pcgcd1 - Intuitionistic Logic Explorer

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Theorem pcgcd1 12981

Description: The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.)

**Assertion**
Ref	Expression
pcgcd1	$/\$ $/\$ $/\$

**Proof of Theorem pcgcd1**
Step	Hyp	Ref	Expression
1		oveq2 6036	. . . 4
2	1	oveq2d 6044	. . 3
3		simp2 1025	. . . . . . 7 $/\$ $/\$
4		gcdid0 12631	. . . . . . 7
5	3, 4	syl 14	. . . . . 6 $/\$ $/\$
6	5	oveq2d 6044	. . . . 5 $/\$ $/\$
7		zq 9921	. . . . . . 7
8		pcabs 12979	. . . . . . 7 $/\$
9	7, 8	sylan2 286	. . . . . 6 $/\$
10	9	3adant3 1044	. . . . 5 $/\$ $/\$
11	6, 10	eqtrd 2264	. . . 4 $/\$ $/\$
12	11	adantr 276	. . 3 $/\$ $/\$ $/\$
13	2, 12	sylan9eqr 2286	. 2 $/\$ $/\$ $/\$ $/\$
14		simpl1 1027	. . . . 5 $/\$ $/\$ $/\$ $/\$
15	3	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
16		simpl3 1029	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
17		simprr 533	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
18		simpr 110	. . . . . . . . 9 $/\$
19	18	necon3ai 2452	. . . . . . . 8 $/\$
20	17, 19	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
21		gcdn0cl 12613	. . . . . . 7 $/\$ $/\$ $/\$
22	15, 16, 20, 21	syl21anc 1273	. . . . . 6 $/\$ $/\$ $/\$ $/\$
23	22	nnzd 9662	. . . . 5 $/\$ $/\$ $/\$ $/\$
24		gcddvds 12614	. . . . . . 7 $/\$ $/\$
25	15, 16, 24	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
26	25	simpld 112	. . . . 5 $/\$ $/\$ $/\$ $/\$
27		pcdvdstr 12980	. . . . 5 $/\$ $/\$ $/\$
28	14, 23, 15, 26, 27	syl13anc 1276	. . . 4 $/\$ $/\$ $/\$ $/\$
29	15, 7	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
30		pcxcl 12964	. . . . . . . . . 10 $/\$
31	14, 29, 30	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
32		pczcl 12951	. . . . . . . . . . 11 $/\$ $/\$
33	14, 16, 17, 32	syl12anc 1272	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34	33	nn0red 9517	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35		pcge0 12966	. . . . . . . . . . 11 $/\$
36	14, 15, 35	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
37		ge0gtmnf 10119	. . . . . . . . . 10 $/\$
38	31, 36, 37	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
39		simprl 531	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
40		xrre 10116	. . . . . . . . 9 $/\$ $/\$ $/\$
41	31, 34, 38, 39, 40	syl22anc 1275	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
42		pnfnre 8280	. . . . . . . . . . . 12
43	42	neli 2500	. . . . . . . . . . 11
44		pc0 12957	. . . . . . . . . . . . 13
45	14, 44	syl 14	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
46	45	eleq1d 2300	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
47	43, 46	mtbiri 682	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
48		oveq2 6036	. . . . . . . . . . . 12
49	48	eleq1d 2300	. . . . . . . . . . 11
50	49	notbid 673	. . . . . . . . . 10
51	47, 50	syl5ibrcom 157	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
52	51	necon2ad 2460	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
53	41, 52	mpd 13	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
54		pczdvds 12967	. . . . . . 7 $/\$ $/\$
55	14, 15, 53, 54	syl12anc 1272	. . . . . 6 $/\$ $/\$ $/\$ $/\$
56		pczcl 12951	. . . . . . . . 9 $/\$ $/\$
57	14, 15, 53, 56	syl12anc 1272	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
58		pcdvdsb 12973	. . . . . . . 8 $/\$ $/\$
59	14, 16, 57, 58	syl3anc 1274	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
60	39, 59	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$ $/\$
61		prmnn 12762	. . . . . . . . . 10
62	14, 61	syl 14	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
63	62, 57	nnexpcld 11020	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
64	63	nnzd 9662	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
65		dvdsgcd 12663	. . . . . . 7 $/\$ $/\$ $/\$
66	64, 15, 16, 65	syl3anc 1274	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
67	55, 60, 66	mp2and 433	. . . . 5 $/\$ $/\$ $/\$ $/\$
68		pcdvdsb 12973	. . . . . 6 $/\$ $/\$
69	14, 23, 57, 68	syl3anc 1274	. . . . 5 $/\$ $/\$ $/\$ $/\$
70	67, 69	mpbird 167	. . . 4 $/\$ $/\$ $/\$ $/\$
71	14, 22	pccld 12953	. . . . . 6 $/\$ $/\$ $/\$ $/\$
72	71	nn0red 9517	. . . . 5 $/\$ $/\$ $/\$ $/\$
73	72, 41	letri3d 8354	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
74	28, 70, 73	mpbir2and 953	. . 3 $/\$ $/\$ $/\$ $/\$
75	74	anassrs 400	. 2 $/\$ $/\$ $/\$ $/\$
76		simpl3 1029	. . . 4 $/\$ $/\$ $/\$
77		0zd 9552	. . . 4 $/\$ $/\$ $/\$
78		zdceq 9616	. . . 4 $/\$ DECID
79	76, 77, 78	syl2anc 411	. . 3 $/\$ $/\$ $/\$ DECID
80		dcne 2414	. . 3 DECID $\/$
81	79, 80	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$
82	13, 75, 81	mpjaodan 806	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716 DECID wdc 842 $/\$ w3a 1005

wceq 1398

wcel 2202

wne 2403 class class class wbr 4093

cfv 5333 (class class class)co 6028

cr 8091

cc0 8092

cpnf 8270

cmnf 8271

cxr 8272

clt 8273

cle 8274

cn 9202

cn0 9461

cz 9540

cq 9914

cexp 10863

cabs 11637

cdvds 12428

cgcd 12604

cprime 12759

cpc 12937

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-stab 839 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-isom 5342 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-1o 6625 df-2o 6626 df-er 6745 df-en 6953 df-sup 7243 df-inf 7244 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-fz 10306 df-fzo 10440 df-fl 10593 df-mod 10648 df-seqfrec 10773 df-exp 10864 df-cj 11482 df-re 11483 df-im 11484 df-rsqrt 11638 df-abs 11639 df-dvds 12429 df-gcd 12605 df-prm 12760 df-pc 12938

This theorem is referenced by: pcgcd 12982

W3C validator