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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 13006 | . . . 4 ⊢ ℚ ∈ V | |
| 2 | 1 | mptex 7229 | . . 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 6683 | . 2 ⊢ 𝐽 Fn ℙ |
| 5 | 3 | padicfval 27836 | . . . . 5 ⊢ (𝑝 ∈ ℙ → (𝐽‘𝑝) = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑝↑-(𝑝 pCnt 𝑥))))) |
| 6 | prmnn 16759 | . . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ) | |
| 7 | 6 | ad2antrr 739 | . . . . . . . . . 10 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑝 ∈ ℕ) |
| 8 | 7 | nncnd 12269 | . . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑝 ∈ ℂ) |
| 9 | 7 | nnne0d 12306 | . . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑝 ≠ 0) |
| 10 | df-ne 2961 | . . . . . . . . . 10 ⊢ (𝑥 ≠ 0 ↔ ¬ 𝑥 = 0) | |
| 11 | pcqcl 16943 | . . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℙ ∧ (𝑥 ∈ ℚ ∧ 𝑥 ≠ 0)) → (𝑝 pCnt 𝑥) ∈ ℤ) | |
| 12 | 11 | anassrs 473 | . . . . . . . . . 10 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ 𝑥 ≠ 0) → (𝑝 pCnt 𝑥) ∈ ℤ) |
| 13 | 10, 12 | sylan2br 607 | . . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑝 pCnt 𝑥) ∈ ℤ) |
| 14 | 8, 9, 13 | expnegd 14212 | . . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑝↑-(𝑝 pCnt 𝑥)) = (1 / (𝑝↑(𝑝 pCnt 𝑥)))) |
| 15 | 8, 9, 13 | exprecd 14213 | . . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → ((1 / 𝑝)↑(𝑝 pCnt 𝑥)) = (1 / (𝑝↑(𝑝 pCnt 𝑥)))) |
| 16 | 14, 15 | eqtr4d 2803 | . . . . . . 7 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑝↑-(𝑝 pCnt 𝑥)) = ((1 / 𝑝)↑(𝑝 pCnt 𝑥))) |
| 17 | 16 | ifeq2da 4522 | . . . . . 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 5206 | . . . . 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 2800 | . . . 4 ⊢ (𝑝 ∈ ℙ → (𝐽‘𝑝) = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥))))) |
| 20 | 6 | nnrecred 12307 | . . . . . 6 ⊢ (𝑝 ∈ ℙ → (1 / 𝑝) ∈ ℝ) |
| 21 | 6 | nnred 12268 | . . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℝ) |
| 22 | prmgt1 16783 | . . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 1 < 𝑝) | |
| 23 | recgt1i 12132 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℝ ∧ 1 < 𝑝) → (0 < (1 / 𝑝) ∧ (1 / 𝑝) < 1)) | |
| 24 | 21, 22, 23 | syl2anc 596 | . . . . . . 7 ⊢ (𝑝 ∈ ℙ → (0 < (1 / 𝑝) ∧ (1 / 𝑝) < 1)) |
| 25 | 24 | simpld 500 | . . . . . 6 ⊢ (𝑝 ∈ ℙ → 0 < (1 / 𝑝)) |
| 26 | 24 | simprd 501 | . . . . . 6 ⊢ (𝑝 ∈ ℙ → (1 / 𝑝) < 1) |
| 27 | 0xr 11276 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
| 28 | 1xr 11288 | . . . . . . 7 ⊢ 1 ∈ ℝ* | |
| 29 | elioo2 13434 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → ((1 / 𝑝) ∈ (0(,)1) ↔ ((1 / 𝑝) ∈ ℝ ∧ 0 < (1 / 𝑝) ∧ (1 / 𝑝) < 1))) | |
| 30 | 27, 28, 29 | mp2an 705 | . . . . . 6 ⊢ ((1 / 𝑝) ∈ (0(,)1) ↔ ((1 / 𝑝) ∈ ℝ ∧ 0 < (1 / 𝑝) ∧ (1 / 𝑝) < 1)) |
| 31 | 20, 25, 26, 30 | syl3anbrc 1362 | . . . . 5 ⊢ (𝑝 ∈ ℙ → (1 / 𝑝) ∈ (0(,)1)) |
| 32 | qrng.q | . . . . . 6 ⊢ 𝑄 = (ℂfld ↾s ℚ) | |
| 33 | qabsabv.a | . . . . . 6 ⊢ 𝐴 = (AbsVal‘𝑄) | |
| 34 | eqid 2765 | . . . . . 6 ⊢ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))) = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))) | |
| 35 | 32, 33, 34 | padicabv 27850 | . . . . 5 ⊢ ((𝑝 ∈ ℙ ∧ (1 / 𝑝) ∈ (0(,)1)) → (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))) ∈ 𝐴) |
| 36 | 31, 35 | mpdan 700 | . . . 4 ⊢ (𝑝 ∈ ℙ → (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))) ∈ 𝐴) |
| 37 | 19, 36 | eqeltrd 2865 | . . 3 ⊢ (𝑝 ∈ ℙ → (𝐽‘𝑝) ∈ 𝐴) |
| 38 | 37 | rgen 3083 | . 2 ⊢ ∀𝑝 ∈ ℙ (𝐽‘𝑝) ∈ 𝐴 |
| 39 | ffnfv 7119 | . 2 ⊢ (𝐽:ℙ⟶𝐴 ↔ (𝐽 Fn ℙ ∧ ∀𝑝 ∈ ℙ (𝐽‘𝑝) ∈ 𝐴)) | |
| 40 | 4, 38, 39 | mpbir2an 724 | 1 ⊢ 𝐽:ℙ⟶𝐴 |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: ¬ wn 3 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 ifcif 4489 class class class wbr 5111 ↦ cmpt 5194 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7420 ℝcr 11119 0cc0 11120 1c1 11121 ℝ*cxr 11262 < clt 11263 -cneg 11462 / cdiv 11891 ℕcn 12253 ℤcz 12611 ℚcq 12993 (,)cioo 13393 ↑cexp 14120 ℙcprime 16756 pCnt cpc 16923 ↾s cress 17317 AbsValcabv 20966 ℂfldccnfld 21577 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7743 ax-cnex 11176 ax-resscn 11177 ax-1cn 11178 ax-icn 11179 ax-addcl 11180 ax-addrcl 11181 ax-mulcl 11182 ax-mulrcl 11183 ax-mulcom 11184 ax-addass 11185 ax-mulass 11186 ax-distr 11187 ax-i2m1 11188 ax-1ne0 11189 ax-1rid 11190 ax-rnegex 11191 ax-rrecex 11192 ax-cnre 11193 ax-pre-lttri 11194 ax-pre-lttrn 11195 ax-pre-ltadd 11196 ax-pre-mulgt0 11197 ax-pre-sup 11198 ax-addf 11199 ax-mulf 11200 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7870 df-1st 7993 df-2nd 7994 df-tpos 8229 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-en 8951 df-dom 8952 df-sdom 8953 df-fin 8954 df-sup 9410 df-inf 9411 df-pnf 11265 df-mnf 11266 df-xr 11267 df-ltxr 11268 df-le 11269 df-sub 11463 df-neg 11464 df-div 11892 df-nn 12254 df-2 12323 df-3 12324 df-4 12325 df-5 12326 df-6 12327 df-7 12328 df-8 12329 df-9 12330 df-n0 12525 df-z 12612 df-dec 12733 df-uz 12884 df-q 12994 df-rp 13038 df-ioo 13397 df-ico 13399 df-fz 13557 df-fl 13848 df-mod 13926 df-seq 14061 df-exp 14121 df-cj 15179 df-re 15180 df-im 15181 df-sqrt 15315 df-abs 15316 df-dvds 16338 df-gcd 16580 df-prm 16757 df-pc 16924 df-struct 17234 df-sets 17251 df-slot 17269 df-ndx 17281 df-base 17297 df-ress 17318 df-plusg 17350 df-mulr 17351 df-starv 17352 df-tset 17356 df-ple 17357 df-ds 17359 df-unif 17360 df-0g 17521 df-mgm 18725 df-sgrp 18814 df-mnd 18830 df-grp 19052 df-minusg 19053 df-subg 19238 df-cmn 19901 df-abl 19902 df-mgp 20266 df-rng 20280 df-ur 20313 df-ring 20366 df-cring 20367 df-oppr 20470 df-dvdsr 20490 df-unit 20491 df-invr 20521 df-dvr 20534 df-subrng 20700 df-subrg 20724 df-drng 20884 df-abv 20967 df-cnfld 21578 |
| This theorem is used by: ostth 27859 |
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