Theorem pcneg 12494

Description: The prime count of a negative number. (Contributed by Mario Carneiro, 13-Mar-2014.)

Assertion

Ref	Expression
pcneg	|- ( ( P e. Prime /\ A e. QQ ) -> ( P pCnt -u A ) = ( P pCnt A ) )

Proof of Theorem pcneg

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

Step	Hyp	Ref	Expression
1		elq 9696	. . 3 |- ( A e. QQ <-> E. x e. ZZ E. y e. NN A = ( x / y ) )
2		zcn 9331	. . . . . . . . 9 |- ( x e. ZZ -> x e. CC )
3	2	ad2antrl 490	. . . . . . . 8 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> x e. CC )
4		nncn 8998	. . . . . . . . 9 |- ( y e. NN -> y e. CC )
5	4	ad2antll 491	. . . . . . . 8 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> y e. CC )
6		nnap0 9019	. . . . . . . . 9 |- ( y e. NN -> y # 0 )
7	6	ad2antll 491	. . . . . . . 8 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> y # 0 )
8	3, 5, 7	divnegapd 8830	. . . . . . 7 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> -u ( x / y ) = ( -u x / y ) )
9	8	oveq2d 5938	. . . . . 6 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( P pCnt -u ( x / y ) ) = ( P pCnt ( -u x / y ) ) )
10		neg0 8272	. . . . . . . . . 10 |- -u 0 = 0
11		simpr 110	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> x = 0 )
12	11	negeqd 8221	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> -u x = -u 0 )
13	10, 12, 11	3eqtr4a 2255	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> -u x = x )
14	13	oveq1d 5937	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> ( -u x / y ) = ( x / y ) )
15	14	oveq2d 5938	. . . . . . 7 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> ( P pCnt ( -u x / y ) ) = ( P pCnt ( x / y ) ) )
16		simpll 527	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> P e. Prime )
17		simplrl 535	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> x e. ZZ )
18	17	znegcld 9450	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> -u x e. ZZ )
19		simpr 110	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> x =/= 0 )
20	2	negne0bd 8330	. . . . . . . . . . . . 13 |- ( x e. ZZ -> ( x =/= 0 <-> -u x =/= 0 ) )
21	17, 20	syl 14	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> ( x =/= 0 <-> -u x =/= 0 ) )
22	19, 21	mpbid 147	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> -u x =/= 0 )
23		eqid 2196	. . . . . . . . . . . 12 |- sup ( { y e. NN0 | ( P ^ y ) || -u x } , RR , < ) = sup ( { y e. NN0 | ( P ^ y ) || -u x } , RR , < )
24	23	pczpre 12466	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( -u x e. ZZ /\ -u x =/= 0 ) ) -> ( P pCnt -u x ) = sup ( { y e. NN0 | ( P ^ y ) || -u x } , RR , < ) )
25	16, 18, 22, 24	syl12anc 1247	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> ( P pCnt -u x ) = sup ( { y e. NN0 | ( P ^ y ) || -u x } , RR , < ) )
26		eqid 2196	. . . . . . . . . . . . 13 |- sup ( { y e. NN0 | ( P ^ y ) || x } , RR , < ) = sup ( { y e. NN0 | ( P ^ y ) || x } , RR , < )
27	26	pczpre 12466	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( P pCnt x ) = sup ( { y e. NN0 | ( P ^ y ) || x } , RR , < ) )
28		prmz 12279	. . . . . . . . . . . . . . . . 17 |- ( P e. Prime -> P e. ZZ )
29		zexpcl 10646	. . . . . . . . . . . . . . . . 17 |- ( ( P e. ZZ /\ y e. NN0 ) -> ( P ^ y ) e. ZZ )
30	28, 29	sylan 283	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ y e. NN0 ) -> ( P ^ y ) e. ZZ )
31		simpl 109	. . . . . . . . . . . . . . . 16 |- ( ( x e. ZZ /\ x =/= 0 ) -> x e. ZZ )
32		dvdsnegb 11973	. . . . . . . . . . . . . . . 16 |- ( ( ( P ^ y ) e. ZZ /\ x e. ZZ ) -> ( ( P ^ y ) || x <-> ( P ^ y ) || -u x ) )
33	30, 31, 32	syl2an 289	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Prime /\ y e. NN0 ) /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( ( P ^ y ) || x <-> ( P ^ y ) || -u x ) )
34	33	an32s 568	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) /\ y e. NN0 ) -> ( ( P ^ y ) || x <-> ( P ^ y ) || -u x ) )
35	34	rabbidva 2751	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> { y e. NN0 | ( P ^ y ) || x } = { y e. NN0 | ( P ^ y ) || -u x } )
36	35	supeq1d 7053	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> sup ( { y e. NN0 | ( P ^ y ) || x } , RR , < ) = sup ( { y e. NN0 | ( P ^ y ) || -u x } , RR , < ) )
37	27, 36	eqtrd 2229	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( P pCnt x ) = sup ( { y e. NN0 | ( P ^ y ) || -u x } , RR , < ) )
38	16, 17, 19, 37	syl12anc 1247	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> ( P pCnt x ) = sup ( { y e. NN0 | ( P ^ y ) || -u x } , RR , < ) )
39	25, 38	eqtr4d 2232	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> ( P pCnt -u x ) = ( P pCnt x ) )
40	39	oveq1d 5937	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> ( ( P pCnt -u x ) - ( P pCnt y ) ) = ( ( P pCnt x ) - ( P pCnt y ) ) )
41		simplrr 536	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> y e. NN )
42		pcdiv 12471	. . . . . . . . 9 |- ( ( P e. Prime /\ ( -u x e. ZZ /\ -u x =/= 0 ) /\ y e. NN ) -> ( P pCnt ( -u x / y ) ) = ( ( P pCnt -u x ) - ( P pCnt y ) ) )
43	16, 18, 22, 41, 42	syl121anc 1254	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> ( P pCnt ( -u x / y ) ) = ( ( P pCnt -u x ) - ( P pCnt y ) ) )
44		pcdiv 12471	. . . . . . . . 9 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt ( x / y ) ) = ( ( P pCnt x ) - ( P pCnt y ) ) )
45	16, 17, 19, 41, 44	syl121anc 1254	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> ( P pCnt ( x / y ) ) = ( ( P pCnt x ) - ( P pCnt y ) ) )
46	40, 43, 45	3eqtr4d 2239	. . . . . . 7 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> ( P pCnt ( -u x / y ) ) = ( P pCnt ( x / y ) ) )
47		simprl 529	. . . . . . . . 9 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> x e. ZZ )
48		0zd 9338	. . . . . . . . 9 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> 0 e. ZZ )
49		zdceq 9401	. . . . . . . . 9 |- ( ( x e. ZZ /\ 0 e. ZZ ) -> DECID x = 0 )
50	47, 48, 49	syl2anc 411	. . . . . . . 8 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> DECID x = 0 )
51		dcne 2378	. . . . . . . 8 |- (DECID x = 0 <-> ( x = 0 \/ x =/= 0 ) )
52	50, 51	sylib 122	. . . . . . 7 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( x = 0 \/ x =/= 0 ) )
53	15, 46, 52	mpjaodan 799	. . . . . 6 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( P pCnt ( -u x / y ) ) = ( P pCnt ( x / y ) ) )
54	9, 53	eqtrd 2229	. . . . 5 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( P pCnt -u ( x / y ) ) = ( P pCnt ( x / y ) ) )
55		negeq 8219	. . . . . . 7 |- ( A = ( x / y ) -> -u A = -u ( x / y ) )
56	55	oveq2d 5938	. . . . . 6 |- ( A = ( x / y ) -> ( P pCnt -u A ) = ( P pCnt -u ( x / y ) ) )
57		oveq2 5930	. . . . . 6 |- ( A = ( x / y ) -> ( P pCnt A ) = ( P pCnt ( x / y ) ) )
58	56, 57	eqeq12d 2211	. . . . 5 |- ( A = ( x / y ) -> ( ( P pCnt -u A ) = ( P pCnt A ) <-> ( P pCnt -u ( x / y ) ) = ( P pCnt ( x / y ) ) ) )
59	54, 58	syl5ibrcom 157	. . . 4 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( A = ( x / y ) -> ( P pCnt -u A ) = ( P pCnt A ) ) )
60	59	rexlimdvva 2622	. . 3 |- ( P e. Prime -> ( E. x e. ZZ E. y e. NN A = ( x / y ) -> ( P pCnt -u A ) = ( P pCnt A ) ) )
61	1, 60	biimtrid 152	. 2 |- ( P e. Prime -> ( A e. QQ -> ( P pCnt -u A ) = ( P pCnt A ) ) )
62	61	imp 124	1 |- ( ( P e. Prime /\ A e. QQ ) -> ( P pCnt -u A ) = ( P pCnt A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2167   =/= wne 2367   E.wrex 2476   {crab 2479   class class class wbr 4033  (class class class)co 5922   supcsup 7048   CCcc 7877   RRcr 7878   0cc0 7879   < clt 8061   - cmin 8197   -ucneg 8198   # cap 8608   / cdiv 8699   NNcn 8990   NN0cn0 9249   ZZcz 9326   QQcq 9693   ^cexp 10630   || cdvds 11952   Primecprime 12275   pCnt cpc 12453

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 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

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 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-1o 6474  df-2o 6475  df-er 6592  df-en 6800  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-dvds 11953  df-gcd 12121  df-prm 12276  df-pc 12454

This theorem is referenced by:  pcabs  12495  pcadd2  12510  lgsneg  15265