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Mirrors > Home > MPE Home > Th. List > ostth1 | Structured version Visualization version GIF version |
Description: - Lemma for ostth 27513: 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 20668 of the absolute value, πΉ is equal to 1 on all the integers, and ostthlem1 27501 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 27494 | . . 3 β’ π β DivRing |
5 | 1 | qrngbas 27493 | . . . 4 β’ β = (Baseβπ) |
6 | 1 | qrng0 27495 | . . . 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 20680 | . . 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 3240 | . . . 4 β’ ((π β§ π β β) β Β¬ (πΉβπ) < 1) |
12 | prmnn 16614 | . . . . 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 3240 | . . . . 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 12945 | . . . . . . 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 20663 | . . . . . 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 11213 | . . . . 5 β’ 1 β β | |
21 | lttri3 11296 | . . . . 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 710 | . . 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 2728 | . . . . . . . 8 β’ (π₯ = π β (π₯ = 0 β π = 0)) | |
26 | 25 | ifbid 4544 | . . . . . . 7 β’ (π₯ = π β if(π₯ = 0, 0, 1) = if(π = 0, 0, 1)) |
27 | c0ex 11207 | . . . . . . . 8 β’ 0 β V | |
28 | 1ex 11209 | . . . . . . . 8 β’ 1 β V | |
29 | 27, 28 | ifex 4571 | . . . . . . 7 β’ if(π = 0, 0, 1) β V |
30 | 26, 7, 29 | fvmpt 6989 | . . . . . 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 12245 | . . . . . . 7 β’ (π β β β π β 0) | |
33 | 32 | neneqd 2937 | . . . . . 6 β’ (π β β β Β¬ π = 0) |
34 | 33 | iffalsed 4532 | . . . . 5 β’ (π β β β if(π = 0, 0, 1) = 1) |
35 | 31, 34 | eqtrd 2764 | . . . 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 2767 | . 2 β’ ((π β§ π β β) β (πΉβπ) = (πΎβπ)) |
38 | 1, 2, 3, 9, 37 | ostthlem2 27502 | 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 1533 β wcel 2098 βwral 3053 ifcif 4521 class class class wbr 5139 β¦ cmpt 5222 βcfv 6534 (class class class)co 7402 βcr 11106 0cc0 11107 1c1 11108 < clt 11247 -cneg 11444 βcn 12211 βcq 12931 βcexp 14028 βcprime 16611 pCnt cpc 16774 βΎs cress 17178 DivRingcdr 20583 AbsValcabv 20655 βfldccnfld 21234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12976 df-ico 13331 df-fz 13486 df-seq 13968 df-exp 14029 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-dvds 16201 df-prm 16612 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-subg 19046 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-subrng 20442 df-subrg 20467 df-drng 20585 df-abv 20656 df-cnfld 21235 |
This theorem is referenced by: ostth 27513 |
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