Theorem pcgcd1 12270

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	|- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> ( P pCnt ( A gcd B ) ) = ( P pCnt A ) )

Proof of Theorem pcgcd1

Step	Hyp	Ref	Expression
1		oveq2 5859	. . . 4 |- ( B = 0 -> ( A gcd B ) = ( A gcd 0 ) )
2	1	oveq2d 5867	. . 3 |- ( B = 0 -> ( P pCnt ( A gcd B ) ) = ( P pCnt ( A gcd 0 ) ) )
3		simp2 993	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> A e. ZZ )
4		gcdid0 11924	. . . . . . 7 |- ( A e. ZZ -> ( A gcd 0 ) = ( abs ` A ) )
5	3, 4	syl 14	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( A gcd 0 ) = ( abs ` A ) )
6	5	oveq2d 5867	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt ( A gcd 0 ) ) = ( P pCnt ( abs ` A ) ) )
7		zq 9574	. . . . . . 7 |- ( A e. ZZ -> A e. QQ )
8		pcabs 12268	. . . . . . 7 |- ( ( P e. Prime /\ A e. QQ ) -> ( P pCnt ( abs ` A ) ) = ( P pCnt A ) )
9	7, 8	sylan2 284	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P pCnt ( abs ` A ) ) = ( P pCnt A ) )
10	9	3adant3 1012	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt ( abs ` A ) ) = ( P pCnt A ) )
11	6, 10	eqtrd 2203	. . . 4 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt ( A gcd 0 ) ) = ( P pCnt A ) )
12	11	adantr 274	. . 3 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> ( P pCnt ( A gcd 0 ) ) = ( P pCnt A ) )
13	2, 12	sylan9eqr 2225	. 2 |- ( ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) /\ B = 0 ) -> ( P pCnt ( A gcd B ) ) = ( P pCnt A ) )
14		simpl1 995	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> P e. Prime )
15	3	adantr 274	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> A e. ZZ )
16		simpl3 997	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> B e. ZZ )
17		simprr 527	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> B =/= 0 )
18		simpr 109	. . . . . . . . 9 |- ( ( A = 0 /\ B = 0 ) -> B = 0 )
19	18	necon3ai 2389	. . . . . . . 8 |- ( B =/= 0 -> -. ( A = 0 /\ B = 0 ) )
20	17, 19	syl 14	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> -. ( A = 0 /\ B = 0 ) )
21		gcdn0cl 11906	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
22	15, 16, 20, 21	syl21anc 1232	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( A gcd B ) e. NN )
23	22	nnzd 9322	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( A gcd B ) e. ZZ )
24		gcddvds 11907	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
25	15, 16, 24	syl2anc 409	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
26	25	simpld 111	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( A gcd B ) || A )
27		pcdvdstr 12269	. . . . 5 |- ( ( P e. Prime /\ ( ( A gcd B ) e. ZZ /\ A e. ZZ /\ ( A gcd B ) || A ) ) -> ( P pCnt ( A gcd B ) ) <_ ( P pCnt A ) )
28	14, 23, 15, 26, 27	syl13anc 1235	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P pCnt ( A gcd B ) ) <_ ( P pCnt A ) )
29	15, 7	syl 14	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> A e. QQ )
30		pcxcl 12254	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. QQ ) -> ( P pCnt A ) e. RR* )
31	14, 29, 30	syl2anc 409	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P pCnt A ) e. RR* )
32		pczcl 12241	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt B ) e. NN0 )
33	14, 16, 17, 32	syl12anc 1231	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P pCnt B ) e. NN0 )
34	33	nn0red 9178	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P pCnt B ) e. RR )
35		pcge0 12255	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ ) -> 0 <_ ( P pCnt A ) )
36	14, 15, 35	syl2anc 409	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> 0 <_ ( P pCnt A ) )
37		ge0gtmnf 9769	. . . . . . . . . 10 |- ( ( ( P pCnt A ) e. RR* /\ 0 <_ ( P pCnt A ) ) -> -oo < ( P pCnt A ) )
38	31, 36, 37	syl2anc 409	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> -oo < ( P pCnt A ) )
39		simprl 526	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P pCnt A ) <_ ( P pCnt B ) )
40		xrre 9766	. . . . . . . . 9 |- ( ( ( ( P pCnt A ) e. RR* /\ ( P pCnt B ) e. RR ) /\ ( -oo < ( P pCnt A ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) ) -> ( P pCnt A ) e. RR )
41	31, 34, 38, 39, 40	syl22anc 1234	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P pCnt A ) e. RR )
42		pnfnre 7950	. . . . . . . . . . . 12 |- +oo e/ RR
43	42	neli 2437	. . . . . . . . . . 11 |- -. +oo e. RR
44		pc0 12247	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( P pCnt 0 ) = +oo )
45	14, 44	syl 14	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P pCnt 0 ) = +oo )
46	45	eleq1d 2239	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( ( P pCnt 0 ) e. RR <-> +oo e. RR ) )
47	43, 46	mtbiri 670	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> -. ( P pCnt 0 ) e. RR )
48		oveq2 5859	. . . . . . . . . . . 12 |- ( A = 0 -> ( P pCnt A ) = ( P pCnt 0 ) )
49	48	eleq1d 2239	. . . . . . . . . . 11 |- ( A = 0 -> ( ( P pCnt A ) e. RR <-> ( P pCnt 0 ) e. RR ) )
50	49	notbid 662	. . . . . . . . . 10 |- ( A = 0 -> ( -. ( P pCnt A ) e. RR <-> -. ( P pCnt 0 ) e. RR ) )
51	47, 50	syl5ibrcom 156	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( A = 0 -> -. ( P pCnt A ) e. RR ) )
52	51	necon2ad 2397	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( ( P pCnt A ) e. RR -> A =/= 0 ) )
53	41, 52	mpd 13	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> A =/= 0 )
54		pczdvds 12256	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P ^ ( P pCnt A ) ) || A )
55	14, 15, 53, 54	syl12anc 1231	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P ^ ( P pCnt A ) ) || A )
56		pczcl 12241	. . . . . . . . 9 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) e. NN0 )
57	14, 15, 53, 56	syl12anc 1231	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P pCnt A ) e. NN0 )
58		pcdvdsb 12262	. . . . . . . 8 |- ( ( P e. Prime /\ B e. ZZ /\ ( P pCnt A ) e. NN0 ) -> ( ( P pCnt A ) <_ ( P pCnt B ) <-> ( P ^ ( P pCnt A ) ) || B ) )
59	14, 16, 57, 58	syl3anc 1233	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( ( P pCnt A ) <_ ( P pCnt B ) <-> ( P ^ ( P pCnt A ) ) || B ) )
60	39, 59	mpbid 146	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P ^ ( P pCnt A ) ) || B )
61		prmnn 12053	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
62	14, 61	syl 14	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> P e. NN )
63	62, 57	nnexpcld 10620	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P ^ ( P pCnt A ) ) e. NN )
64	63	nnzd 9322	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P ^ ( P pCnt A ) ) e. ZZ )
65		dvdsgcd 11956	. . . . . . 7 |- ( ( ( P ^ ( P pCnt A ) ) e. ZZ /\ A e. ZZ /\ B e. ZZ ) -> ( ( ( P ^ ( P pCnt A ) ) || A /\ ( P ^ ( P pCnt A ) ) || B ) -> ( P ^ ( P pCnt A ) ) || ( A gcd B ) ) )
66	64, 15, 16, 65	syl3anc 1233	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( ( ( P ^ ( P pCnt A ) ) || A /\ ( P ^ ( P pCnt A ) ) || B ) -> ( P ^ ( P pCnt A ) ) || ( A gcd B ) ) )
67	55, 60, 66	mp2and 431	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P ^ ( P pCnt A ) ) || ( A gcd B ) )
68		pcdvdsb 12262	. . . . . 6 |- ( ( P e. Prime /\ ( A gcd B ) e. ZZ /\ ( P pCnt A ) e. NN0 ) -> ( ( P pCnt A ) <_ ( P pCnt ( A gcd B ) ) <-> ( P ^ ( P pCnt A ) ) || ( A gcd B ) ) )
69	14, 23, 57, 68	syl3anc 1233	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( ( P pCnt A ) <_ ( P pCnt ( A gcd B ) ) <-> ( P ^ ( P pCnt A ) ) || ( A gcd B ) ) )
70	67, 69	mpbird 166	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P pCnt A ) <_ ( P pCnt ( A gcd B ) ) )
71	14, 22	pccld 12243	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P pCnt ( A gcd B ) ) e. NN0 )
72	71	nn0red 9178	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P pCnt ( A gcd B ) ) e. RR )
73	72, 41	letri3d 8024	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( ( P pCnt ( A gcd B ) ) = ( P pCnt A ) <-> ( ( P pCnt ( A gcd B ) ) <_ ( P pCnt A ) /\ ( P pCnt A ) <_ ( P pCnt ( A gcd B ) ) ) ) )
74	28, 70, 73	mpbir2and 939	. . 3 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( P pCnt ( A gcd B ) ) = ( P pCnt A ) )
75	74	anassrs 398	. 2 |- ( ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) /\ B =/= 0 ) -> ( P pCnt ( A gcd B ) ) = ( P pCnt A ) )
76		simpl3 997	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> B e. ZZ )
77		0zd 9213	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> 0 e. ZZ )
78		zdceq 9276	. . . 4 |- ( ( B e. ZZ /\ 0 e. ZZ ) -> DECID B = 0 )
79	76, 77, 78	syl2anc 409	. . 3 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> DECID B = 0 )
80		dcne 2351	. . 3 |- (DECID B = 0 <-> ( B = 0 \/ B =/= 0 ) )
81	79, 80	sylib 121	. 2 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> ( B = 0 \/ B =/= 0 ) )
82	13, 75, 81	mpjaodan 793	1 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> ( P pCnt ( A gcd B ) ) = ( P pCnt A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703  DECID wdc 829   /\ w3a 973   = wceq 1348   e. wcel 2141   =/= wne 2340   class class class wbr 3987   ` cfv 5196  (class class class)co 5851   RRcr 7762   0cc0 7763   +oocpnf 7940   -oocmnf 7941   RR*cxr 7942   < clt 7943   <_ cle 7944   NNcn 8867   NN0cn0 9124   ZZcz 9201   QQcq 9567   ^cexp 10464   abscabs 10950   || cdvds 11738   gcd cgcd 11886   Primecprime 12050   pCnt cpc 12227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7854  ax-resscn 7855  ax-1cn 7856  ax-1re 7857  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-mulrcl 7862  ax-addcom 7863  ax-mulcom 7864  ax-addass 7865  ax-mulass 7866  ax-distr 7867  ax-i2m1 7868  ax-0lt1 7869  ax-1rid 7870  ax-0id 7871  ax-rnegex 7872  ax-precex 7873  ax-cnre 7874  ax-pre-ltirr 7875  ax-pre-ltwlin 7876  ax-pre-lttrn 7877  ax-pre-apti 7878  ax-pre-ltadd 7879  ax-pre-mulgt0 7880  ax-pre-mulext 7881  ax-arch 7882  ax-caucvg 7883

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-frec 6368  df-1o 6393  df-2o 6394  df-er 6510  df-en 6716  df-sup 6958  df-inf 6959  df-pnf 7945  df-mnf 7946  df-xr 7947  df-ltxr 7948  df-le 7949  df-sub 8081  df-neg 8082  df-reap 8483  df-ap 8490  df-div 8579  df-inn 8868  df-2 8926  df-3 8927  df-4 8928  df-n0 9125  df-z 9202  df-uz 9477  df-q 9568  df-rp 9600  df-fz 9955  df-fzo 10088  df-fl 10215  df-mod 10268  df-seqfrec 10391  df-exp 10465  df-cj 10795  df-re 10796  df-im 10797  df-rsqrt 10951  df-abs 10952  df-dvds 11739  df-gcd 11887  df-prm 12051  df-pc 12228

This theorem is referenced by:  pcgcd  12271