Theorem pcdiv 12658

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 1000	. . 3 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> P e. Prime )
2		simp2l 1026	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A e. ZZ )
3		simp3 1002	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> B e. NN )
4		znq 9747	. . . 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 9498	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A e. CC )
7	3	nncnd 9052	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> B e. CC )
8		simp2r 1027	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> A =/= 0 )
9		0z 9385	. . . . . . 7 |- 0 e. ZZ
10		zapne 9449	. . . . . . 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 9084	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> B # 0 )
14	6, 7, 12, 13	divap0d 8881	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( A / B ) # 0 )
15		zq 9749	. . . . . 6 |- ( 0 e. ZZ -> 0 e. QQ )
16	9, 15	ax-mp 5	. . . . 5 |- 0 e. QQ
17		qapne 9762	. . . . 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 2205	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
21		eqid 2205	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )
22	20, 21	pcval 12652	. . 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 1248	. 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 2205	. . . . . . . 8 |- sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < )
25	24	pczpre 12653	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
26	25	3adant3 1020	. . . . . 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 9393	. . . . . . . . 9 |- ( B e. NN -> B e. ZZ )
28		nnne0 9066	. . . . . . . . 9 |- ( B e. NN -> B =/= 0 )
29	27, 28	jca 306	. . . . . . . 8 |- ( B e. NN -> ( B e. ZZ /\ B =/= 0 ) )
30		eqid 2205	. . . . . . . . 9 |- sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < )
31	30	pczpre 12653	. . . . . . . 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 1019	. . . . . 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 5964	. . . . 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 2205	. . . . 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 5953	. . . . . . 7 |- ( x = A -> ( x / y ) = ( A / y ) )
38	37	eqeq2d 2217	. . . . . 6 |- ( x = A -> ( ( A / B ) = ( x / y ) <-> ( A / B ) = ( A / y ) ) )
39		breq2 4049	. . . . . . . . . 10 |- ( x = A -> ( ( P ^ n ) || x <-> ( P ^ n ) || A ) )
40	39	rabbidv 2761	. . . . . . . . 9 |- ( x = A -> { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || A } )
41	40	supeq1d 7091	. . . . . . . 8 |- ( x = A -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
42	41	oveq1d 5961	. . . . . . 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 2217	. . . . . 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 5954	. . . . . . 7 |- ( y = B -> ( A / y ) = ( A / B ) )
46	45	eqeq2d 2217	. . . . . 6 |- ( y = B -> ( ( A / B ) = ( A / y ) <-> ( A / B ) = ( A / B ) ) )
47		breq2 4049	. . . . . . . . . 10 |- ( y = B -> ( ( P ^ n ) || y <-> ( P ^ n ) || B ) )
48	47	rabbidv 2761	. . . . . . . . 9 |- ( y = B -> { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || B } )
49	48	supeq1d 7091	. . . . . . . 8 |- ( y = B -> sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) )
50	49	oveq2d 5962	. . . . . . 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 2217	. . . . . 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 2892	. . . 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 1250	. . 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 12654	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) e. NN0 )
56	55	3adant3 1020	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt A ) e. NN0 )
57	56	nn0zd 9495	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt A ) e. ZZ )
58	1, 3	pccld 12656	. . . . . 6 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt B ) e. NN0 )
59	58	nn0zd 9495	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt B ) e. ZZ )
60	57, 59	zsubcld 9502	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( ( P pCnt A ) - ( P pCnt B ) ) e. ZZ )
61	20, 21	pceu 12651	. . . . 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 1248	. . . 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 2212	. . . . . . 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 2531	. . . . 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 5262	. . . 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 2238	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 981   = wceq 1373   E!weu 2054   e. wcel 2176   =/= wne 2376   E.wrex 2485   {crab 2488   class class class wbr 4045   iotacio 5231  (class class class)co 5946   supcsup 7086   RRcr 7926   0cc0 7927   < clt 8109   - cmin 8245   # cap 8656   / cdiv 8747   NNcn 9038   NN0cn0 9297   ZZcz 9374   QQcq 9742   ^cexp 10685   || cdvds 12131   Primecprime 12462   pCnt cpc 12640

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-1o 6504  df-2o 6505  df-er 6622  df-en 6830  df-sup 7088  df-inf 7089  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-fz 10133  df-fzo 10267  df-fl 10415  df-mod 10470  df-seqfrec 10595  df-exp 10686  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343  df-dvds 12132  df-gcd 12308  df-prm 12463  df-pc 12641

This theorem is referenced by:  pcqmul  12659  pcqcl  12662  pcid  12680  pcneg  12681  pc2dvds  12686  pcz  12688  pcaddlem  12695  pcadd  12696  pcmpt2  12700  pcbc  12707