . . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = (1 / (2↑𝑖))) |
20 | 19 | sumeq2dv 11242 | . . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) → Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = Σ𝑖 ∈ ℕ (1 / (2↑𝑖))) |
21 | geoihalfsum 11396 | . . . 4 ⊢ Σ𝑖 ∈ ℕ (1 / (2↑𝑖)) = 1 | |
22 | 21 | eqcomi 2158 | . . 3 ⊢ 1 = Σ𝑖 ∈ ℕ (1 / (2↑𝑖)) |
23 | 20, 3, 22 | 3eqtr4g 2212 | . 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) → 𝐴 = 1) |
24 | 5, 23 | impbida 586 | 1 ⊢ (𝜑 → (𝐴 = 1 ↔ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 2125 ∀wral 2432 {cpr 3557 ⟶wf 5159 ‘cfv 5163 (class class class)co 5814 0cc0 7711 1c1 7712 · cmul 7716 / cdiv 8524 ℕcn 8812 2c2 8863 ↑cexp 10396 Σcsu 11227 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-coll 4075 ax-sep 4078 ax-nul 4086 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-iinf 4541 ax-cnex 7802 ax-resscn 7803 ax-1cn 7804 ax-1re 7805 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-mulrcl 7810 ax-addcom 7811 ax-mulcom 7812 ax-addass 7813 ax-mulass 7814 ax-distr 7815 ax-i2m1 7816 ax-0lt1 7817 ax-1rid 7818 ax-0id 7819 ax-rnegex 7820 ax-precex 7821 ax-cnre 7822 ax-pre-ltirr 7823 ax-pre-ltwlin 7824 ax-pre-lttrn 7825 ax-pre-apti 7826 ax-pre-ltadd 7827 ax-pre-mulgt0 7828 ax-pre-mulext 7829 ax-arch 7830 ax-caucvg 7831 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-reu 2439 df-rmo 2440 df-rab 2441 df-v 2711 df-sbc 2934 df-csb 3028 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-nul 3391 df-if 3502 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-int 3804 df-iun 3847 df-br 3962 df-opab 4022 df-mpt 4023 df-tr 4059 df-id 4248 df-po 4251 df-iso 4252 df-iord 4321 df-on 4323 df-ilim 4324 df-suc 4326 df-iom 4544 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-res 4591 df-ima 4592 df-iota 5128 df-fun 5165 df-fn 5166 df-f 5167 df-f1 5168 df-fo 5169 df-f1o 5170 df-fv 5171 df-isom 5172 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-1st 6078 df-2nd 6079 df-recs 6242 df-irdg 6307 df-frec 6328 df-1o 6353 df-oadd 6357 df-er 6469 df-en 6675 df-dom 6676 df-fin 6677 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 df-le 7897 df-sub 8027 df-neg 8028 df-reap 8429 df-ap 8436 df-div 8525 df-inn 8813 df-2 8871 df-3 8872 df-4 8873 df-n0 9070 df-z 9147 df-uz 9419 df-q 9507 df-rp 9539 df-ico 9776 df-fz 9891 df-fzo 10020 df-seqfrec 10323 df-exp 10397 df-ihash 10627 df-cj 10719 df-re 10720 df-im 10721 df-rsqrt 10875 df-abs 10876 df-clim 11153 df-sumdc 11228 |
This theorem is referenced by: redcwlpo 13567 neapmkvlem 13578 |
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