Theorem pcgcd1 13136

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 6096	. . . 4 |- ( B = 0 -> ( A gcd B ) = ( A gcd 0 ) )
2	1	oveq2d 6104	. . 3 |- ( B = 0 -> ( P pCnt ( A gcd B ) ) = ( P pCnt ( A gcd 0 ) ) )
3		simp2 1029	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> A e. ZZ )
4		gcdid0 12782	. . . . . . 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 6104	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt ( A gcd 0 ) ) = ( P pCnt ( abs ` A ) ) )
7		zq 10042	. . . . . . 7 |- ( A e. ZZ -> A e. QQ )
8		pcabs 13134	. . . . . . 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 1048	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt ( abs ` A ) ) = ( P pCnt A ) )
11	6, 10	eqtrd 2271	. . . 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 2293	. 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 1031	. . . . 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 1033	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> B e. ZZ )
17		simprr 537	. . . . . . . 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 2469	. . . . . . . 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 12764	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
22	15, 16, 20, 21	syl21anc 1277	. . . . . 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 9778	. . . . 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 12765	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
25	15, 16, 24	syl2anc 415	. . . . . 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 13135	. . . . 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 1280	. . . 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 13119	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. QQ ) -> ( P pCnt A ) e. RR* )
31	14, 29, 30	syl2anc 415	. . . . . . . . 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 13106	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt B ) e. NN0 )
33	14, 16, 17, 32	syl12anc 1276	. . . . . . . . . 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 9632	. . . . . . . . 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 13121	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ ) -> 0 <_ ( P pCnt A ) )
36	14, 15, 35	syl2anc 415	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> 0 <_ ( P pCnt A ) )
37		ge0gtmnf 10242	. . . . . . . . . 10 |- ( ( ( P pCnt A ) e. RR* /\ 0 <_ ( P pCnt A ) ) -> -oo < ( P pCnt A ) )
38	31, 36, 37	syl2anc 415	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( ( P pCnt A ) <_ ( P pCnt B ) /\ B =/= 0 ) ) -> -oo < ( P pCnt A ) )
39		simprl 535	. . . . . . . . 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 10239	. . . . . . . . 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 1279	. . . . . . . 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 8373	. . . . . . . . . . . 12 |- +oo e/ RR
43	42	neli 2517	. . . . . . . . . . 11 |- -. +oo e. RR
44		pc0 13112	. . . . . . . . . . . . 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 2307	. . . . . . . . . . 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 686	. . . . . . . . . 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 6096	. . . . . . . . . . . 12 |- ( A = 0 -> ( P pCnt A ) = ( P pCnt 0 ) )
49	48	eleq1d 2307	. . . . . . . . . . 11 |- ( A = 0 -> ( ( P pCnt A ) e. RR <-> ( P pCnt 0 ) e. RR ) )
50	49	notbid 677	. . . . . . . . . 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 2477	. . . . . . . 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 13122	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P ^ ( P pCnt A ) ) || A )
55	14, 15, 53, 54	syl12anc 1276	. . . . . 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 13106	. . . . . . . . 9 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) e. NN0 )
57	14, 15, 53, 56	syl12anc 1276	. . . . . . . 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 13128	. . . . . . . 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 1278	. . . . . . 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 12913	. . . . . . . . . 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 11154	. . . . . . . 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 9778	. . . . . . 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 12814	. . . . . . 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 1278	. . . . . 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 437	. . . . 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 13128	. . . . . 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 1278	. . . . 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 13108	. . . . . 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 9632	. . . . 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 8448	. . . 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 957	. . 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 404	. 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 1033	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> B e. ZZ )
77		0zd 9667	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> 0 e. ZZ )
78		zdceq 9731	. . . 4 |- ( ( B e. ZZ /\ 0 e. ZZ ) -> DECID B = 0 )
79	76, 77, 78	syl2anc 415	. . 3 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> DECID B = 0 )
80		dcne 2431	. . 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 810	1 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> ( P pCnt ( A gcd B ) ) = ( P pCnt A ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   RRcr 8184   0cc0 8185   +oocpnf 8363   -oocmnf 8364   RR*cxr 8365   < clt 8366   <_ cle 8367   NNcn 9313   NN0cn0 9574   ZZcz 9655   QQcq 10035   ^cexp 10996   abscabs 11785   || cdvds 12579   gcd cgcd 12755   Primecprime 12910   pCnt cpc 13092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-2o 6692  df-er 6811  df-en 7027  df-sup 7330  df-inf 7331  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fzo 10567  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-dvds 12580  df-gcd 12756  df-prm 12911  df-pc 13093

This theorem is used by:  pcgcd  13137