Theorem pcdiv 12302

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 997	. . 3 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> P e. Prime )
2		simp2l 1023	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A e. ZZ )
3		simp3 999	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> B e. NN )
4		znq 9624	. . . 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 9376	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A e. CC )
7	3	nncnd 8933	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> B e. CC )
8		simp2r 1024	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A =/= 0 )
9		0z 9264	. . . . . . 7 |- 0 e. ZZ
10		zapne 9327	. . . . . . 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 8965	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> B # 0 )
14	6, 7, 12, 13	divap0d 8763	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( A / B ) # 0 )
15		zq 9626	. . . . . 6 |- ( 0 e. ZZ -> 0 e. QQ )
16	9, 15	ax-mp 5	. . . . 5 |- 0 e. QQ
17		qapne 9639	. . . . 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 2177	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
21		eqid 2177	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )
22	20, 21	pcval 12296	. . 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 1236	. 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 2177	. . . . . . . 8 |- sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < )
25	24	pczpre 12297	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
26	25	3adant3 1017	. . . . . 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 9272	. . . . . . . . 9 |- ( B e. NN -> B e. ZZ )
28		nnne0 8947	. . . . . . . . 9 |- ( B e. NN -> B =/= 0 )
29	27, 28	jca 306	. . . . . . . 8 |- ( B e. NN -> ( B e. ZZ /\ B =/= 0 ) )
30		eqid 2177	. . . . . . . . 9 |- sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < )
31	30	pczpre 12297	. . . . . . . 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 1016	. . . . . 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 5893	. . . . 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 2177	. . . . 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 5882	. . . . . . 7 |- ( x = A -> ( x / y ) = ( A / y ) )
38	37	eqeq2d 2189	. . . . . 6 |- ( x = A -> ( ( A / B ) = ( x / y ) <-> ( A / B ) = ( A / y ) ) )
39		breq2 4008	. . . . . . . . . 10 |- ( x = A -> ( ( P ^ n ) || x <-> ( P ^ n ) || A ) )
40	39	rabbidv 2727	. . . . . . . . 9 |- ( x = A -> { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || A } )
41	40	supeq1d 6986	. . . . . . . 8 |- ( x = A -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
42	41	oveq1d 5890	. . . . . . 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 2189	. . . . . 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 5883	. . . . . . 7 |- ( y = B -> ( A / y ) = ( A / B ) )
46	45	eqeq2d 2189	. . . . . 6 |- ( y = B -> ( ( A / B ) = ( A / y ) <-> ( A / B ) = ( A / B ) ) )
47		breq2 4008	. . . . . . . . . 10 |- ( y = B -> ( ( P ^ n ) || y <-> ( P ^ n ) || B ) )
48	47	rabbidv 2727	. . . . . . . . 9 |- ( y = B -> { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || B } )
49	48	supeq1d 6986	. . . . . . . 8 |- ( y = B -> sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) )
50	49	oveq2d 5891	. . . . . . 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 2189	. . . . . 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 2857	. . . 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 1238	. . 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 12298	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) e. NN0 )
56	55	3adant3 1017	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt A ) e. NN0 )
57	56	nn0zd 9373	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt A ) e. ZZ )
58	1, 3	pccld 12300	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt B ) e. NN0 )
59	58	nn0zd 9373	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt B ) e. ZZ )
60	57, 59	zsubcld 9380	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( ( P pCnt A ) - ( P pCnt B ) ) e. ZZ )
61	20, 21	pceu 12295	. . . . 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 1236	. . . 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 2184	. . . . . . 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 2502	. . . . 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 5207	. . . 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 2210	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 978   = wceq 1353   E!weu 2026   e. wcel 2148   =/= wne 2347   E.wrex 2456   {crab 2459   class class class wbr 4004   iotacio 5177  (class class class)co 5875   supcsup 6981   RRcr 7810   0cc0 7811   < clt 7992   - cmin 8128   # cap 8538   / cdiv 8629   NNcn 8919   NN0cn0 9176   ZZcz 9253   QQcq 9619   ^cexp 10519   || cdvds 11794   Primecprime 12107   pCnt cpc 12284

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-isom 5226  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-1o 6417  df-2o 6418  df-er 6535  df-en 6741  df-sup 6983  df-inf 6984  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-fz 10009  df-fzo 10143  df-fl 10270  df-mod 10323  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-dvds 11795  df-gcd 11944  df-prm 12108  df-pc 12285

This theorem is referenced by:  pcqmul  12303  pcqcl  12306  pcid  12323  pcneg  12324  pc2dvds  12329  pcz  12331  pcaddlem  12338  pcadd  12339  pcmpt2  12342  pcbc  12349