pclemub - Intuitionistic Logic Explorer

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Theorem pclemub 12290

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 9273	. 2
2		zcn 9261	. . . . . 6
3	2	abscld 11193	. . . . 5
4	3	ad2antrl 490	. . . 4 $/\$ $/\$
5		eluzelre 9541	. . . . 5
6	5	adantr 276	. . . 4 $/\$ $/\$
7		eluz2gt1 9605	. . . . 5
8	7	adantr 276	. . . 4 $/\$ $/\$
9		expnbnd 10647	. . . 4 $/\$ $/\$
10	4, 6, 8, 9	syl3anc 1238	. . 3 $/\$ $/\$
11		simprr 531	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
12		oveq2 5886	. . . . . . . . . . . . . 14
13	12	breq1d 4015	. . . . . . . . . . . . 13
14		pclem.1	. . . . . . . . . . . . 13
15	13, 14	elrab2 2898	. . . . . . . . . . . 12 $/\$
16	11, 15	sylib 122	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
17	16	simprd 114	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
18		eluz2nn 9569	. . . . . . . . . . . . . 14
19	18	ad2antrr 488	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
20	16	simpld 112	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
21	19, 20	nnexpcld 10679	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
22	21	nnzd 9377	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
23		simplrl 535	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
24		simplrr 536	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
25		dvdsleabs 11854	. . . . . . . . . . 11 $/\$ $/\$
26	22, 23, 24, 25	syl3anc 1238	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
27	17, 26	mpd 13	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
28	21	nnred 8935	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
29	4	adantr 276	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
30	5	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
31		nnnn0 9186	. . . . . . . . . . . 12
32	31	ad2antrl 490	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
33	30, 32	reexpcld 10674	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34		lelttr 8049	. . . . . . . . . 10 $/\$ $/\$ $/\$
35	28, 29, 33, 34	syl3anc 1238	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
36	27, 35	mpand 429	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
37	7	ad2antrr 488	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
38		nn0ltexp2 10692	. . . . . . . . 9 $/\$ $/\$ $/\$
39	30, 20, 32, 37, 38	syl31anc 1241	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
40	36, 39	sylibrd 169	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41	20	nn0red 9233	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
42		nnre 8929	. . . . . . . . 9
43	42	ad2antrl 490	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
44		ltle 8048	. . . . . . . 8 $/\$
45	41, 43, 44	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
46	40, 45	syld 45	. . . . . 6 $/\$ $/\$ $/\$ $/\$
47	46	anassrs 400	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	ralrimdva 2557	. . . 4 $/\$ $/\$ $/\$
49	48	reximdva 2579	. . 3 $/\$ $/\$
50	10, 49	mpd 13	. 2 $/\$ $/\$
51		ssrexv 3222	. 2
52	1, 50, 51	mpsyl 65	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1353

wcel 2148

wne 2347

wral 2455

wrex 2456

crab 2459

wss 3131 class class class wbr 4005

cfv 5218 (class class class)co 5878

cr 7813

cc0 7814

c1 7815

clt 7995

cle 7996

cn 8922

c2 8973

cn0 9179

cz 9256

cuz 9531

cexp 10522

cabs 11009

cdvds 11797

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-mulrcl 7913 ax-addcom 7914 ax-mulcom 7915 ax-addass 7916 ax-mulass 7917 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-1rid 7921 ax-0id 7922 ax-rnegex 7923 ax-precex 7924 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 ax-pre-mulgt0 7931 ax-pre-mulext 7932 ax-arch 7933 ax-caucvg 7934

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-recs 6309 df-frec 6395 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-reap 8535 df-ap 8542 df-div 8633 df-inn 8923 df-2 8981 df-3 8982 df-4 8983 df-n0 9180 df-z 9257 df-uz 9532 df-q 9623 df-rp 9657 df-seqfrec 10449 df-exp 10523 df-cj 10854 df-re 10855 df-im 10856 df-rsqrt 11010 df-abs 11011 df-dvds 11798

This theorem is referenced by: pcprecl 12292 pcprendvds 12293 pcpremul 12296

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