pcgcd1 - Intuitionistic Logic Explorer

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Theorem pcgcd1 12766

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 5975	. . . 4
2	1	oveq2d 5983	. . 3
3		simp2 1001	. . . . . . 7 $/\$ $/\$
4		gcdid0 12416	. . . . . . 7
5	3, 4	syl 14	. . . . . 6 $/\$ $/\$
6	5	oveq2d 5983	. . . . 5 $/\$ $/\$
7		zq 9782	. . . . . . 7
8		pcabs 12764	. . . . . . 7 $/\$
9	7, 8	sylan2 286	. . . . . 6 $/\$
10	9	3adant3 1020	. . . . 5 $/\$ $/\$
11	6, 10	eqtrd 2240	. . . 4 $/\$ $/\$
12	11	adantr 276	. . 3 $/\$ $/\$ $/\$
13	2, 12	sylan9eqr 2262	. 2 $/\$ $/\$ $/\$ $/\$
14		simpl1 1003	. . . . 5 $/\$ $/\$ $/\$ $/\$
15	3	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
16		simpl3 1005	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
17		simprr 531	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
18		simpr 110	. . . . . . . . 9 $/\$
19	18	necon3ai 2427	. . . . . . . 8 $/\$
20	17, 19	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
21		gcdn0cl 12398	. . . . . . 7 $/\$ $/\$ $/\$
22	15, 16, 20, 21	syl21anc 1249	. . . . . 6 $/\$ $/\$ $/\$ $/\$
23	22	nnzd 9529	. . . . 5 $/\$ $/\$ $/\$ $/\$
24		gcddvds 12399	. . . . . . 7 $/\$ $/\$
25	15, 16, 24	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
26	25	simpld 112	. . . . 5 $/\$ $/\$ $/\$ $/\$
27		pcdvdstr 12765	. . . . 5 $/\$ $/\$ $/\$
28	14, 23, 15, 26, 27	syl13anc 1252	. . . 4 $/\$ $/\$ $/\$ $/\$
29	15, 7	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
30		pcxcl 12749	. . . . . . . . . 10 $/\$
31	14, 29, 30	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
32		pczcl 12736	. . . . . . . . . . 11 $/\$ $/\$
33	14, 16, 17, 32	syl12anc 1248	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34	33	nn0red 9384	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35		pcge0 12751	. . . . . . . . . . 11 $/\$
36	14, 15, 35	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
37		ge0gtmnf 9980	. . . . . . . . . 10 $/\$
38	31, 36, 37	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
39		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
40		xrre 9977	. . . . . . . . 9 $/\$ $/\$ $/\$
41	31, 34, 38, 39, 40	syl22anc 1251	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
42		pnfnre 8149	. . . . . . . . . . . 12
43	42	neli 2475	. . . . . . . . . . 11
44		pc0 12742	. . . . . . . . . . . . 13
45	14, 44	syl 14	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
46	45	eleq1d 2276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
47	43, 46	mtbiri 677	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
48		oveq2 5975	. . . . . . . . . . . 12
49	48	eleq1d 2276	. . . . . . . . . . 11
50	49	notbid 669	. . . . . . . . . 10
51	47, 50	syl5ibrcom 157	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
52	51	necon2ad 2435	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
53	41, 52	mpd 13	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
54		pczdvds 12752	. . . . . . 7 $/\$ $/\$
55	14, 15, 53, 54	syl12anc 1248	. . . . . 6 $/\$ $/\$ $/\$ $/\$
56		pczcl 12736	. . . . . . . . 9 $/\$ $/\$
57	14, 15, 53, 56	syl12anc 1248	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
58		pcdvdsb 12758	. . . . . . . 8 $/\$ $/\$
59	14, 16, 57, 58	syl3anc 1250	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
60	39, 59	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$ $/\$
61		prmnn 12547	. . . . . . . . . 10
62	14, 61	syl 14	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
63	62, 57	nnexpcld 10877	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
64	63	nnzd 9529	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
65		dvdsgcd 12448	. . . . . . 7 $/\$ $/\$ $/\$
66	64, 15, 16, 65	syl3anc 1250	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
67	55, 60, 66	mp2and 433	. . . . 5 $/\$ $/\$ $/\$ $/\$
68		pcdvdsb 12758	. . . . . 6 $/\$ $/\$
69	14, 23, 57, 68	syl3anc 1250	. . . . 5 $/\$ $/\$ $/\$ $/\$
70	67, 69	mpbird 167	. . . 4 $/\$ $/\$ $/\$ $/\$
71	14, 22	pccld 12738	. . . . . 6 $/\$ $/\$ $/\$ $/\$
72	71	nn0red 9384	. . . . 5 $/\$ $/\$ $/\$ $/\$
73	72, 41	letri3d 8223	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
74	28, 70, 73	mpbir2and 947	. . 3 $/\$ $/\$ $/\$ $/\$
75	74	anassrs 400	. 2 $/\$ $/\$ $/\$ $/\$
76		simpl3 1005	. . . 4 $/\$ $/\$ $/\$
77		0zd 9419	. . . 4 $/\$ $/\$ $/\$
78		zdceq 9483	. . . 4 $/\$ DECID
79	76, 77, 78	syl2anc 411	. . 3 $/\$ $/\$ $/\$ DECID
80		dcne 2389	. . 3 DECID $\/$
81	79, 80	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$
82	13, 75, 81	mpjaodan 800	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 710 DECID wdc 836 $/\$ w3a 981

wceq 1373

wcel 2178

wne 2378 class class class wbr 4059

cfv 5290 (class class class)co 5967

cr 7959

cc0 7960

cpnf 8139

cmnf 8140

cxr 8141

clt 8142

cle 8143

cn 9071

cn0 9330

cz 9407

cq 9775

cexp 10720

cabs 11423

cdvds 12213

cgcd 12389

cprime 12544

cpc 12722

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4175 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-iinf 4654 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-mulrcl 8059 ax-addcom 8060 ax-mulcom 8061 ax-addass 8062 ax-mulass 8063 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-1rid 8067 ax-0id 8068 ax-rnegex 8069 ax-precex 8070 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-ltwlin 8073 ax-pre-lttrn 8074 ax-pre-apti 8075 ax-pre-ltadd 8076 ax-pre-mulgt0 8077 ax-pre-mulext 8078 ax-arch 8079 ax-caucvg 8080

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rmo 2494 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-if 3580 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-iun 3943 df-br 4060 df-opab 4122 df-mpt 4123 df-tr 4159 df-id 4358 df-po 4361 df-iso 4362 df-iord 4431 df-on 4433 df-ilim 4434 df-suc 4436 df-iom 4657 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-isom 5299 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-1st 6249 df-2nd 6250 df-recs 6414 df-frec 6500 df-1o 6525 df-2o 6526 df-er 6643 df-en 6851 df-sup 7112 df-inf 7113 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147 df-le 8148 df-sub 8280 df-neg 8281 df-reap 8683 df-ap 8690 df-div 8781 df-inn 9072 df-2 9130 df-3 9131 df-4 9132 df-n0 9331 df-z 9408 df-uz 9684 df-q 9776 df-rp 9811 df-fz 10166 df-fzo 10300 df-fl 10450 df-mod 10505 df-seqfrec 10630 df-exp 10721 df-cj 11268 df-re 11269 df-im 11270 df-rsqrt 11424 df-abs 11425 df-dvds 12214 df-gcd 12390 df-prm 12545 df-pc 12723

This theorem is referenced by: pcgcd 12767

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