Theorem pcneg 13054

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 9977	. . 3 |- ( A e. QQ <-> E. x e. ZZ E. y e. NN A = ( x / y ) )
2		zcn 9604	. . . . . . . . 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 9267	. . . . . . . . 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 9288	. . . . . . . . 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 9099	. . . . . . 7 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> -u ( x / y ) = ( -u x / y ) )
9	8	oveq2d 6076	. . . . . 6 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( P pCnt -u ( x / y ) ) = ( P pCnt ( -u x / y ) ) )
10		neg0 8538	. . . . . . . . . 10 |- -u 0 = 0
11		simpr 110	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> x = 0 )
12	11	negeqd 8487	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> -u x = -u 0 )
13	10, 12, 11	3eqtr4a 2293	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> -u x = x )
14	13	oveq1d 6075	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> ( -u x / y ) = ( x / y ) )
15	14	oveq2d 6076	. . . . . . 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 537	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> x e. ZZ )
18	17	znegcld 9725	. . . . . . . . . . 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 8596	. . . . . . . . . . . . 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 2234	. . . . . . . . . . . 12 |- sup ( { y e. NN0 | ( P ^ y ) || -u x } , RR , < ) = sup ( { y e. NN0 | ( P ^ y ) || -u x } , RR , < )
24	23	pczpre 13026	. . . . . . . . . . 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 1272	. . . . . . . . . 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 2234	. . . . . . . . . . . . 13 |- sup ( { y e. NN0 | ( P ^ y ) || x } , RR , < ) = sup ( { y e. NN0 | ( P ^ y ) || x } , RR , < )
27	26	pczpre 13026	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( P pCnt x ) = sup ( { y e. NN0 | ( P ^ y ) || x } , RR , < ) )
28		prmz 12839	. . . . . . . . . . . . . . . . 17 |- ( P e. Prime -> P e. ZZ )
29		zexpcl 10945	. . . . . . . . . . . . . . . . 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 12525	. . . . . . . . . . . . . . . 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 570	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) /\ y e. NN0 ) -> ( ( P ^ y ) || x <-> ( P ^ y ) || -u x ) )
35	34	rabbidva 2803	. . . . . . . . . . . . 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 7293	. . . . . . . . . . . 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 2267	. . . . . . . . . . 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 1272	. . . . . . . . . 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 2270	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> ( P pCnt -u x ) = ( P pCnt x ) )
40	39	oveq1d 6075	. . . . . . . 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 538	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> y e. NN )
42		pcdiv 13031	. . . . . . . . 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 1279	. . . . . . . 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 13031	. . . . . . . . 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 1279	. . . . . . . 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 2277	. . . . . . 7 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> ( P pCnt ( -u x / y ) ) = ( P pCnt ( x / y ) ) )
47		simprl 531	. . . . . . . . 9 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> x e. ZZ )
48		0zd 9611	. . . . . . . . 9 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> 0 e. ZZ )
49		zdceq 9675	. . . . . . . . 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 2425	. . . . . . . 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 806	. . . . . 6 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( P pCnt ( -u x / y ) ) = ( P pCnt ( x / y ) ) )
54	9, 53	eqtrd 2267	. . . . 5 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( P pCnt -u ( x / y ) ) = ( P pCnt ( x / y ) ) )
55		negeq 8485	. . . . . . 7 |- ( A = ( x / y ) -> -u A = -u ( x / y ) )
56	55	oveq2d 6076	. . . . . 6 |- ( A = ( x / y ) -> ( P pCnt -u A ) = ( P pCnt -u ( x / y ) ) )
57		oveq2 6068	. . . . . 6 |- ( A = ( x / y ) -> ( P pCnt A ) = ( P pCnt ( x / y ) ) )
58	56, 57	eqeq12d 2249	. . . . 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 2670	. . 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 716  DECID wdc 842   = wceq 1398   e. wcel 2205   =/= wne 2414   E.wrex 2523   {crab 2526   class class class wbr 4115  (class class class)co 6060   supcsup 7288   CCcc 8143   RRcr 8144   0cc0 8145   < clt 8326   - cmin 8463   -ucneg 8464   # cap 8875   / cdiv 8968   NNcn 9259   NN0cn0 9518   ZZcz 9599   QQcq 9974   ^cexp 10929   || cdvds 12504   Primecprime 12835   pCnt cpc 13013

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-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4666  ax-iinf 4717  ax-cnex 8236  ax-resscn 8237  ax-1cn 8238  ax-1re 8239  ax-icn 8240  ax-addcl 8241  ax-addrcl 8242  ax-mulcl 8243  ax-mulrcl 8244  ax-addcom 8245  ax-mulcom 8246  ax-addass 8247  ax-mulass 8248  ax-distr 8249  ax-i2m1 8250  ax-0lt1 8251  ax-1rid 8252  ax-0id 8253  ax-rnegex 8254  ax-precex 8255  ax-cnre 8256  ax-pre-ltirr 8257  ax-pre-ltwlin 8258  ax-pre-lttrn 8259  ax-pre-apti 8260  ax-pre-ltadd 8261  ax-pre-mulgt0 8262  ax-pre-mulext 8263  ax-arch 8264  ax-caucvg 8265

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 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-po 4423  df-iso 4424  df-iord 4493  df-on 4495  df-ilim 4496  df-suc 4498  df-iom 4720  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-ima 4769  df-iota 5319  df-fun 5361  df-fn 5362  df-f 5363  df-f1 5364  df-fo 5365  df-f1o 5366  df-fv 5367  df-isom 5368  df-riota 6013  df-ov 6063  df-oprab 6064  df-mpo 6065  df-1st 6349  df-2nd 6350  df-recs 6551  df-frec 6637  df-1o 6662  df-2o 6663  df-er 6782  df-en 6991  df-sup 7290  df-inf 7291  df-pnf 8328  df-mnf 8329  df-xr 8330  df-ltxr 8331  df-le 8332  df-sub 8465  df-neg 8466  df-reap 8869  df-ap 8876  df-div 8969  df-inn 9260  df-2 9318  df-3 9319  df-4 9320  df-n0 9519  df-z 9600  df-uz 9877  df-q 9975  df-rp 10010  df-fz 10367  df-fzo 10504  df-fl 10659  df-mod 10714  df-seqfrec 10839  df-exp 10930  df-cj 11557  df-re 11558  df-im 11559  df-rsqrt 11714  df-abs 11715  df-dvds 12505  df-gcd 12681  df-prm 12836  df-pc 13014

This theorem is referenced by:  pcabs  13055  pcadd2  13070  lgsneg  16029