Theorem pcadd2 15594

Description: The inequality of pcadd 15593 becomes an equality when one of the factors has prime count strictly less than the other. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypotheses

Ref	Expression
pcadd2.1	|- ( ph -> P e. Prime )
pcadd2.2	|- ( ph -> A e. QQ )
pcadd2.3	|- ( ph -> B e. QQ )
pcadd2.4	|- ( ph -> ( P pCnt A ) < ( P pCnt B ) )

Assertion

Ref	Expression
pcadd2	|- ( ph -> ( P pCnt A ) = ( P pCnt ( A + B ) ) )

Proof of Theorem pcadd2

Step	Hyp	Ref	Expression
1		pcadd2.1	. . 3 |- ( ph -> P e. Prime )
2		pcadd2.2	. . 3 |- ( ph -> A e. QQ )
3		pcadd2.3	. . 3 |- ( ph -> B e. QQ )
4		pcadd2.4	. . . 4 |- ( ph -> ( P pCnt A ) < ( P pCnt B ) )
5		pcxcl 15565	. . . . . 6 |- ( ( P e. Prime /\ A e. QQ ) -> ( P pCnt A ) e. RR* )
6	1, 2, 5	syl2anc 693	. . . . 5 |- ( ph -> ( P pCnt A ) e. RR* )
7		pcxcl 15565	. . . . . 6 |- ( ( P e. Prime /\ B e. QQ ) -> ( P pCnt B ) e. RR* )
8	1, 3, 7	syl2anc 693	. . . . 5 |- ( ph -> ( P pCnt B ) e. RR* )
9		xrltle 11982	. . . . 5 |- ( ( ( P pCnt A ) e. RR* /\ ( P pCnt B ) e. RR* ) -> ( ( P pCnt A ) < ( P pCnt B ) -> ( P pCnt A ) <_ ( P pCnt B ) ) )
10	6, 8, 9	syl2anc 693	. . . 4 |- ( ph -> ( ( P pCnt A ) < ( P pCnt B ) -> ( P pCnt A ) <_ ( P pCnt B ) ) )
11	4, 10	mpd 15	. . 3 |- ( ph -> ( P pCnt A ) <_ ( P pCnt B ) )
12	1, 2, 3, 11	pcadd 15593	. 2 |- ( ph -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) )
13		qaddcl 11804	. . . . 5 |- ( ( A e. QQ /\ B e. QQ ) -> ( A + B ) e. QQ )
14	2, 3, 13	syl2anc 693	. . . 4 |- ( ph -> ( A + B ) e. QQ )
15		qnegcl 11805	. . . . 5 |- ( B e. QQ -> -u B e. QQ )
16	3, 15	syl 17	. . . 4 |- ( ph -> -u B e. QQ )
17		xrltnle 10105	. . . . . . . . . 10 |- ( ( ( P pCnt A ) e. RR* /\ ( P pCnt B ) e. RR* ) -> ( ( P pCnt A ) < ( P pCnt B ) <-> -. ( P pCnt B ) <_ ( P pCnt A ) ) )
18	6, 8, 17	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( ( P pCnt A ) < ( P pCnt B ) <-> -. ( P pCnt B ) <_ ( P pCnt A ) ) )
19	4, 18	mpbid 222	. . . . . . . 8 |- ( ph -> -. ( P pCnt B ) <_ ( P pCnt A ) )
20	1	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( P pCnt B ) <_ ( P pCnt ( A + B ) ) ) -> P e. Prime )
21	16	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( P pCnt B ) <_ ( P pCnt ( A + B ) ) ) -> -u B e. QQ )
22	14	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( P pCnt B ) <_ ( P pCnt ( A + B ) ) ) -> ( A + B ) e. QQ )
23		pcneg 15578	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ B e. QQ ) -> ( P pCnt -u B ) = ( P pCnt B ) )
24	1, 3, 23	syl2anc 693	. . . . . . . . . . . . 13 |- ( ph -> ( P pCnt -u B ) = ( P pCnt B ) )
25	24	breq1d 4663	. . . . . . . . . . . 12 |- ( ph -> ( ( P pCnt -u B ) <_ ( P pCnt ( A + B ) ) <-> ( P pCnt B ) <_ ( P pCnt ( A + B ) ) ) )
26	25	biimpar 502	. . . . . . . . . . 11 |- ( ( ph /\ ( P pCnt B ) <_ ( P pCnt ( A + B ) ) ) -> ( P pCnt -u B ) <_ ( P pCnt ( A + B ) ) )
27	20, 21, 22, 26	pcadd 15593	. . . . . . . . . 10 |- ( ( ph /\ ( P pCnt B ) <_ ( P pCnt ( A + B ) ) ) -> ( P pCnt -u B ) <_ ( P pCnt ( -u B + ( A + B ) ) ) )
28	27	ex 450	. . . . . . . . 9 |- ( ph -> ( ( P pCnt B ) <_ ( P pCnt ( A + B ) ) -> ( P pCnt -u B ) <_ ( P pCnt ( -u B + ( A + B ) ) ) ) )
29		qcn 11802	. . . . . . . . . . . . . . 15 |- ( B e. QQ -> B e. CC )
30	3, 29	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> B e. CC )
31	30	negcld 10379	. . . . . . . . . . . . 13 |- ( ph -> -u B e. CC )
32		qcn 11802	. . . . . . . . . . . . . 14 |- ( A e. QQ -> A e. CC )
33	2, 32	syl 17	. . . . . . . . . . . . 13 |- ( ph -> A e. CC )
34	31, 33, 30	add12d 10262	. . . . . . . . . . . 12 |- ( ph -> ( -u B + ( A + B ) ) = ( A + ( -u B + B ) ) )
35	31, 30	addcomd 10238	. . . . . . . . . . . . . 14 |- ( ph -> ( -u B + B ) = ( B + -u B ) )
36	30	negidd 10382	. . . . . . . . . . . . . 14 |- ( ph -> ( B + -u B ) = 0 )
37	35, 36	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ph -> ( -u B + B ) = 0 )
38	37	oveq2d 6666	. . . . . . . . . . . 12 |- ( ph -> ( A + ( -u B + B ) ) = ( A + 0 ) )
39	33	addid1d 10236	. . . . . . . . . . . 12 |- ( ph -> ( A + 0 ) = A )
40	34, 38, 39	3eqtrd 2660	. . . . . . . . . . 11 |- ( ph -> ( -u B + ( A + B ) ) = A )
41	40	oveq2d 6666	. . . . . . . . . 10 |- ( ph -> ( P pCnt ( -u B + ( A + B ) ) ) = ( P pCnt A ) )
42	24, 41	breq12d 4666	. . . . . . . . 9 |- ( ph -> ( ( P pCnt -u B ) <_ ( P pCnt ( -u B + ( A + B ) ) ) <-> ( P pCnt B ) <_ ( P pCnt A ) ) )
43	28, 42	sylibd 229	. . . . . . . 8 |- ( ph -> ( ( P pCnt B ) <_ ( P pCnt ( A + B ) ) -> ( P pCnt B ) <_ ( P pCnt A ) ) )
44	19, 43	mtod 189	. . . . . . 7 |- ( ph -> -. ( P pCnt B ) <_ ( P pCnt ( A + B ) ) )
45		pcxcl 15565	. . . . . . . . 9 |- ( ( P e. Prime /\ ( A + B ) e. QQ ) -> ( P pCnt ( A + B ) ) e. RR* )
46	1, 14, 45	syl2anc 693	. . . . . . . 8 |- ( ph -> ( P pCnt ( A + B ) ) e. RR* )
47		xrltnle 10105	. . . . . . . 8 |- ( ( ( P pCnt ( A + B ) ) e. RR* /\ ( P pCnt B ) e. RR* ) -> ( ( P pCnt ( A + B ) ) < ( P pCnt B ) <-> -. ( P pCnt B ) <_ ( P pCnt ( A + B ) ) ) )
48	46, 8, 47	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( P pCnt ( A + B ) ) < ( P pCnt B ) <-> -. ( P pCnt B ) <_ ( P pCnt ( A + B ) ) ) )
49	44, 48	mpbird 247	. . . . . 6 |- ( ph -> ( P pCnt ( A + B ) ) < ( P pCnt B ) )
50		xrltle 11982	. . . . . . 7 |- ( ( ( P pCnt ( A + B ) ) e. RR* /\ ( P pCnt B ) e. RR* ) -> ( ( P pCnt ( A + B ) ) < ( P pCnt B ) -> ( P pCnt ( A + B ) ) <_ ( P pCnt B ) ) )
51	46, 8, 50	syl2anc 693	. . . . . 6 |- ( ph -> ( ( P pCnt ( A + B ) ) < ( P pCnt B ) -> ( P pCnt ( A + B ) ) <_ ( P pCnt B ) ) )
52	49, 51	mpd 15	. . . . 5 |- ( ph -> ( P pCnt ( A + B ) ) <_ ( P pCnt B ) )
53	52, 24	breqtrrd 4681	. . . 4 |- ( ph -> ( P pCnt ( A + B ) ) <_ ( P pCnt -u B ) )
54	1, 14, 16, 53	pcadd 15593	. . 3 |- ( ph -> ( P pCnt ( A + B ) ) <_ ( P pCnt ( ( A + B ) + -u B ) ) )
55	33, 30, 31	addassd 10062	. . . . 5 |- ( ph -> ( ( A + B ) + -u B ) = ( A + ( B + -u B ) ) )
56	36	oveq2d 6666	. . . . 5 |- ( ph -> ( A + ( B + -u B ) ) = ( A + 0 ) )
57	55, 56, 39	3eqtrd 2660	. . . 4 |- ( ph -> ( ( A + B ) + -u B ) = A )
58	57	oveq2d 6666	. . 3 |- ( ph -> ( P pCnt ( ( A + B ) + -u B ) ) = ( P pCnt A ) )
59	54, 58	breqtrd 4679	. 2 |- ( ph -> ( P pCnt ( A + B ) ) <_ ( P pCnt A ) )
60		xrletri3 11985	. . 3 |- ( ( ( P pCnt A ) e. RR* /\ ( P pCnt ( A + B ) ) e. RR* ) -> ( ( P pCnt A ) = ( P pCnt ( A + B ) ) <-> ( ( P pCnt A ) <_ ( P pCnt ( A + B ) ) /\ ( P pCnt ( A + B ) ) <_ ( P pCnt A ) ) ) )
61	6, 46, 60	syl2anc 693	. 2 |- ( ph -> ( ( P pCnt A ) = ( P pCnt ( A + B ) ) <-> ( ( P pCnt A ) <_ ( P pCnt ( A + B ) ) /\ ( P pCnt ( A + B ) ) <_ ( P pCnt A ) ) ) )
62	12, 59, 61	mpbir2and 957	1 |- ( ph -> ( P pCnt A ) = ( P pCnt ( A + B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   CCcc 9934   0cc0 9936   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   -ucneg 10267   QQcq 11788   Primecprime 15385   pCnt cpc 15541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542

This theorem is referenced by:  sylow1lem1  18013