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Theorem padicabvf 27575

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 12896	. . . 4 ⊢ ℚ ∈ V
2	1	mptex 7179	. . 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 6643	. 2 ⊢ 𝐽 Fn ℙ
5	3	padicfval 27560	. . . . 5 ⊢ (𝑝 ∈ ℙ → (𝐽‘𝑝) = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑝↑-(𝑝 pCnt 𝑥)))))
6		prmnn 16620	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
7	6	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑝 ∈ ℕ)
8	7	nncnd 12178	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑝 ∈ ℂ)
9	7	nnne0d 12212	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑝 ≠ 0)
10		df-ne 2926	. . . . . . . . . 10 ⊢ (𝑥 ≠ 0 ↔ ¬ 𝑥 = 0)
11		pcqcl 16803	. . . . . . . . . . 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 14094	. . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑝↑-(𝑝 pCnt 𝑥)) = (1 / (𝑝↑(𝑝 pCnt 𝑥))))
15	8, 9, 13	exprecd 14095	. . . . . . . 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 4517	. . . . . 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 5195	. . . . 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 12213	. . . . . 6 ⊢ (𝑝 ∈ ℙ → (1 / 𝑝) ∈ ℝ)
21	6	nnred 12177	. . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℝ)
22		prmgt1 16643	. . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 1 < 𝑝)
23		recgt1i 12056	. . . . . . . 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 11197	. . . . . . 7 ⊢ 0 ∈ ℝ^*
28		1xr 11209	. . . . . . 7 ⊢ 1 ∈ ℝ^*
29		elioo2 13323	. . . . . . 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 27574	. . . . 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 7073	. 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 4484 class class class wbr 5102 ↦ cmpt 5183 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ℝcr 11043 0cc0 11044 1c1 11045 ℝ^*cxr 11183 < clt 11184 -cneg 11382 / cdiv 11811 ℕcn 12162 ℤcz 12505 ℚcq 12883 (,)cioo 13282 ↑cexp 14002 ℙcprime 16617 pCnt cpc 16783 ↾_s cress 17176 AbsValcabv 20728 ℂ_fldccnfld 21296

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9369 df-inf 9370 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-ioo 13286 df-ico 13288 df-fz 13445 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-dvds 16199 df-gcd 16441 df-prm 16618 df-pc 16784 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-0g 17380 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-grp 18850 df-minusg 18851 df-subg 19037 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-cring 20156 df-oppr 20257 df-dvdsr 20277 df-unit 20278 df-invr 20308 df-dvr 20321 df-subrng 20466 df-subrg 20490 df-drng 20651 df-abv 20729 df-cnfld 21297

This theorem is referenced by: ostth 27583

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