Theorem pcadd2 13140

Description: The inequality of pcadd 13139 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		pcxcl 13110	. . 3 |- ( ( P e. Prime /\ A e. QQ ) -> ( P pCnt A ) e. RR* )
4	1, 2, 3	syl2anc 415	. 2 |- ( ph -> ( P pCnt A ) e. RR* )
5		pcadd2.3	. . . 4 |- ( ph -> B e. QQ )
6		qaddcl 10044	. . . 4 |- ( ( A e. QQ /\ B e. QQ ) -> ( A + B ) e. QQ )
7	2, 5, 6	syl2anc 415	. . 3 |- ( ph -> ( A + B ) e. QQ )
8		pcxcl 13110	. . 3 |- ( ( P e. Prime /\ ( A + B ) e. QQ ) -> ( P pCnt ( A + B ) ) e. RR* )
9	1, 7, 8	syl2anc 415	. 2 |- ( ph -> ( P pCnt ( A + B ) ) e. RR* )
10		pcxcl 13110	. . . . 5 |- ( ( P e. Prime /\ B e. QQ ) -> ( P pCnt B ) e. RR* )
11	1, 5, 10	syl2anc 415	. . . 4 |- ( ph -> ( P pCnt B ) e. RR* )
12		pcadd2.4	. . . 4 |- ( ph -> ( P pCnt A ) < ( P pCnt B ) )
13	4, 11, 12	xrltled 10211	. . 3 |- ( ph -> ( P pCnt A ) <_ ( P pCnt B ) )
14	1, 2, 5, 13	pcadd 13139	. 2 |- ( ph -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) )
15		qnegcl 10045	. . . . 5 |- ( B e. QQ -> -u B e. QQ )
16	5, 15	syl 14	. . . 4 |- ( ph -> -u B e. QQ )
17		pcxqcl 13111	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. QQ ) -> ( ( P pCnt A ) e. ZZ \/ ( P pCnt A ) = +oo ) )
18		zq 10035	. . . . . . . . . . . . 13 |- ( ( P pCnt A ) e. ZZ -> ( P pCnt A ) e. QQ )
19	18	orim1i 772	. . . . . . . . . . . 12 |- ( ( ( P pCnt A ) e. ZZ \/ ( P pCnt A ) = +oo ) -> ( ( P pCnt A ) e. QQ \/ ( P pCnt A ) = +oo ) )
20	17, 19	syl 14	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. QQ ) -> ( ( P pCnt A ) e. QQ \/ ( P pCnt A ) = +oo ) )
21	1, 2, 20	syl2anc 415	. . . . . . . . . 10 |- ( ph -> ( ( P pCnt A ) e. QQ \/ ( P pCnt A ) = +oo ) )
22		pcxqcl 13111	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ B e. QQ ) -> ( ( P pCnt B ) e. ZZ \/ ( P pCnt B ) = +oo ) )
23		zq 10035	. . . . . . . . . . . . 13 |- ( ( P pCnt B ) e. ZZ -> ( P pCnt B ) e. QQ )
24	23	orim1i 772	. . . . . . . . . . . 12 |- ( ( ( P pCnt B ) e. ZZ \/ ( P pCnt B ) = +oo ) -> ( ( P pCnt B ) e. QQ \/ ( P pCnt B ) = +oo ) )
25	22, 24	syl 14	. . . . . . . . . . 11 |- ( ( P e. Prime /\ B e. QQ ) -> ( ( P pCnt B ) e. QQ \/ ( P pCnt B ) = +oo ) )
26	1, 5, 25	syl2anc 415	. . . . . . . . . 10 |- ( ph -> ( ( P pCnt B ) e. QQ \/ ( P pCnt B ) = +oo ) )
27		xqltnle 10712	. . . . . . . . . 10 |- ( ( ( ( P pCnt A ) e. QQ \/ ( P pCnt A ) = +oo ) /\ ( ( P pCnt B ) e. QQ \/ ( P pCnt B ) = +oo ) ) -> ( ( P pCnt A ) < ( P pCnt B ) <-> -. ( P pCnt B ) <_ ( P pCnt A ) ) )
28	21, 26, 27	syl2anc 415	. . . . . . . . 9 |- ( ph -> ( ( P pCnt A ) < ( P pCnt B ) <-> -. ( P pCnt B ) <_ ( P pCnt A ) ) )
29	12, 28	mpbid 147	. . . . . . . 8 |- ( ph -> -. ( P pCnt B ) <_ ( P pCnt A ) )
30	1	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ ( P pCnt B ) <_ ( P pCnt ( A + B ) ) ) -> P e. Prime )
31	16	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ ( P pCnt B ) <_ ( P pCnt ( A + B ) ) ) -> -u B e. QQ )
32	7	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ ( P pCnt B ) <_ ( P pCnt ( A + B ) ) ) -> ( A + B ) e. QQ )
33		pcneg 13124	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ B e. QQ ) -> ( P pCnt -u B ) = ( P pCnt B ) )
34	1, 5, 33	syl2anc 415	. . . . . . . . . . . . 13 |- ( ph -> ( P pCnt -u B ) = ( P pCnt B ) )
35	34	breq1d 4140	. . . . . . . . . . . 12 |- ( ph -> ( ( P pCnt -u B ) <_ ( P pCnt ( A + B ) ) <-> ( P pCnt B ) <_ ( P pCnt ( A + B ) ) ) )
36	35	biimpar 297	. . . . . . . . . . 11 |- ( ( ph /\ ( P pCnt B ) <_ ( P pCnt ( A + B ) ) ) -> ( P pCnt -u B ) <_ ( P pCnt ( A + B ) ) )
37	30, 31, 32, 36	pcadd 13139	. . . . . . . . . 10 |- ( ( ph /\ ( P pCnt B ) <_ ( P pCnt ( A + B ) ) ) -> ( P pCnt -u B ) <_ ( P pCnt ( -u B + ( A + B ) ) ) )
38	37	ex 115	. . . . . . . . 9 |- ( ph -> ( ( P pCnt B ) <_ ( P pCnt ( A + B ) ) -> ( P pCnt -u B ) <_ ( P pCnt ( -u B + ( A + B ) ) ) ) )
39		qcn 10043	. . . . . . . . . . . . . . 15 |- ( B e. QQ -> B e. CC )
40	5, 39	syl 14	. . . . . . . . . . . . . 14 |- ( ph -> B e. CC )
41	40	negcld 8625	. . . . . . . . . . . . 13 |- ( ph -> -u B e. CC )
42		qcn 10043	. . . . . . . . . . . . . 14 |- ( A e. QQ -> A e. CC )
43	2, 42	syl 14	. . . . . . . . . . . . 13 |- ( ph -> A e. CC )
44	41, 43, 40	add12d 8494	. . . . . . . . . . . 12 |- ( ph -> ( -u B + ( A + B ) ) = ( A + ( -u B + B ) ) )
45	41, 40	addcomd 8478	. . . . . . . . . . . . . 14 |- ( ph -> ( -u B + B ) = ( B + -u B ) )
46	40	negidd 8628	. . . . . . . . . . . . . 14 |- ( ph -> ( B + -u B ) = 0 )
47	45, 46	eqtrd 2271	. . . . . . . . . . . . 13 |- ( ph -> ( -u B + B ) = 0 )
48	47	oveq2d 6101	. . . . . . . . . . . 12 |- ( ph -> ( A + ( -u B + B ) ) = ( A + 0 ) )
49	43	addridd 8476	. . . . . . . . . . . 12 |- ( ph -> ( A + 0 ) = A )
50	44, 48, 49	3eqtrd 2275	. . . . . . . . . . 11 |- ( ph -> ( -u B + ( A + B ) ) = A )
51	50	oveq2d 6101	. . . . . . . . . 10 |- ( ph -> ( P pCnt ( -u B + ( A + B ) ) ) = ( P pCnt A ) )
52	34, 51	breq12d 4143	. . . . . . . . 9 |- ( ph -> ( ( P pCnt -u B ) <_ ( P pCnt ( -u B + ( A + B ) ) ) <-> ( P pCnt B ) <_ ( P pCnt A ) ) )
53	38, 52	sylibd 149	. . . . . . . 8 |- ( ph -> ( ( P pCnt B ) <_ ( P pCnt ( A + B ) ) -> ( P pCnt B ) <_ ( P pCnt A ) ) )
54	29, 53	mtod 673	. . . . . . 7 |- ( ph -> -. ( P pCnt B ) <_ ( P pCnt ( A + B ) ) )
55		pcxqcl 13111	. . . . . . . . . 10 |- ( ( P e. Prime /\ ( A + B ) e. QQ ) -> ( ( P pCnt ( A + B ) ) e. ZZ \/ ( P pCnt ( A + B ) ) = +oo ) )
56		zq 10035	. . . . . . . . . . 11 |- ( ( P pCnt ( A + B ) ) e. ZZ -> ( P pCnt ( A + B ) ) e. QQ )
57	56	orim1i 772	. . . . . . . . . 10 |- ( ( ( P pCnt ( A + B ) ) e. ZZ \/ ( P pCnt ( A + B ) ) = +oo ) -> ( ( P pCnt ( A + B ) ) e. QQ \/ ( P pCnt ( A + B ) ) = +oo ) )
58	55, 57	syl 14	. . . . . . . . 9 |- ( ( P e. Prime /\ ( A + B ) e. QQ ) -> ( ( P pCnt ( A + B ) ) e. QQ \/ ( P pCnt ( A + B ) ) = +oo ) )
59	1, 7, 58	syl2anc 415	. . . . . . . 8 |- ( ph -> ( ( P pCnt ( A + B ) ) e. QQ \/ ( P pCnt ( A + B ) ) = +oo ) )
60		xqltnle 10712	. . . . . . . 8 |- ( ( ( ( P pCnt ( A + B ) ) e. QQ \/ ( P pCnt ( A + B ) ) = +oo ) /\ ( ( P pCnt B ) e. QQ \/ ( P pCnt B ) = +oo ) ) -> ( ( P pCnt ( A + B ) ) < ( P pCnt B ) <-> -. ( P pCnt B ) <_ ( P pCnt ( A + B ) ) ) )
61	59, 26, 60	syl2anc 415	. . . . . . 7 |- ( ph -> ( ( P pCnt ( A + B ) ) < ( P pCnt B ) <-> -. ( P pCnt B ) <_ ( P pCnt ( A + B ) ) ) )
62	54, 61	mpbird 167	. . . . . 6 |- ( ph -> ( P pCnt ( A + B ) ) < ( P pCnt B ) )
63	9, 11, 62	xrltled 10211	. . . . 5 |- ( ph -> ( P pCnt ( A + B ) ) <_ ( P pCnt B ) )
64	63, 34	breqtrrd 4158	. . . 4 |- ( ph -> ( P pCnt ( A + B ) ) <_ ( P pCnt -u B ) )
65	1, 7, 16, 64	pcadd 13139	. . 3 |- ( ph -> ( P pCnt ( A + B ) ) <_ ( P pCnt ( ( A + B ) + -u B ) ) )
66	43, 40, 41	addassd 8348	. . . . 5 |- ( ph -> ( ( A + B ) + -u B ) = ( A + ( B + -u B ) ) )
67	46	oveq2d 6101	. . . . 5 |- ( ph -> ( A + ( B + -u B ) ) = ( A + 0 ) )
68	66, 67, 49	3eqtrd 2275	. . . 4 |- ( ph -> ( ( A + B ) + -u B ) = A )
69	68	oveq2d 6101	. . 3 |- ( ph -> ( P pCnt ( ( A + B ) + -u B ) ) = ( P pCnt A ) )
70	65, 69	breqtrd 4156	. 2 |- ( ph -> ( P pCnt ( A + B ) ) <_ ( P pCnt A ) )
71	4, 9, 14, 70	xrletrid 10217	1 |- ( ph -> ( P pCnt A ) = ( P pCnt ( A + B ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   CCcc 8177   0cc0 8179   + caddc 8182   +oocpnf 8357   RR*cxr 8359   < clt 8360   <_ cle 8361   -ucneg 8499   ZZcz 9648   QQcq 10028   Primecprime 12901   pCnt cpc 13083

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-2 9365  df-3 9366  df-4 9367  df-n0 9568  df-z 9649  df-uz 9931  df-q 10029  df-rp 10065  df-fz 10422  df-fzo 10560  df-fl 10715  df-mod 10773  df-seqfrec 10898  df-exp 10989  df-cj 11621  df-re 11622  df-im 11623  df-rsqrt 11778  df-abs 11779  df-dvds 12571  df-gcd 12747  df-prm 12902  df-pc 13084

This theorem is used by: (None)