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Mirrors > Home > MPE Home > Th. List > ostth1 | Structured version Visualization version GIF version |
Description: - Lemma for ostth 27010: trivial case. (Not that the proof is trivial, but that we are proving that the function is trivial.) If πΉ is equal to 1 on the primes, then by complete induction and the multiplicative property abvmul 20331 of the absolute value, πΉ is equal to 1 on all the integers, and ostthlem1 26998 extends this to the other rational numbers. (Contributed by Mario Carneiro, 10-Sep-2014.) |
Ref | Expression |
---|---|
qrng.q | β’ π = (βfld βΎs β) |
qabsabv.a | β’ π΄ = (AbsValβπ) |
padic.j | β’ π½ = (π β β β¦ (π₯ β β β¦ if(π₯ = 0, 0, (πβ-(π pCnt π₯))))) |
ostth.k | β’ πΎ = (π₯ β β β¦ if(π₯ = 0, 0, 1)) |
ostth.1 | β’ (π β πΉ β π΄) |
ostth1.2 | β’ (π β βπ β β Β¬ 1 < (πΉβπ)) |
ostth1.3 | β’ (π β βπ β β Β¬ (πΉβπ) < 1) |
Ref | Expression |
---|---|
ostth1 | β’ (π β πΉ = πΎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qrng.q | . 2 β’ π = (βfld βΎs β) | |
2 | qabsabv.a | . 2 β’ π΄ = (AbsValβπ) | |
3 | ostth.1 | . 2 β’ (π β πΉ β π΄) | |
4 | 1 | qdrng 26991 | . . 3 β’ π β DivRing |
5 | 1 | qrngbas 26990 | . . . 4 β’ β = (Baseβπ) |
6 | 1 | qrng0 26992 | . . . 4 β’ 0 = (0gβπ) |
7 | ostth.k | . . . 4 β’ πΎ = (π₯ β β β¦ if(π₯ = 0, 0, 1)) | |
8 | 2, 5, 6, 7 | abvtriv 20343 | . . 3 β’ (π β DivRing β πΎ β π΄) |
9 | 4, 8 | mp1i 13 | . 2 β’ (π β πΎ β π΄) |
10 | ostth1.3 | . . . . 5 β’ (π β βπ β β Β¬ (πΉβπ) < 1) | |
11 | 10 | r19.21bi 3233 | . . . 4 β’ ((π β§ π β β) β Β¬ (πΉβπ) < 1) |
12 | prmnn 16558 | . . . . 5 β’ (π β β β π β β) | |
13 | ostth1.2 | . . . . . 6 β’ (π β βπ β β Β¬ 1 < (πΉβπ)) | |
14 | 13 | r19.21bi 3233 | . . . . 5 β’ ((π β§ π β β) β Β¬ 1 < (πΉβπ)) |
15 | 12, 14 | sylan2 594 | . . . 4 β’ ((π β§ π β β) β Β¬ 1 < (πΉβπ)) |
16 | nnq 12895 | . . . . . . 7 β’ (π β β β π β β) | |
17 | 12, 16 | syl 17 | . . . . . 6 β’ (π β β β π β β) |
18 | 2, 5 | abvcl 20326 | . . . . . 6 β’ ((πΉ β π΄ β§ π β β) β (πΉβπ) β β) |
19 | 3, 17, 18 | syl2an 597 | . . . . 5 β’ ((π β§ π β β) β (πΉβπ) β β) |
20 | 1re 11163 | . . . . 5 β’ 1 β β | |
21 | lttri3 11246 | . . . . 5 β’ (((πΉβπ) β β β§ 1 β β) β ((πΉβπ) = 1 β (Β¬ (πΉβπ) < 1 β§ Β¬ 1 < (πΉβπ)))) | |
22 | 19, 20, 21 | sylancl 587 | . . . 4 β’ ((π β§ π β β) β ((πΉβπ) = 1 β (Β¬ (πΉβπ) < 1 β§ Β¬ 1 < (πΉβπ)))) |
23 | 11, 15, 22 | mpbir2and 712 | . . 3 β’ ((π β§ π β β) β (πΉβπ) = 1) |
24 | 12 | adantl 483 | . . . 4 β’ ((π β§ π β β) β π β β) |
25 | eqeq1 2737 | . . . . . . . 8 β’ (π₯ = π β (π₯ = 0 β π = 0)) | |
26 | 25 | ifbid 4513 | . . . . . . 7 β’ (π₯ = π β if(π₯ = 0, 0, 1) = if(π = 0, 0, 1)) |
27 | c0ex 11157 | . . . . . . . 8 β’ 0 β V | |
28 | 1ex 11159 | . . . . . . . 8 β’ 1 β V | |
29 | 27, 28 | ifex 4540 | . . . . . . 7 β’ if(π = 0, 0, 1) β V |
30 | 26, 7, 29 | fvmpt 6952 | . . . . . 6 β’ (π β β β (πΎβπ) = if(π = 0, 0, 1)) |
31 | 16, 30 | syl 17 | . . . . 5 β’ (π β β β (πΎβπ) = if(π = 0, 0, 1)) |
32 | nnne0 12195 | . . . . . . 7 β’ (π β β β π β 0) | |
33 | 32 | neneqd 2945 | . . . . . 6 β’ (π β β β Β¬ π = 0) |
34 | 33 | iffalsed 4501 | . . . . 5 β’ (π β β β if(π = 0, 0, 1) = 1) |
35 | 31, 34 | eqtrd 2773 | . . . 4 β’ (π β β β (πΎβπ) = 1) |
36 | 24, 35 | syl 17 | . . 3 β’ ((π β§ π β β) β (πΎβπ) = 1) |
37 | 23, 36 | eqtr4d 2776 | . 2 β’ ((π β§ π β β) β (πΉβπ) = (πΎβπ)) |
38 | 1, 2, 3, 9, 37 | ostthlem2 26999 | 1 β’ (π β πΉ = πΎ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 ifcif 4490 class class class wbr 5109 β¦ cmpt 5192 βcfv 6500 (class class class)co 7361 βcr 11058 0cc0 11059 1c1 11060 < clt 11197 -cneg 11394 βcn 12161 βcq 12881 βcexp 13976 βcprime 16555 pCnt cpc 16716 βΎs cress 17120 DivRingcdr 20219 AbsValcabv 20318 βfldccnfld 20819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-q 12882 df-rp 12924 df-ico 13279 df-fz 13434 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-dvds 16145 df-prm 16556 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-subg 18933 df-cmn 19572 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-dvr 20120 df-drng 20221 df-subrg 20262 df-abv 20319 df-cnfld 20820 |
This theorem is referenced by: ostth 27010 |
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