pcgcd1 - Intuitionistic Logic Explorer

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Theorem pcgcd1 12964

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 12614	. . . . . . 7
5	3, 4	syl 14	. . . . . 6 $/\$ $/\$
6	5	oveq2d 6044	. . . . 5 $/\$ $/\$
7		zq 9905	. . . . . . 7
8		pcabs 12962	. . . . . . 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 12596	. . . . . . 7 $/\$ $/\$ $/\$
22	15, 16, 20, 21	syl21anc 1273	. . . . . 6 $/\$ $/\$ $/\$ $/\$
23	22	nnzd 9646	. . . . 5 $/\$ $/\$ $/\$ $/\$
24		gcddvds 12597	. . . . . . 7 $/\$ $/\$
25	15, 16, 24	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
26	25	simpld 112	. . . . 5 $/\$ $/\$ $/\$ $/\$
27		pcdvdstr 12963	. . . . 5 $/\$ $/\$ $/\$
28	14, 23, 15, 26, 27	syl13anc 1276	. . . 4 $/\$ $/\$ $/\$ $/\$
29	15, 7	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
30		pcxcl 12947	. . . . . . . . . 10 $/\$
31	14, 29, 30	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
32		pczcl 12934	. . . . . . . . . . 11 $/\$ $/\$
33	14, 16, 17, 32	syl12anc 1272	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34	33	nn0red 9501	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35		pcge0 12949	. . . . . . . . . . 11 $/\$
36	14, 15, 35	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
37		ge0gtmnf 10103	. . . . . . . . . 10 $/\$
38	31, 36, 37	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
39		simprl 531	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
40		xrre 10100	. . . . . . . . 9 $/\$ $/\$ $/\$
41	31, 34, 38, 39, 40	syl22anc 1275	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
42		pnfnre 8264	. . . . . . . . . . . 12
43	42	neli 2500	. . . . . . . . . . 11
44		pc0 12940	. . . . . . . . . . . . 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 12950	. . . . . . 7 $/\$ $/\$
55	14, 15, 53, 54	syl12anc 1272	. . . . . 6 $/\$ $/\$ $/\$ $/\$
56		pczcl 12934	. . . . . . . . 9 $/\$ $/\$
57	14, 15, 53, 56	syl12anc 1272	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
58		pcdvdsb 12956	. . . . . . . 8 $/\$ $/\$
59	14, 16, 57, 58	syl3anc 1274	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
60	39, 59	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$ $/\$
61		prmnn 12745	. . . . . . . . . 10
62	14, 61	syl 14	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
63	62, 57	nnexpcld 11003	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
64	63	nnzd 9646	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
65		dvdsgcd 12646	. . . . . . 7 $/\$ $/\$ $/\$
66	64, 15, 16, 65	syl3anc 1274	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
67	55, 60, 66	mp2and 433	. . . . 5 $/\$ $/\$ $/\$ $/\$
68		pcdvdsb 12956	. . . . . 6 $/\$ $/\$
69	14, 23, 57, 68	syl3anc 1274	. . . . 5 $/\$ $/\$ $/\$ $/\$
70	67, 69	mpbird 167	. . . 4 $/\$ $/\$ $/\$ $/\$
71	14, 22	pccld 12936	. . . . . 6 $/\$ $/\$ $/\$ $/\$
72	71	nn0red 9501	. . . . 5 $/\$ $/\$ $/\$ $/\$
73	72, 41	letri3d 8338	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
74	28, 70, 73	mpbir2and 953	. . 3 $/\$ $/\$ $/\$ $/\$
75	74	anassrs 400	. 2 $/\$ $/\$ $/\$ $/\$
76		simpl3 1029	. . . 4 $/\$ $/\$ $/\$
77		0zd 9536	. . . 4 $/\$ $/\$ $/\$
78		zdceq 9600	. . . 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 8074

cc0 8075

cpnf 8254

cmnf 8255

cxr 8256

clt 8257

cle 8258

cn 9186

cn0 9445

cz 9524

cq 9898

cexp 10846

cabs 11620

cdvds 12411

cgcd 12587

cprime 12742

cpc 12920

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 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194 ax-caucvg 8195

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 7226 df-inf 7227 df-pnf 8259 df-mnf 8260 df-xr 8261 df-ltxr 8262 df-le 8263 df-sub 8395 df-neg 8396 df-reap 8798 df-ap 8805 df-div 8896 df-inn 9187 df-2 9245 df-3 9246 df-4 9247 df-n0 9446 df-z 9525 df-uz 9801 df-q 9899 df-rp 9934 df-fz 10289 df-fzo 10423 df-fl 10576 df-mod 10631 df-seqfrec 10756 df-exp 10847 df-cj 11465 df-re 11466 df-im 11467 df-rsqrt 11621 df-abs 11622 df-dvds 12412 df-gcd 12588 df-prm 12743 df-pc 12921

This theorem is referenced by: pcgcd 12965

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