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Theorem padicabvf 25893

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 12214	. . . 4 ⊢ ℚ ∈ V
2	1	mptex 6859	. . 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 6366	. 2 ⊢ 𝐽 Fn ℙ
5	3	padicfval 25878	. . . . 5 ⊢ (𝑝 ∈ ℙ → (𝐽‘𝑝) = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑝↑-(𝑝 pCnt 𝑥)))))
6		prmnn 15851	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
7	6	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑝 ∈ ℕ)
8	7	nncnd 11508	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑝 ∈ ℂ)
9	7	nnne0d 11541	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑝 ≠ 0)
10		df-ne 2987	. . . . . . . . . 10 ⊢ (𝑥 ≠ 0 ↔ ¬ 𝑥 = 0)
11		pcqcl 16026	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℙ ∧ (𝑥 ∈ ℚ ∧ 𝑥 ≠ 0)) → (𝑝 pCnt 𝑥) ∈ ℤ)
12	11	anassrs 468	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ 𝑥 ≠ 0) → (𝑝 pCnt 𝑥) ∈ ℤ)
13	10, 12	sylan2br 594	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑝 pCnt 𝑥) ∈ ℤ)
14	8, 9, 13	expnegd 13371	. . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑝↑-(𝑝 pCnt 𝑥)) = (1 / (𝑝↑(𝑝 pCnt 𝑥))))
15	8, 9, 13	exprecd 13372	. . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → ((1 / 𝑝)↑(𝑝 pCnt 𝑥)) = (1 / (𝑝↑(𝑝 pCnt 𝑥))))
16	14, 15	eqtr4d 2836	. . . . . . 7 ⊢ (((𝑝 ∈ ℙ ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑝↑-(𝑝 pCnt 𝑥)) = ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))
17	16	ifeq2da 4418	. . . . . 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 5062	. . . . 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 2833	. . . 4 ⊢ (𝑝 ∈ ℙ → (𝐽‘𝑝) = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))))
20	6	nnrecred 11542	. . . . . 6 ⊢ (𝑝 ∈ ℙ → (1 / 𝑝) ∈ ℝ)
21	6	nnred 11507	. . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℝ)
22		prmgt1 15874	. . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 1 < 𝑝)
23		recgt1i 11391	. . . . . . . 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 495	. . . . . 6 ⊢ (𝑝 ∈ ℙ → 0 < (1 / 𝑝))
26	24	simprd 496	. . . . . 6 ⊢ (𝑝 ∈ ℙ → (1 / 𝑝) < 1)
27		0xr 10541	. . . . . . 7 ⊢ 0 ∈ ℝ^*
28		1xr 10553	. . . . . . 7 ⊢ 1 ∈ ℝ^*
29		elioo2 12633	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((1 / 𝑝) ∈ (0(,)1) ↔ ((1 / 𝑝) ∈ ℝ ∧ 0 < (1 / 𝑝) ∧ (1 / 𝑝) < 1)))
30	27, 28, 29	mp2an 688	. . . . . 6 ⊢ ((1 / 𝑝) ∈ (0(,)1) ↔ ((1 / 𝑝) ∈ ℝ ∧ 0 < (1 / 𝑝) ∧ (1 / 𝑝) < 1))
31	20, 25, 26, 30	syl3anbrc 1336	. . . . 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 2797	. . . . . 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 25892	. . . . 5 ⊢ ((𝑝 ∈ ℙ ∧ (1 / 𝑝) ∈ (0(,)1)) → (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))) ∈ 𝐴)
36	31, 35	mpdan 683	. . . 4 ⊢ (𝑝 ∈ ℙ → (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, ((1 / 𝑝)↑(𝑝 pCnt 𝑥)))) ∈ 𝐴)
37	19, 36	eqeltrd 2885	. . 3 ⊢ (𝑝 ∈ ℙ → (𝐽‘𝑝) ∈ 𝐴)
38	37	rgen 3117	. 2 ⊢ ∀𝑝 ∈ ℙ (𝐽‘𝑝) ∈ 𝐴
39		ffnfv 6752	. 2 ⊢ (𝐽:ℙ⟶𝐴 ↔ (𝐽 Fn ℙ ∧ ∀𝑝 ∈ ℙ (𝐽‘𝑝) ∈ 𝐴))
40	4, 38, 39	mpbir2an 707	1 ⊢ 𝐽:ℙ⟶𝐴

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ≠ wne 2986 ∀wral 3107 ifcif 4387 class class class wbr 4968 ↦ cmpt 5047 Fn wfn 6227 ⟶wf 6228 ‘cfv 6232 (class class class)co 7023 ℝcr 10389 0cc0 10390 1c1 10391 ℝ^*cxr 10527 < clt 10528 -cneg 10724 / cdiv 11151 ℕcn 11492 ℤcz 11835 ℚcq 12201 (,)cioo 12592 ↑cexp 13283 ℙcprime 15848 pCnt cpc 16006 ↾_s cress 16317 AbsValcabv 19281 ℂ_fldccnfld 20231

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 ax-addf 10469 ax-mulf 10470

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-tpos 7750 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-2o 7961 df-oadd 7964 df-er 8146 df-map 8265 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-sup 8759 df-inf 8760 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-z 11836 df-dec 11953 df-uz 12098 df-q 12202 df-rp 12244 df-ioo 12596 df-ico 12598 df-fz 12747 df-fl 13016 df-mod 13092 df-seq 13224 df-exp 13284 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-dvds 15445 df-gcd 15681 df-prm 15849 df-pc 16007 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-starv 16413 df-tset 16417 df-ple 16418 df-ds 16420 df-unif 16421 df-0g 16548 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-grp 17868 df-minusg 17869 df-subg 18034 df-cmn 18639 df-mgp 18934 df-ur 18946 df-ring 18993 df-cring 18994 df-oppr 19067 df-dvdsr 19085 df-unit 19086 df-invr 19116 df-dvr 19127 df-drng 19198 df-subrg 19227 df-abv 19282 df-cnfld 20232

This theorem is referenced by: ostth 25901

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