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Mirrors > Home > MPE Home > Th. List > padicabvf | Structured version Visualization version GIF version |
Description: The p-adic absolute value is an absolute value. (Contributed by Mario Carneiro, 9-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))))) |
Ref | Expression |
---|---|
padicabvf | ⊢ 𝐽:ℙ⟶𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qex 12013 | . . . 4 ⊢ ℚ ∈ V | |
2 | 1 | mptex 6651 | . . 3 ⊢ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥)))) ∈ V |
3 | padic.j | . . 3 ⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥))))) | |
4 | 2, 3 | fnmpti 6183 | . 2 ⊢ 𝐽 Fn ℙ |
5 | 3 | padicfval 25525 | . . . . 5 ⊢ (𝑝 ∈ ℙ → (𝐽‘𝑝) = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑝↑-(𝑝 pCnt 𝑥))))) |
6 | prmnn 15610 | . . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ) | |
7 | 6 | ad2antrr 764 | . . . . . . . . . 10 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑝 ∈ ℕ) |
8 | 7 | nncnd 11248 | . . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑝 ∈ ℂ) |
9 | 7 | nnne0d 11277 | . . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑝 ≠ 0) |
10 | df-ne 2933 | . . . . . . . . . 10 ⊢ (𝑥 ≠ 0 ↔ ¬ 𝑥 = 0) | |
11 | pcqcl 15783 | . . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℙ ∧ (𝑥 ∈ ℚ ∧ 𝑥 ≠ 0)) → (𝑝 pCnt 𝑥) ∈ ℤ) | |
12 | 11 | anassrs 683 | . . . . . . . . . 10 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ 𝑥 ≠ 0) → (𝑝 pCnt 𝑥) ∈ ℤ) |
13 | 10, 12 | sylan2br 494 | . . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑝 pCnt 𝑥) ∈ ℤ) |
14 | 8, 9, 13 | expnegd 13229 | . . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑝↑-(𝑝 pCnt 𝑥)) = (1 / (𝑝↑(𝑝 pCnt 𝑥)))) |
15 | 8, 9, 13 | exprecd 13230 | . . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → ((1 / 𝑝)↑(𝑝 pCnt 𝑥)) = (1 / (𝑝↑(𝑝 pCnt 𝑥)))) |
16 | 14, 15 | eqtr4d 2797 | . . . . . . 7 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑝↑-(𝑝 pCnt 𝑥)) = ((1 / 𝑝)↑(𝑝 pCnt 𝑥))) |
17 | 16 | ifeq2da 4261 | . . . . . 6 ⊢ ((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) → if(𝑥 = 0, 0, (𝑝↑-(𝑝 pCnt 𝑥))) = if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))) |
18 | 17 | mpteq2dva 4896 | . . . . 5 ⊢ (𝑝 ∈ ℙ → (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑝↑-(𝑝 pCnt 𝑥)))) = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥))))) |
19 | 5, 18 | eqtrd 2794 | . . . 4 ⊢ (𝑝 ∈ ℙ → (𝐽‘𝑝) = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥))))) |
20 | 6 | nnrecred 11278 | . . . . . 6 ⊢ (𝑝 ∈ ℙ → (1 / 𝑝) ∈ ℝ) |
21 | 6 | nnred 11247 | . . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℝ) |
22 | prmgt1 15631 | . . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 1 < 𝑝) | |
23 | recgt1i 11132 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℝ ∧ 1 < 𝑝) → (0 < (1 / 𝑝) ∧ (1 / 𝑝) < 1)) | |
24 | 21, 22, 23 | syl2anc 696 | . . . . . . 7 ⊢ (𝑝 ∈ ℙ → (0 < (1 / 𝑝) ∧ (1 / 𝑝) < 1)) |
25 | 24 | simpld 477 | . . . . . 6 ⊢ (𝑝 ∈ ℙ → 0 < (1 / 𝑝)) |
26 | 24 | simprd 482 | . . . . . 6 ⊢ (𝑝 ∈ ℙ → (1 / 𝑝) < 1) |
27 | 0xr 10298 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
28 | 1re 10251 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
29 | 28 | rexri 10309 | . . . . . . 7 ⊢ 1 ∈ ℝ* |
30 | elioo2 12429 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → ((1 / 𝑝) ∈ (0(,)1) ↔ ((1 / 𝑝) ∈ ℝ ∧ 0 < (1 / 𝑝) ∧ (1 / 𝑝) < 1))) | |
31 | 27, 29, 30 | mp2an 710 | . . . . . 6 ⊢ ((1 / 𝑝) ∈ (0(,)1) ↔ ((1 / 𝑝) ∈ ℝ ∧ 0 < (1 / 𝑝) ∧ (1 / 𝑝) < 1)) |
32 | 20, 25, 26, 31 | syl3anbrc 1429 | . . . . 5 ⊢ (𝑝 ∈ ℙ → (1 / 𝑝) ∈ (0(,)1)) |
33 | qrng.q | . . . . . 6 ⊢ 𝑄 = (ℂfld ↾s ℚ) | |
34 | qabsabv.a | . . . . . 6 ⊢ 𝐴 = (AbsVal‘𝑄) | |
35 | eqid 2760 | . . . . . 6 ⊢ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))) = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))) | |
36 | 33, 34, 35 | padicabv 25539 | . . . . 5 ⊢ ((𝑝 ∈ ℙ ∧ (1 / 𝑝) ∈ (0(,)1)) → (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))) ∈ 𝐴) |
37 | 32, 36 | mpdan 705 | . . . 4 ⊢ (𝑝 ∈ ℙ → (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))) ∈ 𝐴) |
38 | 19, 37 | eqeltrd 2839 | . . 3 ⊢ (𝑝 ∈ ℙ → (𝐽‘𝑝) ∈ 𝐴) |
39 | 38 | rgen 3060 | . 2 ⊢ ∀𝑝 ∈ ℙ (𝐽‘𝑝) ∈ 𝐴 |
40 | ffnfv 6552 | . 2 ⊢ (𝐽:ℙ⟶𝐴 ↔ (𝐽 Fn ℙ ∧ ∀𝑝 ∈ ℙ (𝐽‘𝑝) ∈ 𝐴)) | |
41 | 4, 39, 40 | mpbir2an 993 | 1 ⊢ 𝐽:ℙ⟶𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∀wral 3050 ifcif 4230 class class class wbr 4804 ↦ cmpt 4881 Fn wfn 6044 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 ℝcr 10147 0cc0 10148 1c1 10149 ℝ*cxr 10285 < clt 10286 -cneg 10479 / cdiv 10896 ℕcn 11232 ℤcz 11589 ℚcq 12001 (,)cioo 12388 ↑cexp 13074 ℙcprime 15607 pCnt cpc 15763 ↾s cress 16080 AbsValcabv 19038 ℂfldccnfld 19968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 ax-addf 10227 ax-mulf 10228 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-tpos 7522 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-2o 7731 df-oadd 7734 df-er 7913 df-map 8027 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-sup 8515 df-inf 8516 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-q 12002 df-rp 12046 df-ioo 12392 df-ico 12394 df-fz 12540 df-fl 12807 df-mod 12883 df-seq 13016 df-exp 13075 df-cj 14058 df-re 14059 df-im 14060 df-sqrt 14194 df-abs 14195 df-dvds 15203 df-gcd 15439 df-prm 15608 df-pc 15764 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-starv 16178 df-tset 16182 df-ple 16183 df-ds 16186 df-unif 16187 df-0g 16324 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-grp 17646 df-minusg 17647 df-subg 17812 df-cmn 18415 df-mgp 18710 df-ur 18722 df-ring 18769 df-cring 18770 df-oppr 18843 df-dvdsr 18861 df-unit 18862 df-invr 18892 df-dvr 18903 df-drng 18971 df-subrg 19000 df-abv 19039 df-cnfld 19969 |
This theorem is referenced by: ostth 25548 |
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