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Mirrors > Home > MPE Home > Th. List > ostth1 | Structured version Visualization version GIF version |
Description: - Lemma for ostth 25328: 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 18829 of the absolute value, 𝐹 is equal to 1 on all the integers, and ostthlem1 25316 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 25309 | . . 3 ⊢ 𝑄 ∈ DivRing |
5 | 1 | qrngbas 25308 | . . . 4 ⊢ ℚ = (Base‘𝑄) |
6 | 1 | qrng0 25310 | . . . 4 ⊢ 0 = (0g‘𝑄) |
7 | ostth.k | . . . 4 ⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1)) | |
8 | 2, 5, 6, 7 | abvtriv 18841 | . . 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 2932 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → ¬ (𝐹‘𝑛) < 1) |
12 | prmnn 15388 | . . . . 5 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℕ) | |
13 | ostth1.2 | . . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ¬ 1 < (𝐹‘𝑛)) | |
14 | 13 | r19.21bi 2932 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ 1 < (𝐹‘𝑛)) |
15 | 12, 14 | sylan2 491 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → ¬ 1 < (𝐹‘𝑛)) |
16 | nnq 11801 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℚ) | |
17 | 12, 16 | syl 17 | . . . . . 6 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℚ) |
18 | 2, 5 | abvcl 18824 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑛 ∈ ℚ) → (𝐹‘𝑛) ∈ ℝ) |
19 | 3, 17, 18 | syl2an 494 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐹‘𝑛) ∈ ℝ) |
20 | 1re 10039 | . . . . 5 ⊢ 1 ∈ ℝ | |
21 | lttri3 10121 | . . . . 5 ⊢ (((𝐹‘𝑛) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝐹‘𝑛) = 1 ↔ (¬ (𝐹‘𝑛) < 1 ∧ ¬ 1 < (𝐹‘𝑛)))) | |
22 | 19, 20, 21 | sylancl 694 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → ((𝐹‘𝑛) = 1 ↔ (¬ (𝐹‘𝑛) < 1 ∧ ¬ 1 < (𝐹‘𝑛)))) |
23 | 11, 15, 22 | mpbir2and 957 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐹‘𝑛) = 1) |
24 | 12 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℕ) |
25 | eqeq1 2626 | . . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝑥 = 0 ↔ 𝑛 = 0)) | |
26 | 25 | ifbid 4108 | . . . . . . 7 ⊢ (𝑥 = 𝑛 → if(𝑥 = 0, 0, 1) = if(𝑛 = 0, 0, 1)) |
27 | c0ex 10034 | . . . . . . . 8 ⊢ 0 ∈ V | |
28 | 1ex 10035 | . . . . . . . 8 ⊢ 1 ∈ V | |
29 | 27, 28 | ifex 4156 | . . . . . . 7 ⊢ if(𝑛 = 0, 0, 1) ∈ V |
30 | 26, 7, 29 | fvmpt 6282 | . . . . . 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 11053 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0) | |
33 | 32 | neneqd 2799 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0) |
34 | 33 | iffalsed 4097 | . . . . 5 ⊢ (𝑛 ∈ ℕ → if(𝑛 = 0, 0, 1) = 1) |
35 | 31, 34 | eqtrd 2656 | . . . 4 ⊢ (𝑛 ∈ ℕ → (𝐾‘𝑛) = 1) |
36 | 24, 35 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐾‘𝑛) = 1) |
37 | 23, 36 | eqtr4d 2659 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐹‘𝑛) = (𝐾‘𝑛)) |
38 | 1, 2, 3, 9, 37 | ostthlem2 25317 | 1 ⊢ (𝜑 → 𝐹 = 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ifcif 4086 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 < clt 10074 -cneg 10267 ℕcn 11020 ℚcq 11788 ↑cexp 12860 ℙcprime 15385 pCnt cpc 15541 ↾s cress 15858 DivRingcdr 18747 AbsValcabv 18816 ℂfldccnfld 19746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-ico 12181 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-prm 15386 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-subg 17591 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-subrg 18778 df-abv 18817 df-cnfld 19747 |
This theorem is referenced by: ostth 25328 |
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