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Mirrors > Home > MPE Home > Th. List > ostth1 | Structured version Visualization version GIF version |
Description: - Lemma for ostth 26142: trivial case. (Not that the proof is trivial, but that we are proving that the function is trivial.) If 𝐹 is equal to 1 on the primes, then by complete induction and the multiplicative property abvmul 19529 of the absolute value, 𝐹 is equal to 1 on all the integers, and ostthlem1 26130 extends this to the other rational numbers. (Contributed by Mario Carneiro, 10-Sep-2014.) |
Ref | Expression |
---|---|
qrng.q | ⊢ 𝑄 = (ℂfld ↾s ℚ) |
qabsabv.a | ⊢ 𝐴 = (AbsVal‘𝑄) |
padic.j | ⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥))))) |
ostth.k | ⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1)) |
ostth.1 | ⊢ (𝜑 → 𝐹 ∈ 𝐴) |
ostth1.2 | ⊢ (𝜑 → ∀𝑛 ∈ ℕ ¬ 1 < (𝐹‘𝑛)) |
ostth1.3 | ⊢ (𝜑 → ∀𝑛 ∈ ℙ ¬ (𝐹‘𝑛) < 1) |
Ref | Expression |
---|---|
ostth1 | ⊢ (𝜑 → 𝐹 = 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qrng.q | . 2 ⊢ 𝑄 = (ℂfld ↾s ℚ) | |
2 | qabsabv.a | . 2 ⊢ 𝐴 = (AbsVal‘𝑄) | |
3 | ostth.1 | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐴) | |
4 | 1 | qdrng 26123 | . . 3 ⊢ 𝑄 ∈ DivRing |
5 | 1 | qrngbas 26122 | . . . 4 ⊢ ℚ = (Base‘𝑄) |
6 | 1 | qrng0 26124 | . . . 4 ⊢ 0 = (0g‘𝑄) |
7 | ostth.k | . . . 4 ⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1)) | |
8 | 2, 5, 6, 7 | abvtriv 19541 | . . 3 ⊢ (𝑄 ∈ DivRing → 𝐾 ∈ 𝐴) |
9 | 4, 8 | mp1i 13 | . 2 ⊢ (𝜑 → 𝐾 ∈ 𝐴) |
10 | ostth1.3 | . . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℙ ¬ (𝐹‘𝑛) < 1) | |
11 | 10 | r19.21bi 3205 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → ¬ (𝐹‘𝑛) < 1) |
12 | prmnn 16006 | . . . . 5 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℕ) | |
13 | ostth1.2 | . . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ¬ 1 < (𝐹‘𝑛)) | |
14 | 13 | r19.21bi 3205 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ 1 < (𝐹‘𝑛)) |
15 | 12, 14 | sylan2 592 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → ¬ 1 < (𝐹‘𝑛)) |
16 | nnq 12349 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℚ) | |
17 | 12, 16 | syl 17 | . . . . . 6 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℚ) |
18 | 2, 5 | abvcl 19524 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑛 ∈ ℚ) → (𝐹‘𝑛) ∈ ℝ) |
19 | 3, 17, 18 | syl2an 595 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐹‘𝑛) ∈ ℝ) |
20 | 1re 10629 | . . . . 5 ⊢ 1 ∈ ℝ | |
21 | lttri3 10712 | . . . . 5 ⊢ (((𝐹‘𝑛) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝐹‘𝑛) = 1 ↔ (¬ (𝐹‘𝑛) < 1 ∧ ¬ 1 < (𝐹‘𝑛)))) | |
22 | 19, 20, 21 | sylancl 586 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → ((𝐹‘𝑛) = 1 ↔ (¬ (𝐹‘𝑛) < 1 ∧ ¬ 1 < (𝐹‘𝑛)))) |
23 | 11, 15, 22 | mpbir2and 709 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐹‘𝑛) = 1) |
24 | 12 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℕ) |
25 | eqeq1 2822 | . . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝑥 = 0 ↔ 𝑛 = 0)) | |
26 | 25 | ifbid 4485 | . . . . . . 7 ⊢ (𝑥 = 𝑛 → if(𝑥 = 0, 0, 1) = if(𝑛 = 0, 0, 1)) |
27 | c0ex 10623 | . . . . . . . 8 ⊢ 0 ∈ V | |
28 | 1ex 10625 | . . . . . . . 8 ⊢ 1 ∈ V | |
29 | 27, 28 | ifex 4511 | . . . . . . 7 ⊢ if(𝑛 = 0, 0, 1) ∈ V |
30 | 26, 7, 29 | fvmpt 6761 | . . . . . 6 ⊢ (𝑛 ∈ ℚ → (𝐾‘𝑛) = if(𝑛 = 0, 0, 1)) |
31 | 16, 30 | syl 17 | . . . . 5 ⊢ (𝑛 ∈ ℕ → (𝐾‘𝑛) = if(𝑛 = 0, 0, 1)) |
32 | nnne0 11659 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0) | |
33 | 32 | neneqd 3018 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0) |
34 | 33 | iffalsed 4474 | . . . . 5 ⊢ (𝑛 ∈ ℕ → if(𝑛 = 0, 0, 1) = 1) |
35 | 31, 34 | eqtrd 2853 | . . . 4 ⊢ (𝑛 ∈ ℕ → (𝐾‘𝑛) = 1) |
36 | 24, 35 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐾‘𝑛) = 1) |
37 | 23, 36 | eqtr4d 2856 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐹‘𝑛) = (𝐾‘𝑛)) |
38 | 1, 2, 3, 9, 37 | ostthlem2 26131 | 1 ⊢ (𝜑 → 𝐹 = 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ifcif 4463 class class class wbr 5057 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 0cc0 10525 1c1 10526 < clt 10663 -cneg 10859 ℕcn 11626 ℚcq 12336 ↑cexp 13417 ℙcprime 16003 pCnt cpc 16161 ↾s cress 16472 DivRingcdr 19431 AbsValcabv 19516 ℂfldccnfld 20473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-ico 12732 df-fz 12881 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-prm 16004 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-subg 18214 df-cmn 18837 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-subrg 19462 df-abv 19517 df-cnfld 20474 |
This theorem is referenced by: ostth 26142 |
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