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Mirrors > Home > MPE Home > Th. List > ostth1 | Structured version Visualization version GIF version |
Description: - Lemma for ostth 27585: 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 20709 of the absolute value, πΉ is equal to 1 on all the integers, and ostthlem1 27573 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 27566 | . . 3 β’ π β DivRing |
5 | 1 | qrngbas 27565 | . . . 4 β’ β = (Baseβπ) |
6 | 1 | qrng0 27567 | . . . 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 20721 | . . 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 3245 | . . . 4 β’ ((π β§ π β β) β Β¬ (πΉβπ) < 1) |
12 | prmnn 16645 | . . . . 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 3245 | . . . . 5 β’ ((π β§ π β β) β Β¬ 1 < (πΉβπ)) |
15 | 12, 14 | sylan2 592 | . . . 4 β’ ((π β§ π β β) β Β¬ 1 < (πΉβπ)) |
16 | nnq 12977 | . . . . . . 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 20704 | . . . . . 6 β’ ((πΉ β π΄ β§ π β β) β (πΉβπ) β β) |
19 | 3, 17, 18 | syl2an 595 | . . . . 5 β’ ((π β§ π β β) β (πΉβπ) β β) |
20 | 1re 11245 | . . . . 5 β’ 1 β β | |
21 | lttri3 11328 | . . . . 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 585 | . . . 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 481 | . . . 4 β’ ((π β§ π β β) β π β β) |
25 | eqeq1 2732 | . . . . . . . 8 β’ (π₯ = π β (π₯ = 0 β π = 0)) | |
26 | 25 | ifbid 4552 | . . . . . . 7 β’ (π₯ = π β if(π₯ = 0, 0, 1) = if(π = 0, 0, 1)) |
27 | c0ex 11239 | . . . . . . . 8 β’ 0 β V | |
28 | 1ex 11241 | . . . . . . . 8 β’ 1 β V | |
29 | 27, 28 | ifex 4579 | . . . . . . 7 β’ if(π = 0, 0, 1) β V |
30 | 26, 7, 29 | fvmpt 7005 | . . . . . 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 12277 | . . . . . . 7 β’ (π β β β π β 0) | |
33 | 32 | neneqd 2942 | . . . . . 6 β’ (π β β β Β¬ π = 0) |
34 | 33 | iffalsed 4540 | . . . . 5 β’ (π β β β if(π = 0, 0, 1) = 1) |
35 | 31, 34 | eqtrd 2768 | . . . 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 2771 | . 2 β’ ((π β§ π β β) β (πΉβπ) = (πΎβπ)) |
38 | 1, 2, 3, 9, 37 | ostthlem2 27574 | 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 395 = wceq 1534 β wcel 2099 βwral 3058 ifcif 4529 class class class wbr 5148 β¦ cmpt 5231 βcfv 6548 (class class class)co 7420 βcr 11138 0cc0 11139 1c1 11140 < clt 11279 -cneg 11476 βcn 12243 βcq 12963 βcexp 14059 βcprime 16642 pCnt cpc 16805 βΎs cress 17209 DivRingcdr 20624 AbsValcabv 20696 βfldccnfld 21279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-ico 13363 df-fz 13518 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-dvds 16232 df-prm 16643 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-subg 19078 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-dvr 20340 df-subrng 20483 df-subrg 20508 df-drng 20626 df-abv 20697 df-cnfld 21280 |
This theorem is referenced by: ostth 27585 |
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