Theorem pcdiv 13025

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 1024	. . 3 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> P e. Prime )
2		simp2l 1050	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A e. ZZ )
3		simp3 1026	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> B e. NN )
4		znq 9974	. . . 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 9719	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A e. CC )
7	3	nncnd 9268	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> B e. CC )
8		simp2r 1051	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A =/= 0 )
9		0z 9605	. . . . . . 7 |- 0 e. ZZ
10		zapne 9669	. . . . . . 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 9300	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> B # 0 )
14	6, 7, 12, 13	divap0d 9097	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( A / B ) # 0 )
15		zq 9976	. . . . . 6 |- ( 0 e. ZZ -> 0 e. QQ )
16	9, 15	ax-mp 5	. . . . 5 |- 0 e. QQ
17		qapne 9989	. . . . 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 2234	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
21		eqid 2234	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )
22	20, 21	pcval 13019	. . 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 1272	. 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 2234	. . . . . . . 8 |- sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < )
25	24	pczpre 13020	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
26	25	3adant3 1044	. . . . . 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 9613	. . . . . . . . 9 |- ( B e. NN -> B e. ZZ )
28		nnne0 9282	. . . . . . . . 9 |- ( B e. NN -> B =/= 0 )
29	27, 28	jca 306	. . . . . . . 8 |- ( B e. NN -> ( B e. ZZ /\ B =/= 0 ) )
30		eqid 2234	. . . . . . . . 9 |- sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < )
31	30	pczpre 13020	. . . . . . . 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 1043	. . . . . 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 6076	. . . . 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 2234	. . . . 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 6065	. . . . . . 7 |- ( x = A -> ( x / y ) = ( A / y ) )
38	37	eqeq2d 2246	. . . . . 6 |- ( x = A -> ( ( A / B ) = ( x / y ) <-> ( A / B ) = ( A / y ) ) )
39		breq2 4118	. . . . . . . . . 10 |- ( x = A -> ( ( P ^ n ) || x <-> ( P ^ n ) || A ) )
40	39	rabbidv 2804	. . . . . . . . 9 |- ( x = A -> { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || A } )
41	40	supeq1d 7291	. . . . . . . 8 |- ( x = A -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
42	41	oveq1d 6073	. . . . . . 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 2246	. . . . . 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 6066	. . . . . . 7 |- ( y = B -> ( A / y ) = ( A / B ) )
46	45	eqeq2d 2246	. . . . . 6 |- ( y = B -> ( ( A / B ) = ( A / y ) <-> ( A / B ) = ( A / B ) ) )
47		breq2 4118	. . . . . . . . . 10 |- ( y = B -> ( ( P ^ n ) || y <-> ( P ^ n ) || B ) )
48	47	rabbidv 2804	. . . . . . . . 9 |- ( y = B -> { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || B } )
49	48	supeq1d 7291	. . . . . . . 8 |- ( y = B -> sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) )
50	49	oveq2d 6074	. . . . . . 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 2246	. . . . . 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 2939	. . . 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 1274	. . 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 13021	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) e. NN0 )
56	55	3adant3 1044	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt A ) e. NN0 )
57	56	nn0zd 9716	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt A ) e. ZZ )
58	1, 3	pccld 13023	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt B ) e. NN0 )
59	58	nn0zd 9716	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt B ) e. ZZ )
60	57, 59	zsubcld 9723	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( ( P pCnt A ) - ( P pCnt B ) ) e. ZZ )
61	20, 21	pceu 13018	. . . . 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 1272	. . . 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 2241	. . . . . . 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 2569	. . . . 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 5347	. . . 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 2267	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 1005   = wceq 1398   E!weu 2082   e. wcel 2205   =/= wne 2414   E.wrex 2523   {crab 2526   class class class wbr 4114   iotacio 5315  (class class class)co 6058   supcsup 7286   RRcr 8142   0cc0 8143   < clt 8324   - cmin 8460   # cap 8872   / cdiv 8963   NNcn 9254   NN0cn0 9513   ZZcz 9594   QQcq 9969   ^cexp 10924   || cdvds 12498   Primecprime 12829   pCnt cpc 13007

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-2o 6661  df-er 6780  df-en 6989  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-gcd 12675  df-prm 12830  df-pc 13008

This theorem is referenced by:  pcqmul  13026  pcqcl  13029  pcid  13047  pcneg  13048  pc2dvds  13053  pcz  13055  pcaddlem  13062  pcadd  13063  pcmpt2  13067  pcbc  13074