padicabvf - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > padicabvf		Structured version Visualization version GIF version

Theorem padicabvf 27540

Description: The p-adic absolute value is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)

**Hypotheses**
Ref	Expression
qrng.q	⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
qabsabv.a	⊢ 𝐴 = (AbsVal‘𝑄)
padic.j	⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥)))))

**Assertion**
Ref	Expression
padicabvf	⊢ 𝐽:ℙ⟶𝐴

Distinct variable groups: 𝑥,𝑞,𝐴 𝑥,𝑄 Allowed substitution hints: 𝑄(𝑞) 𝐽(𝑥,𝑞)

Proof of Theorem padicabvf Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qex 12862	. . . 4 ⊢ ℚ ∈ V
2	1	mptex 7159	. . 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 6625	. 2 ⊢ 𝐽 Fn ℙ
5	3	padicfval 27525	. . . . 5 ⊢ (𝑝 ∈ ℙ → (𝐽‘𝑝) = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑝↑-(𝑝 pCnt 𝑥)))))
6		prmnn 16585	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
7	6	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑝 ∈ ℕ)
8	7	nncnd 12144	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑝 ∈ ℂ)
9	7	nnne0d 12178	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑝 ≠ 0)
10		df-ne 2926	. . . . . . . . . 10 ⊢ (𝑥 ≠ 0 ↔ ¬ 𝑥 = 0)
11		pcqcl 16768	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℙ ∧ (𝑥 ∈ ℚ ∧ 𝑥 ≠ 0)) → (𝑝 pCnt 𝑥) ∈ ℤ)
12	11	anassrs 467	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ 𝑥 ≠ 0) → (𝑝 pCnt 𝑥) ∈ ℤ)
13	10, 12	sylan2br 595	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑝 pCnt 𝑥) ∈ ℤ)
14	8, 9, 13	expnegd 14060	. . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑝↑-(𝑝 pCnt 𝑥)) = (1 / (𝑝↑(𝑝 pCnt 𝑥))))
15	8, 9, 13	exprecd 14061	. . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → ((1 / 𝑝)↑(𝑝 pCnt 𝑥)) = (1 / (𝑝↑(𝑝 pCnt 𝑥))))
16	14, 15	eqtr4d 2767	. . . . . . 7 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑝↑-(𝑝 pCnt 𝑥)) = ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))
17	16	ifeq2da 4509	. . . . . 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 5185	. . . . 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 2764	. . . 4 ⊢ (𝑝 ∈ ℙ → (𝐽‘𝑝) = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))))
20	6	nnrecred 12179	. . . . . 6 ⊢ (𝑝 ∈ ℙ → (1 / 𝑝) ∈ ℝ)
21	6	nnred 12143	. . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℝ)
22		prmgt1 16608	. . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 1 < 𝑝)
23		recgt1i 12022	. . . . . . . 8 ⊢ ((𝑝 ∈ ℝ ∧ 1 < 𝑝) → (0 < (1 / 𝑝) ∧ (1 / 𝑝) < 1))
24	21, 22, 23	syl2anc 584	. . . . . . 7 ⊢ (𝑝 ∈ ℙ → (0 < (1 / 𝑝) ∧ (1 / 𝑝) < 1))
25	24	simpld 494	. . . . . 6 ⊢ (𝑝 ∈ ℙ → 0 < (1 / 𝑝))
26	24	simprd 495	. . . . . 6 ⊢ (𝑝 ∈ ℙ → (1 / 𝑝) < 1)
27		0xr 11162	. . . . . . 7 ⊢ 0 ∈ ℝ^*
28		1xr 11174	. . . . . . 7 ⊢ 1 ∈ ℝ^*
29		elioo2 13289	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((1 / 𝑝) ∈ (0(,)1) ↔ ((1 / 𝑝) ∈ ℝ ∧ 0 < (1 / 𝑝) ∧ (1 / 𝑝) < 1)))
30	27, 28, 29	mp2an 692	. . . . . 6 ⊢ ((1 / 𝑝) ∈ (0(,)1) ↔ ((1 / 𝑝) ∈ ℝ ∧ 0 < (1 / 𝑝) ∧ (1 / 𝑝) < 1))
31	20, 25, 26, 30	syl3anbrc 1344	. . . . 5 ⊢ (𝑝 ∈ ℙ → (1 / 𝑝) ∈ (0(,)1))
32		qrng.q	. . . . . 6 ⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
33		qabsabv.a	. . . . . 6 ⊢ 𝐴 = (AbsVal‘𝑄)
34		eqid 2729	. . . . . 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 27539	. . . . 5 ⊢ ((𝑝 ∈ ℙ ∧ (1 / 𝑝) ∈ (0(,)1)) → (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))) ∈ 𝐴)
36	31, 35	mpdan 687	. . . 4 ⊢ (𝑝 ∈ ℙ → (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))) ∈ 𝐴)
37	19, 36	eqeltrd 2828	. . 3 ⊢ (𝑝 ∈ ℙ → (𝐽‘𝑝) ∈ 𝐴)
38	37	rgen 3046	. 2 ⊢ ∀𝑝 ∈ ℙ (𝐽‘𝑝) ∈ 𝐴
39		ffnfv 7053	. 2 ⊢ (𝐽:ℙ⟶𝐴 ↔ (𝐽 Fn ℙ ∧ ∀𝑝 ∈ ℙ (𝐽‘𝑝) ∈ 𝐴))
40	4, 38, 39	mpbir2an 711	1 ⊢ 𝐽:ℙ⟶𝐴

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ifcif 4476 class class class wbr 5092 ↦ cmpt 5173 Fn wfn 6477 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ℝcr 11008 0cc0 11009 1c1 11010 ℝ^*cxr 11148 < clt 11149 -cneg 11348 / cdiv 11777 ℕcn 12128 ℤcz 12471 ℚcq 12849 (,)cioo 13248 ↑cexp 13968 ℙcprime 16582 pCnt cpc 16748 ↾_s cress 17141 AbsValcabv 20693 ℂ_fldccnfld 21261

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-tpos 8159 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-inf 9333 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-ioo 13252 df-ico 13254 df-fz 13411 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-dvds 16164 df-gcd 16406 df-prm 16583 df-pc 16749 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-0g 17345 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-grp 18815 df-minusg 18816 df-subg 19002 df-cmn 19661 df-abl 19662 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-subrng 20431 df-subrg 20455 df-drng 20616 df-abv 20694 df-cnfld 21262

This theorem is referenced by: ostth 27548

W3C validator