pclemub - Intuitionistic Logic Explorer

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Theorem pclemub 12940

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 9557	. 2
2		zcn 9545	. . . . . 6
3	2	abscld 11821	. . . . 5
4	3	ad2antrl 490	. . . 4 $/\$ $/\$
5		eluzelre 9827	. . . . 5
6	5	adantr 276	. . . 4 $/\$ $/\$
7		eluz2gt1 9897	. . . . 5
8	7	adantr 276	. . . 4 $/\$ $/\$
9		expnbnd 10988	. . . 4 $/\$ $/\$
10	4, 6, 8, 9	syl3anc 1274	. . 3 $/\$ $/\$
11		simprr 533	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
12		oveq2 6036	. . . . . . . . . . . . . 14
13	12	breq1d 4103	. . . . . . . . . . . . 13
14		pclem.1	. . . . . . . . . . . . 13
15	13, 14	elrab2 2966	. . . . . . . . . . . 12 $/\$
16	11, 15	sylib 122	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
17	16	simprd 114	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
18		eluz2nn 9861	. . . . . . . . . . . . . 14
19	18	ad2antrr 488	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
20	16	simpld 112	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
21	19, 20	nnexpcld 11020	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
22	21	nnzd 9662	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
23		simplrl 537	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
24		simplrr 538	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
25		dvdsleabs 12486	. . . . . . . . . . 11 $/\$ $/\$
26	22, 23, 24, 25	syl3anc 1274	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
27	17, 26	mpd 13	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
28	21	nnred 9215	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
29	4	adantr 276	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
30	5	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
31		nnnn0 9468	. . . . . . . . . . . 12
32	31	ad2antrl 490	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
33	30, 32	reexpcld 11015	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34		lelttr 8327	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	28, 29, 33, 34	syl3anc 1274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
36	27, 35	mpand 429	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
37	7	ad2antrr 488	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
38		nn0ltexp2 11034	. . . . . . . . 9 $/\$ $/\$ $/\$
39	30, 20, 32, 37, 38	syl31anc 1277	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
40	36, 39	sylibrd 169	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41	20	nn0red 9517	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
42		nnre 9209	. . . . . . . . 9
43	42	ad2antrl 490	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
44		ltle 8326	. . . . . . . 8 $/\$
45	41, 43, 44	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
46	40, 45	syld 45	. . . . . 6 $/\$ $/\$ $/\$ $/\$
47	46	anassrs 400	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	ralrimdva 2613	. . . 4 $/\$ $/\$ $/\$
49	48	reximdva 2635	. . 3 $/\$ $/\$
50	10, 49	mpd 13	. 2 $/\$ $/\$
51		ssrexv 3293	. 2
52	1, 50, 51	mpsyl 65	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1398

wcel 2202

wne 2403

wral 2511

wrex 2512

crab 2515

wss 3201 class class class wbr 4093

cfv 5333 (class class class)co 6028

cr 8091

cc0 8092

c1 8093

clt 8273

cle 8274

cn 9202

c2 9253

cn0 9461

cz 9540

cuz 9816

cexp 10863

cabs 11637

cdvds 12428

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8183 ax-resscn 8184 ax-1cn 8185 ax-1re 8186 ax-icn 8187 ax-addcl 8188 ax-addrcl 8189 ax-mulcl 8190 ax-mulrcl 8191 ax-addcom 8192 ax-mulcom 8193 ax-addass 8194 ax-mulass 8195 ax-distr 8196 ax-i2m1 8197 ax-0lt1 8198 ax-1rid 8199 ax-0id 8200 ax-rnegex 8201 ax-precex 8202 ax-cnre 8203 ax-pre-ltirr 8204 ax-pre-ltwlin 8205 ax-pre-lttrn 8206 ax-pre-apti 8207 ax-pre-ltadd 8208 ax-pre-mulgt0 8209 ax-pre-mulext 8210 ax-arch 8211 ax-caucvg 8212

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-pnf 8275 df-mnf 8276 df-xr 8277 df-ltxr 8278 df-le 8279 df-sub 8411 df-neg 8412 df-reap 8814 df-ap 8821 df-div 8912 df-inn 9203 df-2 9261 df-3 9262 df-4 9263 df-n0 9462 df-z 9541 df-uz 9817 df-q 9915 df-rp 9950 df-seqfrec 10773 df-exp 10864 df-cj 11482 df-re 11483 df-im 11484 df-rsqrt 11638 df-abs 11639 df-dvds 12429

This theorem is referenced by: pcprecl 12942 pcprendvds 12943 pcpremul 12946

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