pcgcd1 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > pcgcd1		Unicode version

Theorem pcgcd1 12851

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 6009	. . . 4
2	1	oveq2d 6017	. . 3
3		simp2 1022	. . . . . . 7 $/\$ $/\$
4		gcdid0 12501	. . . . . . 7
5	3, 4	syl 14	. . . . . 6 $/\$ $/\$
6	5	oveq2d 6017	. . . . 5 $/\$ $/\$
7		zq 9821	. . . . . . 7
8		pcabs 12849	. . . . . . 7 $/\$
9	7, 8	sylan2 286	. . . . . 6 $/\$
10	9	3adant3 1041	. . . . 5 $/\$ $/\$
11	6, 10	eqtrd 2262	. . . 4 $/\$ $/\$
12	11	adantr 276	. . 3 $/\$ $/\$ $/\$
13	2, 12	sylan9eqr 2284	. 2 $/\$ $/\$ $/\$ $/\$
14		simpl1 1024	. . . . 5 $/\$ $/\$ $/\$ $/\$
15	3	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
16		simpl3 1026	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
17		simprr 531	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
18		simpr 110	. . . . . . . . 9 $/\$
19	18	necon3ai 2449	. . . . . . . 8 $/\$
20	17, 19	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
21		gcdn0cl 12483	. . . . . . 7 $/\$ $/\$ $/\$
22	15, 16, 20, 21	syl21anc 1270	. . . . . 6 $/\$ $/\$ $/\$ $/\$
23	22	nnzd 9568	. . . . 5 $/\$ $/\$ $/\$ $/\$
24		gcddvds 12484	. . . . . . 7 $/\$ $/\$
25	15, 16, 24	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
26	25	simpld 112	. . . . 5 $/\$ $/\$ $/\$ $/\$
27		pcdvdstr 12850	. . . . 5 $/\$ $/\$ $/\$
28	14, 23, 15, 26, 27	syl13anc 1273	. . . 4 $/\$ $/\$ $/\$ $/\$
29	15, 7	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
30		pcxcl 12834	. . . . . . . . . 10 $/\$
31	14, 29, 30	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
32		pczcl 12821	. . . . . . . . . . 11 $/\$ $/\$
33	14, 16, 17, 32	syl12anc 1269	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34	33	nn0red 9423	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35		pcge0 12836	. . . . . . . . . . 11 $/\$
36	14, 15, 35	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
37		ge0gtmnf 10019	. . . . . . . . . 10 $/\$
38	31, 36, 37	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
39		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
40		xrre 10016	. . . . . . . . 9 $/\$ $/\$ $/\$
41	31, 34, 38, 39, 40	syl22anc 1272	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
42		pnfnre 8188	. . . . . . . . . . . 12
43	42	neli 2497	. . . . . . . . . . 11
44		pc0 12827	. . . . . . . . . . . . 13
45	14, 44	syl 14	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
46	45	eleq1d 2298	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
47	43, 46	mtbiri 679	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
48		oveq2 6009	. . . . . . . . . . . 12
49	48	eleq1d 2298	. . . . . . . . . . 11
50	49	notbid 671	. . . . . . . . . 10
51	47, 50	syl5ibrcom 157	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
52	51	necon2ad 2457	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
53	41, 52	mpd 13	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
54		pczdvds 12837	. . . . . . 7 $/\$ $/\$
55	14, 15, 53, 54	syl12anc 1269	. . . . . 6 $/\$ $/\$ $/\$ $/\$
56		pczcl 12821	. . . . . . . . 9 $/\$ $/\$
57	14, 15, 53, 56	syl12anc 1269	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
58		pcdvdsb 12843	. . . . . . . 8 $/\$ $/\$
59	14, 16, 57, 58	syl3anc 1271	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
60	39, 59	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$ $/\$
61		prmnn 12632	. . . . . . . . . 10
62	14, 61	syl 14	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
63	62, 57	nnexpcld 10917	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
64	63	nnzd 9568	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
65		dvdsgcd 12533	. . . . . . 7 $/\$ $/\$ $/\$
66	64, 15, 16, 65	syl3anc 1271	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
67	55, 60, 66	mp2and 433	. . . . 5 $/\$ $/\$ $/\$ $/\$
68		pcdvdsb 12843	. . . . . 6 $/\$ $/\$
69	14, 23, 57, 68	syl3anc 1271	. . . . 5 $/\$ $/\$ $/\$ $/\$
70	67, 69	mpbird 167	. . . 4 $/\$ $/\$ $/\$ $/\$
71	14, 22	pccld 12823	. . . . . 6 $/\$ $/\$ $/\$ $/\$
72	71	nn0red 9423	. . . . 5 $/\$ $/\$ $/\$ $/\$
73	72, 41	letri3d 8262	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
74	28, 70, 73	mpbir2and 950	. . 3 $/\$ $/\$ $/\$ $/\$
75	74	anassrs 400	. 2 $/\$ $/\$ $/\$ $/\$
76		simpl3 1026	. . . 4 $/\$ $/\$ $/\$
77		0zd 9458	. . . 4 $/\$ $/\$ $/\$
78		zdceq 9522	. . . 4 $/\$ DECID
79	76, 77, 78	syl2anc 411	. . 3 $/\$ $/\$ $/\$ DECID
80		dcne 2411	. . 3 DECID $\/$
81	79, 80	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$
82	13, 75, 81	mpjaodan 803	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 713 DECID wdc 839 $/\$ w3a 1002

wceq 1395

wcel 2200

wne 2400 class class class wbr 4083

cfv 5318 (class class class)co 6001

cr 7998

cc0 7999

cpnf 8178

cmnf 8179

cxr 8180

clt 8181

cle 8182

cn 9110

cn0 9369

cz 9446

cq 9814

cexp 10760

cabs 11508

cdvds 12298

cgcd 12474

cprime 12629

cpc 12807

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116 ax-pre-mulext 8117 ax-arch 8118 ax-caucvg 8119

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-1st 6286 df-2nd 6287 df-recs 6451 df-frec 6537 df-1o 6562 df-2o 6563 df-er 6680 df-en 6888 df-sup 7151 df-inf 7152 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-reap 8722 df-ap 8729 df-div 8820 df-inn 9111 df-2 9169 df-3 9170 df-4 9171 df-n0 9370 df-z 9447 df-uz 9723 df-q 9815 df-rp 9850 df-fz 10205 df-fzo 10339 df-fl 10490 df-mod 10545 df-seqfrec 10670 df-exp 10761 df-cj 11353 df-re 11354 df-im 11355 df-rsqrt 11509 df-abs 11510 df-dvds 12299 df-gcd 12475 df-prm 12630 df-pc 12808

This theorem is referenced by: pcgcd 12852

W3C validator