Theorem pcgcd1 13026

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 6058	. . . 4 |- ( B = 0 -> ( A gcd B ) = ( A gcd 0 ) )
2	1	oveq2d 6066	. . 3 |- ( B = 0 -> ( P pCnt ( A gcd B ) ) = ( P pCnt ( A gcd 0 ) ) )
3		simp2 1025	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> A e. ZZ )
4		gcdid0 12676	. . . . . . 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 6066	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt ( A gcd 0 ) ) = ( P pCnt ( abs ` A ) ) )
7		zq 9958	. . . . . . 7 |- ( A e. ZZ -> A e. QQ )
8		pcabs 13024	. . . . . . 7 |- ( ( P e. Prime /\ A e. QQ ) -> ( P pCnt ( abs ` A ) ) = ( P pCnt A ) )
9	7, 8	sylan2 286	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P pCnt ( abs ` A ) ) = ( P pCnt A ) )
10	9	3adant3 1044	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt ( abs ` A ) ) = ( P pCnt A ) )
11	6, 10	eqtrd 2265	. . . 4 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt ( A gcd 0 ) ) = ( P pCnt A ) )
12	11	adantr 276	. . 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 2287	. 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 1027	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> P e. Prime )
15	3	adantr 276	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> A e. ZZ )
16		simpl3 1029	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> B e. ZZ )
17		simprr 533	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> B =/= 0 )
18		simpr 110	. . . . . . . . 9 |- ( ( A = 0 /\ B = 0 ) -> B = 0 )
19	18	necon3ai 2461	. . . . . . . 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 12658	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
22	15, 16, 20, 21	syl21anc 1273	. . . . . 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 9699	. . . . 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 12659	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
25	15, 16, 24	syl2anc 411	. . . . . 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 112	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> ( A gcd B ) || A )
27		pcdvdstr 13025	. . . . 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 1276	. . . 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 13009	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. QQ ) -> ( P pCnt A ) e. RR* )
31	14, 29, 30	syl2anc 411	. . . . . . . . 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 12996	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt B ) e. NN0 )
33	14, 16, 17, 32	syl12anc 1272	. . . . . . . . . 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 9554	. . . . . . . . 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 13011	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ ) -> 0 <_ ( P pCnt A ) )
36	14, 15, 35	syl2anc 411	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> 0 <_ ( P pCnt A ) )
37		ge0gtmnf 10156	. . . . . . . . . 10 |- ( ( ( P pCnt A ) e. RR* /\ 0 <_ ( P pCnt A ) ) -> -oo < ( P pCnt A ) )
38	31, 36, 37	syl2anc 411	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> -oo < ( P pCnt A ) )
39		simprl 531	. . . . . . . . 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 10153	. . . . . . . . 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 1275	. . . . . . . 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 8315	. . . . . . . . . . . 12 |- +oo e/ RR
43	42	neli 2509	. . . . . . . . . . 11 |- -. +oo e. RR
44		pc0 13002	. . . . . . . . . . . . 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 2301	. . . . . . . . . . 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 682	. . . . . . . . . 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 6058	. . . . . . . . . . . 12 |- ( A = 0 -> ( P pCnt A ) = ( P pCnt 0 ) )
49	48	eleq1d 2301	. . . . . . . . . . 11 |- ( A = 0 -> ( ( P pCnt A ) e. RR <-> ( P pCnt 0 ) e. RR ) )
50	49	notbid 673	. . . . . . . . . 10 |- ( A = 0 -> ( -. ( P pCnt A ) e. RR <-> -. ( P pCnt 0 ) e. RR ) )
51	47, 50	syl5ibrcom 157	. . . . . . . . 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 2469	. . . . . . . 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 13012	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P ^ ( P pCnt A ) ) || A )
55	14, 15, 53, 54	syl12anc 1272	. . . . . 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 12996	. . . . . . . . 9 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) e. NN0 )
57	14, 15, 53, 56	syl12anc 1272	. . . . . . . 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 13018	. . . . . . . 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 1274	. . . . . . 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 147	. . . . . 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 12807	. . . . . . . . . 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 11057	. . . . . . . 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 9699	. . . . . . 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 12708	. . . . . . 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 1274	. . . . . 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 433	. . . . 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 13018	. . . . . 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 1274	. . . . 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 167	. . . 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 12998	. . . . . 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 9554	. . . . 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 8389	. . . 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 953	. . 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 400	. 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 1029	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> B e. ZZ )
77		0zd 9589	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> 0 e. ZZ )
78		zdceq 9653	. . . 4 |- ( ( B e. ZZ /\ 0 e. ZZ ) -> DECID B = 0 )
79	76, 77, 78	syl2anc 411	. . 3 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> DECID B = 0 )
80		dcne 2423	. . 3 |- (DECID B = 0 <-> ( B = 0 \/ B =/= 0 ) )
81	79, 80	sylib 122	. 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 806	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 104   <-> wb 105   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   RRcr 8126   0cc0 8127   +oocpnf 8305   -oocmnf 8306   RR*cxr 8307   < clt 8308   <_ cle 8309   NNcn 9237   NN0cn0 9496   ZZcz 9577   QQcq 9951   ^cexp 10900   abscabs 11682   || cdvds 12473   gcd cgcd 12649   Primecprime 12804   pCnt cpc 12982

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-2o 6648  df-er 6767  df-en 6976  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-dvds 12474  df-gcd 12650  df-prm 12805  df-pc 12983

This theorem is referenced by:  pcgcd  13027