Theorem pcdiv 12243

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 992	. . 3 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> P e. Prime )
2		simp2l 1018	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A e. ZZ )
3		simp3 994	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> B e. NN )
4		znq 9570	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A / B ) e. QQ )
5	2, 3, 4	syl2anc 409	. . 3 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( A / B ) e. QQ )
6	2	zcnd 9322	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A e. CC )
7	3	nncnd 8879	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> B e. CC )
8		simp2r 1019	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A =/= 0 )
9		0z 9210	. . . . . . 7 |- 0 e. ZZ
10		zapne 9273	. . . . . . 7 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> ( A # 0 <-> A =/= 0 ) )
11	2, 9, 10	sylancl 411	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( A # 0 <-> A =/= 0 ) )
12	8, 11	mpbird 166	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A # 0 )
13	3	nnap0d 8911	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> B # 0 )
14	6, 7, 12, 13	divap0d 8710	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( A / B ) # 0 )
15		zq 9572	. . . . . 6 |- ( 0 e. ZZ -> 0 e. QQ )
16	9, 15	ax-mp 5	. . . . 5 |- 0 e. QQ
17		qapne 9585	. . . . 5 |- ( ( ( A / B ) e. QQ /\ 0 e. QQ ) -> ( ( A / B ) # 0 <-> ( A / B ) =/= 0 ) )
18	5, 16, 17	sylancl 411	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( ( A / B ) # 0 <-> ( A / B ) =/= 0 ) )
19	14, 18	mpbid 146	. . 3 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( A / B ) =/= 0 )
20		eqid 2170	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
21		eqid 2170	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )
22	20, 21	pcval 12237	. . 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 1231	. 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 2170	. . . . . . . 8 |- sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < )
25	24	pczpre 12238	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
26	25	3adant3 1012	. . . . . 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 9218	. . . . . . . . 9 |- ( B e. NN -> B e. ZZ )
28		nnne0 8893	. . . . . . . . 9 |- ( B e. NN -> B =/= 0 )
29	27, 28	jca 304	. . . . . . . 8 |- ( B e. NN -> ( B e. ZZ /\ B =/= 0 ) )
30		eqid 2170	. . . . . . . . 9 |- sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < )
31	30	pczpre 12238	. . . . . . . 8 |- ( ( P e. Prime /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt B ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) )
32	29, 31	sylan2 284	. . . . . . 7 |- ( ( P e. Prime /\ B e. NN ) -> ( P pCnt B ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) )
33	32	3adant2 1011	. . . . . 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 5868	. . . . 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 2170	. . . . 5 |- ( A / B ) = ( A / B )
36	34, 35	jctil 310	. . . 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 5857	. . . . . . 7 |- ( x = A -> ( x / y ) = ( A / y ) )
38	37	eqeq2d 2182	. . . . . 6 |- ( x = A -> ( ( A / B ) = ( x / y ) <-> ( A / B ) = ( A / y ) ) )
39		breq2 3991	. . . . . . . . . 10 |- ( x = A -> ( ( P ^ n ) || x <-> ( P ^ n ) || A ) )
40	39	rabbidv 2719	. . . . . . . . 9 |- ( x = A -> { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || A } )
41	40	supeq1d 6960	. . . . . . . 8 |- ( x = A -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
42	41	oveq1d 5865	. . . . . . 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 2182	. . . . . 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 470	. . . . 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 5858	. . . . . . 7 |- ( y = B -> ( A / y ) = ( A / B ) )
46	45	eqeq2d 2182	. . . . . 6 |- ( y = B -> ( ( A / B ) = ( A / y ) <-> ( A / B ) = ( A / B ) ) )
47		breq2 3991	. . . . . . . . . 10 |- ( y = B -> ( ( P ^ n ) || y <-> ( P ^ n ) || B ) )
48	47	rabbidv 2719	. . . . . . . . 9 |- ( y = B -> { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || B } )
49	48	supeq1d 6960	. . . . . . . 8 |- ( y = B -> sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) )
50	49	oveq2d 5866	. . . . . . 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 2182	. . . . . 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 470	. . . . 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 2849	. . . 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 1233	. . 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 12239	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) e. NN0 )
56	55	3adant3 1012	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt A ) e. NN0 )
57	56	nn0zd 9319	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt A ) e. ZZ )
58	1, 3	pccld 12241	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt B ) e. NN0 )
59	58	nn0zd 9319	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt B ) e. ZZ )
60	57, 59	zsubcld 9326	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( ( P pCnt A ) - ( P pCnt B ) ) e. ZZ )
61	20, 21	pceu 12236	. . . . 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 1231	. . . 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 2177	. . . . . . 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 461	. . . . . 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 2495	. . . . 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 5186	. . . 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 409	. . 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 146	. 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 2203	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 103   <-> wb 104   /\ w3a 973   = wceq 1348   E!weu 2019   e. wcel 2141   =/= wne 2340   E.wrex 2449   {crab 2452   class class class wbr 3987   iotacio 5156  (class class class)co 5850   supcsup 6955   RRcr 7760   0cc0 7761   < clt 7941   - cmin 8077   # cap 8487   / cdiv 8576   NNcn 8865   NN0cn0 9122   ZZcz 9199   QQcq 9565   ^cexp 10462   || cdvds 11736   Primecprime 12048   pCnt cpc 12225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880  ax-caucvg 7881

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-1o 6392  df-2o 6393  df-er 6509  df-en 6715  df-sup 6957  df-inf 6958  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-rp 9598  df-fz 9953  df-fzo 10086  df-fl 10213  df-mod 10266  df-seqfrec 10389  df-exp 10463  df-cj 10793  df-re 10794  df-im 10795  df-rsqrt 10949  df-abs 10950  df-dvds 11737  df-gcd 11885  df-prm 12049  df-pc 12226

This theorem is referenced by:  pcqmul  12244  pcqcl  12247  pcid  12264  pcneg  12265  pc2dvds  12270  pcz  12272  pcaddlem  12279  pcadd  12280  pcmpt2  12283  pcbc  12290