Theorem pcneg 12864

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 9829	. . 3 |- ( A e. QQ <-> E. x e. ZZ E. y e. NN A = ( x / y ) )
2		zcn 9462	. . . . . . . . 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 9129	. . . . . . . . 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 9150	. . . . . . . . 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 8961	. . . . . . 7 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> -u ( x / y ) = ( -u x / y ) )
9	8	oveq2d 6023	. . . . . 6 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( P pCnt -u ( x / y ) ) = ( P pCnt ( -u x / y ) ) )
10		neg0 8403	. . . . . . . . . 10 |- -u 0 = 0
11		simpr 110	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> x = 0 )
12	11	negeqd 8352	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> -u x = -u 0 )
13	10, 12, 11	3eqtr4a 2288	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> -u x = x )
14	13	oveq1d 6022	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> ( -u x / y ) = ( x / y ) )
15	14	oveq2d 6023	. . . . . . 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 9582	. . . . . . . . . . 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 8461	. . . . . . . . . . . . 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 2229	. . . . . . . . . . . 12 |- sup ( { y e. NN0 | ( P ^ y ) || -u x } , RR , < ) = sup ( { y e. NN0 | ( P ^ y ) || -u x } , RR , < )
24	23	pczpre 12836	. . . . . . . . . . 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 1269	. . . . . . . . . 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 2229	. . . . . . . . . . . . 13 |- sup ( { y e. NN0 | ( P ^ y ) || x } , RR , < ) = sup ( { y e. NN0 | ( P ^ y ) || x } , RR , < )
27	26	pczpre 12836	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( P pCnt x ) = sup ( { y e. NN0 | ( P ^ y ) || x } , RR , < ) )
28		prmz 12649	. . . . . . . . . . . . . . . . 17 |- ( P e. Prime -> P e. ZZ )
29		zexpcl 10788	. . . . . . . . . . . . . . . . 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 12335	. . . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . 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 7165	. . . . . . . . . . . 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 2262	. . . . . . . . . . 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 1269	. . . . . . . . . 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 2265	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> ( P pCnt -u x ) = ( P pCnt x ) )
40	39	oveq1d 6022	. . . . . . . 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 12841	. . . . . . . . 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 1276	. . . . . . . 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 12841	. . . . . . . . 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 1276	. . . . . . . 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 2272	. . . . . . 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 9469	. . . . . . . . 9 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> 0 e. ZZ )
49		zdceq 9533	. . . . . . . . 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 2411	. . . . . . . 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 803	. . . . . 6 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( P pCnt ( -u x / y ) ) = ( P pCnt ( x / y ) ) )
54	9, 53	eqtrd 2262	. . . . 5 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( P pCnt -u ( x / y ) ) = ( P pCnt ( x / y ) ) )
55		negeq 8350	. . . . . . 7 |- ( A = ( x / y ) -> -u A = -u ( x / y ) )
56	55	oveq2d 6023	. . . . . 6 |- ( A = ( x / y ) -> ( P pCnt -u A ) = ( P pCnt -u ( x / y ) ) )
57		oveq2 6015	. . . . . 6 |- ( A = ( x / y ) -> ( P pCnt A ) = ( P pCnt ( x / y ) ) )
58	56, 57	eqeq12d 2244	. . . . 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 2656	. . 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 713  DECID wdc 839   = wceq 1395   e. wcel 2200   =/= wne 2400   E.wrex 2509   {crab 2512   class class class wbr 4083  (class class class)co 6007   supcsup 7160   CCcc 8008   RRcr 8009   0cc0 8010   < clt 8192   - cmin 8328   -ucneg 8329   # cap 8739   / cdiv 8830   NNcn 9121   NN0cn0 9380   ZZcz 9457   QQcq 9826   ^cexp 10772   || cdvds 12314   Primecprime 12645   pCnt cpc 12823

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-2o 6569  df-er 6688  df-en 6896  df-sup 7162  df-inf 7163  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fz 10217  df-fzo 10351  df-fl 10502  df-mod 10557  df-seqfrec 10682  df-exp 10773  df-cj 11369  df-re 11370  df-im 11371  df-rsqrt 11525  df-abs 11526  df-dvds 12315  df-gcd 12491  df-prm 12646  df-pc 12824

This theorem is referenced by:  pcabs  12865  pcadd2  12880  lgsneg  15719