Theorem pcgcd1 12919

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 6026	. . . 4 |- ( B = 0 -> ( A gcd B ) = ( A gcd 0 ) )
2	1	oveq2d 6034	. . 3 |- ( B = 0 -> ( P pCnt ( A gcd B ) ) = ( P pCnt ( A gcd 0 ) ) )
3		simp2 1024	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> A e. ZZ )
4		gcdid0 12569	. . . . . . 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 6034	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt ( A gcd 0 ) ) = ( P pCnt ( abs ` A ) ) )
7		zq 9860	. . . . . . 7 |- ( A e. ZZ -> A e. QQ )
8		pcabs 12917	. . . . . . 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 1043	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt ( abs ` A ) ) = ( P pCnt A ) )
11	6, 10	eqtrd 2264	. . . 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 2286	. 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 1026	. . . . 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 1028	. . . . . . 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 2451	. . . . . . . 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 12551	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
22	15, 16, 20, 21	syl21anc 1272	. . . . . 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 9601	. . . . 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 12552	. . . . . . 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 12918	. . . . 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 1275	. . . 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 12902	. . . . . . . . . 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 12889	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt B ) e. NN0 )
33	14, 16, 17, 32	syl12anc 1271	. . . . . . . . . 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 9456	. . . . . . . . 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 12904	. . . . . . . . . . 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 10058	. . . . . . . . . 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 10055	. . . . . . . . 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 1274	. . . . . . . 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 8221	. . . . . . . . . . . 12 |- +oo e/ RR
43	42	neli 2499	. . . . . . . . . . 11 |- -. +oo e. RR
44		pc0 12895	. . . . . . . . . . . . 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 2300	. . . . . . . . . . 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 681	. . . . . . . . . 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 6026	. . . . . . . . . . . 12 |- ( A = 0 -> ( P pCnt A ) = ( P pCnt 0 ) )
49	48	eleq1d 2300	. . . . . . . . . . 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 2459	. . . . . . . 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 12905	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P ^ ( P pCnt A ) ) || A )
55	14, 15, 53, 54	syl12anc 1271	. . . . . 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 12889	. . . . . . . . 9 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) e. NN0 )
57	14, 15, 53, 56	syl12anc 1271	. . . . . . . 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 12911	. . . . . . . 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 1273	. . . . . . 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 12700	. . . . . . . . . 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 10958	. . . . . . . 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 9601	. . . . . . 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 12601	. . . . . . 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 1273	. . . . . 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 12911	. . . . . 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 1273	. . . . 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 12891	. . . . . 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 9456	. . . . 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 8295	. . . 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 952	. . 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 1028	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> B e. ZZ )
77		0zd 9491	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> 0 e. ZZ )
78		zdceq 9555	. . . 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 2413	. . 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 805	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 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   RRcr 8031   0cc0 8032   +oocpnf 8211   -oocmnf 8212   RR*cxr 8213   < clt 8214   <_ cle 8215   NNcn 9143   NN0cn0 9402   ZZcz 9479   QQcq 9853   ^cexp 10801   abscabs 11575   || cdvds 12366   gcd cgcd 12542   Primecprime 12697   pCnt cpc 12875

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-2o 6583  df-er 6702  df-en 6910  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-cj 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577  df-dvds 12367  df-gcd 12543  df-prm 12698  df-pc 12876

This theorem is referenced by:  pcgcd  12920