Theorem pcdiv 12498

Description: Division property of the prime power function. (Contributed by Mario Carneiro, 1-Mar-2014.)

Assertion

Ref	Expression
pcdiv	|- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt ( A / B ) ) = ( ( P pCnt A ) - ( P pCnt B ) ) )

Proof of Theorem pcdiv

Dummy variables x n y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 999	. . 3 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> P e. Prime )
2		simp2l 1025	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A e. ZZ )
3		simp3 1001	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> B e. NN )
4		znq 9717	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A / B ) e. QQ )
5	2, 3, 4	syl2anc 411	. . 3 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( A / B ) e. QQ )
6	2	zcnd 9468	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A e. CC )
7	3	nncnd 9023	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> B e. CC )
8		simp2r 1026	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A =/= 0 )
9		0z 9356	. . . . . . 7 |- 0 e. ZZ
10		zapne 9419	. . . . . . 7 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> ( A # 0 <-> A =/= 0 ) )
11	2, 9, 10	sylancl 413	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( A # 0 <-> A =/= 0 ) )
12	8, 11	mpbird 167	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A # 0 )
13	3	nnap0d 9055	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> B # 0 )
14	6, 7, 12, 13	divap0d 8852	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( A / B ) # 0 )
15		zq 9719	. . . . . 6 |- ( 0 e. ZZ -> 0 e. QQ )
16	9, 15	ax-mp 5	. . . . 5 |- 0 e. QQ
17		qapne 9732	. . . . 5 |- ( ( ( A / B ) e. QQ /\ 0 e. QQ ) -> ( ( A / B ) # 0 <-> ( A / B ) =/= 0 ) )
18	5, 16, 17	sylancl 413	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( ( A / B ) # 0 <-> ( A / B ) =/= 0 ) )
19	14, 18	mpbid 147	. . 3 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( A / B ) =/= 0 )
20		eqid 2196	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
21		eqid 2196	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )
22	20, 21	pcval 12492	. . 3 |- ( ( P e. Prime /\ ( ( A / B ) e. QQ /\ ( A / B ) =/= 0 ) ) -> ( P pCnt ( A / B ) ) = ( iota z E. x e. ZZ E. y e. NN ( ( A / B ) = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) )
23	1, 5, 19, 22	syl12anc 1247	. 2 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt ( A / B ) ) = ( iota z E. x e. ZZ E. y e. NN ( ( A / B ) = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) )
24		eqid 2196	. . . . . . . 8 |- sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < )
25	24	pczpre 12493	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
26	25	3adant3 1019	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt A ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
27		nnz 9364	. . . . . . . . 9 |- ( B e. NN -> B e. ZZ )
28		nnne0 9037	. . . . . . . . 9 |- ( B e. NN -> B =/= 0 )
29	27, 28	jca 306	. . . . . . . 8 |- ( B e. NN -> ( B e. ZZ /\ B =/= 0 ) )
30		eqid 2196	. . . . . . . . 9 |- sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < )
31	30	pczpre 12493	. . . . . . . 8 |- ( ( P e. Prime /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt B ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) )
32	29, 31	sylan2 286	. . . . . . 7 |- ( ( P e. Prime /\ B e. NN ) -> ( P pCnt B ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) )
33	32	3adant2 1018	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt B ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) )
34	26, 33	oveq12d 5943	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) ) )
35		eqid 2196	. . . . 5 |- ( A / B ) = ( A / B )
36	34, 35	jctil 312	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( ( A / B ) = ( A / B ) /\ ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) ) ) )
37		oveq1 5932	. . . . . . 7 |- ( x = A -> ( x / y ) = ( A / y ) )
38	37	eqeq2d 2208	. . . . . 6 |- ( x = A -> ( ( A / B ) = ( x / y ) <-> ( A / B ) = ( A / y ) ) )
39		breq2 4038	. . . . . . . . . 10 |- ( x = A -> ( ( P ^ n ) || x <-> ( P ^ n ) || A ) )
40	39	rabbidv 2752	. . . . . . . . 9 |- ( x = A -> { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || A } )
41	40	supeq1d 7062	. . . . . . . 8 |- ( x = A -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
42	41	oveq1d 5940	. . . . . . 7 |- ( x = A -> ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) )
43	42	eqeq2d 2208	. . . . . 6 |- ( x = A -> ( ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) <-> ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) )
44	38, 43	anbi12d 473	. . . . 5 |- ( x = A -> ( ( ( A / B ) = ( x / y ) /\ ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) <-> ( ( A / B ) = ( A / y ) /\ ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) )
45		oveq2 5933	. . . . . . 7 |- ( y = B -> ( A / y ) = ( A / B ) )
46	45	eqeq2d 2208	. . . . . 6 |- ( y = B -> ( ( A / B ) = ( A / y ) <-> ( A / B ) = ( A / B ) ) )
47		breq2 4038	. . . . . . . . . 10 |- ( y = B -> ( ( P ^ n ) || y <-> ( P ^ n ) || B ) )
48	47	rabbidv 2752	. . . . . . . . 9 |- ( y = B -> { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || B } )
49	48	supeq1d 7062	. . . . . . . 8 |- ( y = B -> sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) )
50	49	oveq2d 5941	. . . . . . 7 |- ( y = B -> ( sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) ) )
51	50	eqeq2d 2208	. . . . . 6 |- ( y = B -> ( ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) <-> ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) ) ) )
52	46, 51	anbi12d 473	. . . . 5 |- ( y = B -> ( ( ( A / B ) = ( A / y ) /\ ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) <-> ( ( A / B ) = ( A / B ) /\ ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) ) ) ) )
53	44, 52	rspc2ev 2883	. . . 4 |- ( ( A e. ZZ /\ B e. NN /\ ( ( A / B ) = ( A / B ) /\ ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) ) ) ) -> E. x e. ZZ E. y e. NN ( ( A / B ) = ( x / y ) /\ ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) )
54	2, 3, 36, 53	syl3anc 1249	. . 3 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> E. x e. ZZ E. y e. NN ( ( A / B ) = ( x / y ) /\ ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) )
55		pczcl 12494	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) e. NN0 )
56	55	3adant3 1019	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt A ) e. NN0 )
57	56	nn0zd 9465	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt A ) e. ZZ )
58	1, 3	pccld 12496	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt B ) e. NN0 )
59	58	nn0zd 9465	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt B ) e. ZZ )
60	57, 59	zsubcld 9472	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( ( P pCnt A ) - ( P pCnt B ) ) e. ZZ )
61	20, 21	pceu 12491	. . . . 5 |- ( ( P e. Prime /\ ( ( A / B ) e. QQ /\ ( A / B ) =/= 0 ) ) -> E! z E. x e. ZZ E. y e. NN ( ( A / B ) = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) )
62	1, 5, 19, 61	syl12anc 1247	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> E! z E. x e. ZZ E. y e. NN ( ( A / B ) = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) )
63		eqeq1 2203	. . . . . . 7 |- ( z = ( ( P pCnt A ) - ( P pCnt B ) ) -> ( z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) <-> ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) )
64	63	anbi2d 464	. . . . . 6 |- ( z = ( ( P pCnt A ) - ( P pCnt B ) ) -> ( ( ( A / B ) = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) <-> ( ( A / B ) = ( x / y ) /\ ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) )
65	64	2rexbidv 2522	. . . . 5 |- ( z = ( ( P pCnt A ) - ( P pCnt B ) ) -> ( E. x e. ZZ E. y e. NN ( ( A / B ) = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) <-> E. x e. ZZ E. y e. NN ( ( A / B ) = ( x / y ) /\ ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) )
66	65	iota2 5249	. . . 4 |- ( ( ( ( P pCnt A ) - ( P pCnt B ) ) e. ZZ /\ E! z E. x e. ZZ E. y e. NN ( ( A / B ) = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) -> ( E. x e. ZZ E. y e. NN ( ( A / B ) = ( x / y ) /\ ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) <-> ( iota z E. x e. ZZ E. y e. NN ( ( A / B ) = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) = ( ( P pCnt A ) - ( P pCnt B ) ) ) )
67	60, 62, 66	syl2anc 411	. . 3 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( E. x e. ZZ E. y e. NN ( ( A / B ) = ( x / y ) /\ ( ( P pCnt A ) - ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) <-> ( iota z E. x e. ZZ E. y e. NN ( ( A / B ) = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) = ( ( P pCnt A ) - ( P pCnt B ) ) ) )
68	54, 67	mpbid 147	. 2 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( iota z E. x e. ZZ E. y e. NN ( ( A / B ) = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) = ( ( P pCnt A ) - ( P pCnt B ) ) )
69	23, 68	eqtrd 2229	1 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt ( A / B ) ) = ( ( P pCnt A ) - ( P pCnt B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   E!weu 2045   e. wcel 2167   =/= wne 2367   E.wrex 2476   {crab 2479   class class class wbr 4034   iotacio 5218  (class class class)co 5925   supcsup 7057   RRcr 7897   0cc0 7898   < clt 8080   - cmin 8216   # cap 8627   / cdiv 8718   NNcn 9009   NN0cn0 9268   ZZcz 9345   QQcq 9712   ^cexp 10649   || cdvds 11971   Primecprime 12302   pCnt cpc 12480

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-mulrcl 7997  ax-addcom 7998  ax-mulcom 7999  ax-addass 8000  ax-mulass 8001  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-1rid 8005  ax-0id 8006  ax-rnegex 8007  ax-precex 8008  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014  ax-pre-mulgt0 8015  ax-pre-mulext 8016  ax-arch 8017  ax-caucvg 8018

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-2o 6484  df-er 6601  df-en 6809  df-sup 7059  df-inf 7060  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-reap 8621  df-ap 8628  df-div 8719  df-inn 9010  df-2 9068  df-3 9069  df-4 9070  df-n0 9269  df-z 9346  df-uz 9621  df-q 9713  df-rp 9748  df-fz 10103  df-fzo 10237  df-fl 10379  df-mod 10434  df-seqfrec 10559  df-exp 10650  df-cj 11026  df-re 11027  df-im 11028  df-rsqrt 11182  df-abs 11183  df-dvds 11972  df-gcd 12148  df-prm 12303  df-pc 12481

This theorem is referenced by:  pcqmul  12499  pcqcl  12502  pcid  12520  pcneg  12521  pc2dvds  12526  pcz  12528  pcaddlem  12535  pcadd  12536  pcmpt2  12540  pcbc  12547