pcgcd1 - Intuitionistic Logic Explorer

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Theorem pcgcd1 12651

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 5952	. . . 4
2	1	oveq2d 5960	. . 3
3		simp2 1001	. . . . . . 7 $/\$ $/\$
4		gcdid0 12301	. . . . . . 7
5	3, 4	syl 14	. . . . . 6 $/\$ $/\$
6	5	oveq2d 5960	. . . . 5 $/\$ $/\$
7		zq 9747	. . . . . . 7
8		pcabs 12649	. . . . . . 7 $/\$
9	7, 8	sylan2 286	. . . . . 6 $/\$
10	9	3adant3 1020	. . . . 5 $/\$ $/\$
11	6, 10	eqtrd 2238	. . . 4 $/\$ $/\$
12	11	adantr 276	. . 3 $/\$ $/\$ $/\$
13	2, 12	sylan9eqr 2260	. 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 2425	. . . . . . . 8 $/\$
20	17, 19	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
21		gcdn0cl 12283	. . . . . . 7 $/\$ $/\$ $/\$
22	15, 16, 20, 21	syl21anc 1249	. . . . . 6 $/\$ $/\$ $/\$ $/\$
23	22	nnzd 9494	. . . . 5 $/\$ $/\$ $/\$ $/\$
24		gcddvds 12284	. . . . . . 7 $/\$ $/\$
25	15, 16, 24	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
26	25	simpld 112	. . . . 5 $/\$ $/\$ $/\$ $/\$
27		pcdvdstr 12650	. . . . 5 $/\$ $/\$ $/\$
28	14, 23, 15, 26, 27	syl13anc 1252	. . . 4 $/\$ $/\$ $/\$ $/\$
29	15, 7	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
30		pcxcl 12634	. . . . . . . . . 10 $/\$
31	14, 29, 30	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
32		pczcl 12621	. . . . . . . . . . 11 $/\$ $/\$
33	14, 16, 17, 32	syl12anc 1248	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34	33	nn0red 9349	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35		pcge0 12636	. . . . . . . . . . 11 $/\$
36	14, 15, 35	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
37		ge0gtmnf 9945	. . . . . . . . . 10 $/\$
38	31, 36, 37	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
39		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
40		xrre 9942	. . . . . . . . 9 $/\$ $/\$ $/\$
41	31, 34, 38, 39, 40	syl22anc 1251	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
42		pnfnre 8114	. . . . . . . . . . . 12
43	42	neli 2473	. . . . . . . . . . 11
44		pc0 12627	. . . . . . . . . . . . 13
45	14, 44	syl 14	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
46	45	eleq1d 2274	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
47	43, 46	mtbiri 677	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
48		oveq2 5952	. . . . . . . . . . . 12
49	48	eleq1d 2274	. . . . . . . . . . 11
50	49	notbid 669	. . . . . . . . . 10
51	47, 50	syl5ibrcom 157	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
52	51	necon2ad 2433	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
53	41, 52	mpd 13	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
54		pczdvds 12637	. . . . . . 7 $/\$ $/\$
55	14, 15, 53, 54	syl12anc 1248	. . . . . 6 $/\$ $/\$ $/\$ $/\$
56		pczcl 12621	. . . . . . . . 9 $/\$ $/\$
57	14, 15, 53, 56	syl12anc 1248	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
58		pcdvdsb 12643	. . . . . . . 8 $/\$ $/\$
59	14, 16, 57, 58	syl3anc 1250	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
60	39, 59	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$ $/\$
61		prmnn 12432	. . . . . . . . . 10
62	14, 61	syl 14	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
63	62, 57	nnexpcld 10840	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
64	63	nnzd 9494	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
65		dvdsgcd 12333	. . . . . . 7 $/\$ $/\$ $/\$
66	64, 15, 16, 65	syl3anc 1250	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
67	55, 60, 66	mp2and 433	. . . . 5 $/\$ $/\$ $/\$ $/\$
68		pcdvdsb 12643	. . . . . 6 $/\$ $/\$
69	14, 23, 57, 68	syl3anc 1250	. . . . 5 $/\$ $/\$ $/\$ $/\$
70	67, 69	mpbird 167	. . . 4 $/\$ $/\$ $/\$ $/\$
71	14, 22	pccld 12623	. . . . . 6 $/\$ $/\$ $/\$ $/\$
72	71	nn0red 9349	. . . . 5 $/\$ $/\$ $/\$ $/\$
73	72, 41	letri3d 8188	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
74	28, 70, 73	mpbir2and 947	. . 3 $/\$ $/\$ $/\$ $/\$
75	74	anassrs 400	. 2 $/\$ $/\$ $/\$ $/\$
76		simpl3 1005	. . . 4 $/\$ $/\$ $/\$
77		0zd 9384	. . . 4 $/\$ $/\$ $/\$
78		zdceq 9448	. . . 4 $/\$ DECID
79	76, 77, 78	syl2anc 411	. . 3 $/\$ $/\$ $/\$ DECID
80		dcne 2387	. . 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 2176

wne 2376 class class class wbr 4044

cfv 5271 (class class class)co 5944

cr 7924

cc0 7925

cpnf 8104

cmnf 8105

cxr 8106

clt 8107

cle 8108

cn 9036

cn0 9295

cz 9372

cq 9740

cexp 10683

cabs 11308

cdvds 12098

cgcd 12274

cprime 12429

cpc 12607

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 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043 ax-arch 8044 ax-caucvg 8045

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 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-po 4343 df-iso 4344 df-iord 4413 df-on 4415 df-ilim 4416 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-isom 5280 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-recs 6391 df-frec 6477 df-1o 6502 df-2o 6503 df-er 6620 df-en 6828 df-sup 7086 df-inf 7087 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 df-inn 9037 df-2 9095 df-3 9096 df-4 9097 df-n0 9296 df-z 9373 df-uz 9649 df-q 9741 df-rp 9776 df-fz 10131 df-fzo 10265 df-fl 10413 df-mod 10468 df-seqfrec 10593 df-exp 10684 df-cj 11153 df-re 11154 df-im 11155 df-rsqrt 11309 df-abs 11310 df-dvds 12099 df-gcd 12275 df-prm 12430 df-pc 12608

This theorem is referenced by: pcgcd 12652

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