Theorem pcneg 12271

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 9574	. . 3 |- ( A e. QQ <-> E. x e. ZZ E. y e. NN A = ( x / y ) )
2		zcn 9210	. . . . . . . . 9 |- ( x e. ZZ -> x e. CC )
3	2	ad2antrl 487	. . . . . . . 8 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> x e. CC )
4		nncn 8879	. . . . . . . . 9 |- ( y e. NN -> y e. CC )
5	4	ad2antll 488	. . . . . . . 8 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> y e. CC )
6		nnap0 8900	. . . . . . . . 9 |- ( y e. NN -> y # 0 )
7	6	ad2antll 488	. . . . . . . 8 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> y # 0 )
8	3, 5, 7	divnegapd 8713	. . . . . . 7 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> -u ( x / y ) = ( -u x / y ) )
9	8	oveq2d 5867	. . . . . 6 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( P pCnt -u ( x / y ) ) = ( P pCnt ( -u x / y ) ) )
10		neg0 8158	. . . . . . . . . 10 |- -u 0 = 0
11		simpr 109	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> x = 0 )
12	11	negeqd 8107	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> -u x = -u 0 )
13	10, 12, 11	3eqtr4a 2229	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> -u x = x )
14	13	oveq1d 5866	. . . . . . . 8 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> ( -u x / y ) = ( x / y ) )
15	14	oveq2d 5867	. . . . . . 7 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x = 0 ) -> ( P pCnt ( -u x / y ) ) = ( P pCnt ( x / y ) ) )
16		simpll 524	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> P e. Prime )
17		simplrl 530	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> x e. ZZ )
18	17	znegcld 9329	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> -u x e. ZZ )
19		simpr 109	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> x =/= 0 )
20	2	negne0bd 8216	. . . . . . . . . . . . 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 146	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> -u x =/= 0 )
23		eqid 2170	. . . . . . . . . . . 12 |- sup ( { y e. NN0 | ( P ^ y ) || -u x } , RR , < ) = sup ( { y e. NN0 | ( P ^ y ) || -u x } , RR , < )
24	23	pczpre 12244	. . . . . . . . . . 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 1231	. . . . . . . . . 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 2170	. . . . . . . . . . . . 13 |- sup ( { y e. NN0 | ( P ^ y ) || x } , RR , < ) = sup ( { y e. NN0 | ( P ^ y ) || x } , RR , < )
27	26	pczpre 12244	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( P pCnt x ) = sup ( { y e. NN0 | ( P ^ y ) || x } , RR , < ) )
28		prmz 12058	. . . . . . . . . . . . . . . . 17 |- ( P e. Prime -> P e. ZZ )
29		zexpcl 10484	. . . . . . . . . . . . . . . . 17 |- ( ( P e. ZZ /\ y e. NN0 ) -> ( P ^ y ) e. ZZ )
30	28, 29	sylan 281	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ y e. NN0 ) -> ( P ^ y ) e. ZZ )
31		simpl 108	. . . . . . . . . . . . . . . 16 |- ( ( x e. ZZ /\ x =/= 0 ) -> x e. ZZ )
32		dvdsnegb 11763	. . . . . . . . . . . . . . . 16 |- ( ( ( P ^ y ) e. ZZ /\ x e. ZZ ) -> ( ( P ^ y ) || x <-> ( P ^ y ) || -u x ) )
33	30, 31, 32	syl2an 287	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Prime /\ y e. NN0 ) /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( ( P ^ y ) || x <-> ( P ^ y ) || -u x ) )
34	33	an32s 563	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) /\ y e. NN0 ) -> ( ( P ^ y ) || x <-> ( P ^ y ) || -u x ) )
35	34	rabbidva 2718	. . . . . . . . . . . . 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 6962	. . . . . . . . . . . 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 2203	. . . . . . . . . . 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 1231	. . . . . . . . . 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 2206	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> ( P pCnt -u x ) = ( P pCnt x ) )
40	39	oveq1d 5866	. . . . . . . 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 531	. . . . . . . . 9 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> y e. NN )
42		pcdiv 12249	. . . . . . . . 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 1238	. . . . . . . 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 12249	. . . . . . . . 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 1238	. . . . . . . 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 2213	. . . . . . 7 |- ( ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) /\ x =/= 0 ) -> ( P pCnt ( -u x / y ) ) = ( P pCnt ( x / y ) ) )
47		simprl 526	. . . . . . . . 9 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> x e. ZZ )
48		0zd 9217	. . . . . . . . 9 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> 0 e. ZZ )
49		zdceq 9280	. . . . . . . . 9 |- ( ( x e. ZZ /\ 0 e. ZZ ) -> DECID x = 0 )
50	47, 48, 49	syl2anc 409	. . . . . . . 8 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> DECID x = 0 )
51		dcne 2351	. . . . . . . 8 |- (DECID x = 0 <-> ( x = 0 \/ x =/= 0 ) )
52	50, 51	sylib 121	. . . . . . 7 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( x = 0 \/ x =/= 0 ) )
53	15, 46, 52	mpjaodan 793	. . . . . 6 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( P pCnt ( -u x / y ) ) = ( P pCnt ( x / y ) ) )
54	9, 53	eqtrd 2203	. . . . 5 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( P pCnt -u ( x / y ) ) = ( P pCnt ( x / y ) ) )
55		negeq 8105	. . . . . . 7 |- ( A = ( x / y ) -> -u A = -u ( x / y ) )
56	55	oveq2d 5867	. . . . . 6 |- ( A = ( x / y ) -> ( P pCnt -u A ) = ( P pCnt -u ( x / y ) ) )
57		oveq2 5859	. . . . . 6 |- ( A = ( x / y ) -> ( P pCnt A ) = ( P pCnt ( x / y ) ) )
58	56, 57	eqeq12d 2185	. . . . 5 |- ( A = ( x / y ) -> ( ( P pCnt -u A ) = ( P pCnt A ) <-> ( P pCnt -u ( x / y ) ) = ( P pCnt ( x / y ) ) ) )
59	54, 58	syl5ibrcom 156	. . . 4 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( A = ( x / y ) -> ( P pCnt -u A ) = ( P pCnt A ) ) )
60	59	rexlimdvva 2595	. . 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	syl5bi 151	. 2 |- ( P e. Prime -> ( A e. QQ -> ( P pCnt -u A ) = ( P pCnt A ) ) )
62	61	imp 123	1 |- ( ( P e. Prime /\ A e. QQ ) -> ( P pCnt -u A ) = ( P pCnt A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703  DECID wdc 829   = wceq 1348   e. wcel 2141   =/= wne 2340   E.wrex 2449   {crab 2452   class class class wbr 3987  (class class class)co 5851   supcsup 6957   CCcc 7765   RRcr 7766   0cc0 7767   < clt 7947   - cmin 8083   -ucneg 8084   # cap 8493   / cdiv 8582   NNcn 8871   NN0cn0 9128   ZZcz 9205   QQcq 9571   ^cexp 10468   || cdvds 11742   Primecprime 12054   pCnt cpc 12231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-mulrcl 7866  ax-addcom 7867  ax-mulcom 7868  ax-addass 7869  ax-mulass 7870  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-1rid 7874  ax-0id 7875  ax-rnegex 7876  ax-precex 7877  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883  ax-pre-mulgt0 7884  ax-pre-mulext 7885  ax-arch 7886  ax-caucvg 7887

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-frec 6368  df-1o 6393  df-2o 6394  df-er 6511  df-en 6717  df-sup 6959  df-inf 6960  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-reap 8487  df-ap 8494  df-div 8583  df-inn 8872  df-2 8930  df-3 8931  df-4 8932  df-n0 9129  df-z 9206  df-uz 9481  df-q 9572  df-rp 9604  df-fz 9959  df-fzo 10092  df-fl 10219  df-mod 10272  df-seqfrec 10395  df-exp 10469  df-cj 10799  df-re 10800  df-im 10801  df-rsqrt 10955  df-abs 10956  df-dvds 11743  df-gcd 11891  df-prm 12055  df-pc 12232

This theorem is referenced by:  pcabs  12272  lgsneg  13684