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Theorem pclemub 12178

Description: Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Jim Kingdon, 7-Oct-2024.)

**Hypothesis**
Ref	Expression
pclem.1

**Assertion**
Ref	Expression
pclemub	$/\$ $/\$

Distinct variable groups:

Allowed substitution hints:

(

)

**Proof of Theorem pclemub**
Step	Hyp	Ref	Expression
1		nnssz 9190	. 2
2		zcn 9178	. . . . . 6
3	2	abscld 11093	. . . . 5
4	3	ad2antrl 482	. . . 4 $/\$ $/\$
5		eluzelre 9455	. . . . 5
6	5	adantr 274	. . . 4 $/\$ $/\$
7		eluz2gt1 9519	. . . . 5
8	7	adantr 274	. . . 4 $/\$ $/\$
9		expnbnd 10551	. . . 4 $/\$ $/\$
10	4, 6, 8, 9	syl3anc 1220	. . 3 $/\$ $/\$
11		simprr 522	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
12		oveq2 5835	. . . . . . . . . . . . . 14
13	12	breq1d 3977	. . . . . . . . . . . . 13
14		pclem.1	. . . . . . . . . . . . 13
15	13, 14	elrab2 2871	. . . . . . . . . . . 12 $/\$
16	11, 15	sylib 121	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
17	16	simprd 113	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
18		eluz2nn 9483	. . . . . . . . . . . . . 14
19	18	ad2antrr 480	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
20	16	simpld 111	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
21	19, 20	nnexpcld 10583	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
22	21	nnzd 9291	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
23		simplrl 525	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
24		simplrr 526	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
25		dvdsleabs 11750	. . . . . . . . . . 11 $/\$ $/\$
26	22, 23, 24, 25	syl3anc 1220	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
27	17, 26	mpd 13	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
28	21	nnred 8852	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
29	4	adantr 274	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
30	5	ad2antrr 480	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
31		nnnn0 9103	. . . . . . . . . . . 12
32	31	ad2antrl 482	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
33	30, 32	reexpcld 10578	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34		lelttr 7969	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	28, 29, 33, 34	syl3anc 1220	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
36	27, 35	mpand 426	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
37	7	ad2antrr 480	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
38		nn0ltexp2 10596	. . . . . . . . 9 $/\$ $/\$ $/\$
39	30, 20, 32, 37, 38	syl31anc 1223	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
40	36, 39	sylibrd 168	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41	20	nn0red 9150	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
42		nnre 8846	. . . . . . . . 9
43	42	ad2antrl 482	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
44		ltle 7968	. . . . . . . 8 $/\$
45	41, 43, 44	syl2anc 409	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
46	40, 45	syld 45	. . . . . 6 $/\$ $/\$ $/\$ $/\$
47	46	anassrs 398	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	ralrimdva 2537	. . . 4 $/\$ $/\$ $/\$
49	48	reximdva 2559	. . 3 $/\$ $/\$
50	10, 49	mpd 13	. 2 $/\$ $/\$
51		ssrexv 3193	. 2
52	1, 50, 51	mpsyl 65	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104

wceq 1335

wcel 2128

wne 2327

wral 2435

wrex 2436

crab 2439

wss 3102 class class class wbr 3967

cfv 5173 (class class class)co 5827

cr 7734

cc0 7735

c1 7736

clt 7915

cle 7916

cn 8839

c2 8890

cn0 9096

cz 9173

cuz 9445

cexp 10428

cabs 10909

cdvds 11695

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4082 ax-sep 4085 ax-nul 4093 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-iinf 4550 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-mulrcl 7834 ax-addcom 7835 ax-mulcom 7836 ax-addass 7837 ax-mulass 7838 ax-distr 7839 ax-i2m1 7840 ax-0lt1 7841 ax-1rid 7842 ax-0id 7843 ax-rnegex 7844 ax-precex 7845 ax-cnre 7846 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-lttrn 7849 ax-pre-apti 7850 ax-pre-ltadd 7851 ax-pre-mulgt0 7852 ax-pre-mulext 7853 ax-arch 7854 ax-caucvg 7855

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4029 df-mpt 4030 df-tr 4066 df-id 4256 df-po 4259 df-iso 4260 df-iord 4329 df-on 4331 df-ilim 4332 df-suc 4334 df-iom 4553 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-f1 5178 df-fo 5179 df-f1o 5180 df-fv 5181 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-1st 6091 df-2nd 6092 df-recs 6255 df-frec 6341 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 df-sub 8053 df-neg 8054 df-reap 8455 df-ap 8462 df-div 8551 df-inn 8840 df-2 8898 df-3 8899 df-4 8900 df-n0 9097 df-z 9174 df-uz 9446 df-q 9536 df-rp 9568 df-seqfrec 10355 df-exp 10429 df-cj 10754 df-re 10755 df-im 10756 df-rsqrt 10910 df-abs 10911 df-dvds 11696

This theorem is referenced by: pcprecl 12180 pcprendvds 12181 pcpremul 12184

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