pcgcd1 - Intuitionistic Logic Explorer

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Theorem pcgcd1 12345

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 5899	. . . 4
2	1	oveq2d 5907	. . 3
3		simp2 1000	. . . . . . 7 $/\$ $/\$
4		gcdid0 11999	. . . . . . 7
5	3, 4	syl 14	. . . . . 6 $/\$ $/\$
6	5	oveq2d 5907	. . . . 5 $/\$ $/\$
7		zq 9644	. . . . . . 7
8		pcabs 12343	. . . . . . 7 $/\$
9	7, 8	sylan2 286	. . . . . 6 $/\$
10	9	3adant3 1019	. . . . 5 $/\$ $/\$
11	6, 10	eqtrd 2222	. . . 4 $/\$ $/\$
12	11	adantr 276	. . 3 $/\$ $/\$ $/\$
13	2, 12	sylan9eqr 2244	. 2 $/\$ $/\$ $/\$ $/\$
14		simpl1 1002	. . . . 5 $/\$ $/\$ $/\$ $/\$
15	3	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
16		simpl3 1004	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
17		simprr 531	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
18		simpr 110	. . . . . . . . 9 $/\$
19	18	necon3ai 2409	. . . . . . . 8 $/\$
20	17, 19	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
21		gcdn0cl 11981	. . . . . . 7 $/\$ $/\$ $/\$
22	15, 16, 20, 21	syl21anc 1248	. . . . . 6 $/\$ $/\$ $/\$ $/\$
23	22	nnzd 9392	. . . . 5 $/\$ $/\$ $/\$ $/\$
24		gcddvds 11982	. . . . . . 7 $/\$ $/\$
25	15, 16, 24	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
26	25	simpld 112	. . . . 5 $/\$ $/\$ $/\$ $/\$
27		pcdvdstr 12344	. . . . 5 $/\$ $/\$ $/\$
28	14, 23, 15, 26, 27	syl13anc 1251	. . . 4 $/\$ $/\$ $/\$ $/\$
29	15, 7	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
30		pcxcl 12329	. . . . . . . . . 10 $/\$
31	14, 29, 30	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
32		pczcl 12316	. . . . . . . . . . 11 $/\$ $/\$
33	14, 16, 17, 32	syl12anc 1247	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34	33	nn0red 9248	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35		pcge0 12330	. . . . . . . . . . 11 $/\$
36	14, 15, 35	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
37		ge0gtmnf 9841	. . . . . . . . . 10 $/\$
38	31, 36, 37	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
39		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
40		xrre 9838	. . . . . . . . 9 $/\$ $/\$ $/\$
41	31, 34, 38, 39, 40	syl22anc 1250	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
42		pnfnre 8017	. . . . . . . . . . . 12
43	42	neli 2457	. . . . . . . . . . 11
44		pc0 12322	. . . . . . . . . . . . 13
45	14, 44	syl 14	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
46	45	eleq1d 2258	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
47	43, 46	mtbiri 676	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
48		oveq2 5899	. . . . . . . . . . . 12
49	48	eleq1d 2258	. . . . . . . . . . 11
50	49	notbid 668	. . . . . . . . . 10
51	47, 50	syl5ibrcom 157	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
52	51	necon2ad 2417	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
53	41, 52	mpd 13	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
54		pczdvds 12331	. . . . . . 7 $/\$ $/\$
55	14, 15, 53, 54	syl12anc 1247	. . . . . 6 $/\$ $/\$ $/\$ $/\$
56		pczcl 12316	. . . . . . . . 9 $/\$ $/\$
57	14, 15, 53, 56	syl12anc 1247	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
58		pcdvdsb 12337	. . . . . . . 8 $/\$ $/\$
59	14, 16, 57, 58	syl3anc 1249	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
60	39, 59	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$ $/\$
61		prmnn 12128	. . . . . . . . . 10
62	14, 61	syl 14	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
63	62, 57	nnexpcld 10694	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
64	63	nnzd 9392	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
65		dvdsgcd 12031	. . . . . . 7 $/\$ $/\$ $/\$
66	64, 15, 16, 65	syl3anc 1249	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
67	55, 60, 66	mp2and 433	. . . . 5 $/\$ $/\$ $/\$ $/\$
68		pcdvdsb 12337	. . . . . 6 $/\$ $/\$
69	14, 23, 57, 68	syl3anc 1249	. . . . 5 $/\$ $/\$ $/\$ $/\$
70	67, 69	mpbird 167	. . . 4 $/\$ $/\$ $/\$ $/\$
71	14, 22	pccld 12318	. . . . . 6 $/\$ $/\$ $/\$ $/\$
72	71	nn0red 9248	. . . . 5 $/\$ $/\$ $/\$ $/\$
73	72, 41	letri3d 8091	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
74	28, 70, 73	mpbir2and 946	. . 3 $/\$ $/\$ $/\$ $/\$
75	74	anassrs 400	. 2 $/\$ $/\$ $/\$ $/\$
76		simpl3 1004	. . . 4 $/\$ $/\$ $/\$
77		0zd 9283	. . . 4 $/\$ $/\$ $/\$
78		zdceq 9346	. . . 4 $/\$ DECID
79	76, 77, 78	syl2anc 411	. . 3 $/\$ $/\$ $/\$ DECID
80		dcne 2371	. . 3 DECID $\/$
81	79, 80	sylib 122	. 2 $/\$ $/\$ $/\$ $\/$
82	13, 75, 81	mpjaodan 799	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709 DECID wdc 835 $/\$ w3a 980

wceq 1364

wcel 2160

wne 2360 class class class wbr 4018

cfv 5231 (class class class)co 5891

cr 7828

cc0 7829

cpnf 8007

cmnf 8008

cxr 8009

clt 8010

cle 8011

cn 8937

cn0 9194

cz 9271

cq 9637

cexp 10537

cabs 11024

cdvds 11812

cgcd 11961

cprime 12125

cpc 12302

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-mulrcl 7928 ax-addcom 7929 ax-mulcom 7930 ax-addass 7931 ax-mulass 7932 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-1rid 7936 ax-0id 7937 ax-rnegex 7938 ax-precex 7939 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-apti 7944 ax-pre-ltadd 7945 ax-pre-mulgt0 7946 ax-pre-mulext 7947 ax-arch 7948 ax-caucvg 7949

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-isom 5240 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-frec 6410 df-1o 6435 df-2o 6436 df-er 6553 df-en 6759 df-sup 7001 df-inf 7002 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-reap 8550 df-ap 8557 df-div 8648 df-inn 8938 df-2 8996 df-3 8997 df-4 8998 df-n0 9195 df-z 9272 df-uz 9547 df-q 9638 df-rp 9672 df-fz 10027 df-fzo 10161 df-fl 10288 df-mod 10341 df-seqfrec 10464 df-exp 10538 df-cj 10869 df-re 10870 df-im 10871 df-rsqrt 11025 df-abs 11026 df-dvds 11813 df-gcd 11962 df-prm 12126 df-pc 12303

This theorem is referenced by: pcgcd 12346

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