Theorem pcadd2 12904

Description: The inequality of pcadd 12903 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 12874	. . 3 |- ( ( P e. Prime /\ A e. QQ ) -> ( P pCnt A ) e. RR* )
4	1, 2, 3	syl2anc 411	. 2 |- ( ph -> ( P pCnt A ) e. RR* )
5		pcadd2.3	. . . 4 |- ( ph -> B e. QQ )
6		qaddcl 9859	. . . 4 |- ( ( A e. QQ /\ B e. QQ ) -> ( A + B ) e. QQ )
7	2, 5, 6	syl2anc 411	. . 3 |- ( ph -> ( A + B ) e. QQ )
8		pcxcl 12874	. . 3 |- ( ( P e. Prime /\ ( A + B ) e. QQ ) -> ( P pCnt ( A + B ) ) e. RR* )
9	1, 7, 8	syl2anc 411	. 2 |- ( ph -> ( P pCnt ( A + B ) ) e. RR* )
10		pcxcl 12874	. . . . 5 |- ( ( P e. Prime /\ B e. QQ ) -> ( P pCnt B ) e. RR* )
11	1, 5, 10	syl2anc 411	. . . 4 |- ( ph -> ( P pCnt B ) e. RR* )
12		pcadd2.4	. . . 4 |- ( ph -> ( P pCnt A ) < ( P pCnt B ) )
13	4, 11, 12	xrltled 10024	. . 3 |- ( ph -> ( P pCnt A ) <_ ( P pCnt B ) )
14	1, 2, 5, 13	pcadd 12903	. 2 |- ( ph -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) )
15		qnegcl 9860	. . . . 5 |- ( B e. QQ -> -u B e. QQ )
16	5, 15	syl 14	. . . 4 |- ( ph -> -u B e. QQ )
17		pcxqcl 12875	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. QQ ) -> ( ( P pCnt A ) e. ZZ \/ ( P pCnt A ) = +oo ) )
18		zq 9850	. . . . . . . . . . . . 13 |- ( ( P pCnt A ) e. ZZ -> ( P pCnt A ) e. QQ )
19	18	orim1i 765	. . . . . . . . . . . 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 411	. . . . . . . . . 10 |- ( ph -> ( ( P pCnt A ) e. QQ \/ ( P pCnt A ) = +oo ) )
22		pcxqcl 12875	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ B e. QQ ) -> ( ( P pCnt B ) e. ZZ \/ ( P pCnt B ) = +oo ) )
23		zq 9850	. . . . . . . . . . . . 13 |- ( ( P pCnt B ) e. ZZ -> ( P pCnt B ) e. QQ )
24	23	orim1i 765	. . . . . . . . . . . 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 411	. . . . . . . . . 10 |- ( ph -> ( ( P pCnt B ) e. QQ \/ ( P pCnt B ) = +oo ) )
27		xqltnle 10517	. . . . . . . . . 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 411	. . . . . . . . 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 12888	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ B e. QQ ) -> ( P pCnt -u B ) = ( P pCnt B ) )
34	1, 5, 33	syl2anc 411	. . . . . . . . . . . . 13 |- ( ph -> ( P pCnt -u B ) = ( P pCnt B ) )
35	34	breq1d 4096	. . . . . . . . . . . 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 12903	. . . . . . . . . 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 9858	. . . . . . . . . . . . . . 15 |- ( B e. QQ -> B e. CC )
40	5, 39	syl 14	. . . . . . . . . . . . . 14 |- ( ph -> B e. CC )
41	40	negcld 8467	. . . . . . . . . . . . 13 |- ( ph -> -u B e. CC )
42		qcn 9858	. . . . . . . . . . . . . 14 |- ( A e. QQ -> A e. CC )
43	2, 42	syl 14	. . . . . . . . . . . . 13 |- ( ph -> A e. CC )
44	41, 43, 40	add12d 8336	. . . . . . . . . . . 12 |- ( ph -> ( -u B + ( A + B ) ) = ( A + ( -u B + B ) ) )
45	41, 40	addcomd 8320	. . . . . . . . . . . . . 14 |- ( ph -> ( -u B + B ) = ( B + -u B ) )
46	40	negidd 8470	. . . . . . . . . . . . . 14 |- ( ph -> ( B + -u B ) = 0 )
47	45, 46	eqtrd 2262	. . . . . . . . . . . . 13 |- ( ph -> ( -u B + B ) = 0 )
48	47	oveq2d 6029	. . . . . . . . . . . 12 |- ( ph -> ( A + ( -u B + B ) ) = ( A + 0 ) )
49	43	addridd 8318	. . . . . . . . . . . 12 |- ( ph -> ( A + 0 ) = A )
50	44, 48, 49	3eqtrd 2266	. . . . . . . . . . 11 |- ( ph -> ( -u B + ( A + B ) ) = A )
51	50	oveq2d 6029	. . . . . . . . . 10 |- ( ph -> ( P pCnt ( -u B + ( A + B ) ) ) = ( P pCnt A ) )
52	34, 51	breq12d 4099	. . . . . . . . 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 667	. . . . . . 7 |- ( ph -> -. ( P pCnt B ) <_ ( P pCnt ( A + B ) ) )
55		pcxqcl 12875	. . . . . . . . . 10 |- ( ( P e. Prime /\ ( A + B ) e. QQ ) -> ( ( P pCnt ( A + B ) ) e. ZZ \/ ( P pCnt ( A + B ) ) = +oo ) )
56		zq 9850	. . . . . . . . . . 11 |- ( ( P pCnt ( A + B ) ) e. ZZ -> ( P pCnt ( A + B ) ) e. QQ )
57	56	orim1i 765	. . . . . . . . . 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 411	. . . . . . . 8 |- ( ph -> ( ( P pCnt ( A + B ) ) e. QQ \/ ( P pCnt ( A + B ) ) = +oo ) )
60		xqltnle 10517	. . . . . . . 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 411	. . . . . . 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 10024	. . . . 5 |- ( ph -> ( P pCnt ( A + B ) ) <_ ( P pCnt B ) )
64	63, 34	breqtrrd 4114	. . . 4 |- ( ph -> ( P pCnt ( A + B ) ) <_ ( P pCnt -u B ) )
65	1, 7, 16, 64	pcadd 12903	. . 3 |- ( ph -> ( P pCnt ( A + B ) ) <_ ( P pCnt ( ( A + B ) + -u B ) ) )
66	43, 40, 41	addassd 8192	. . . . 5 |- ( ph -> ( ( A + B ) + -u B ) = ( A + ( B + -u B ) ) )
67	46	oveq2d 6029	. . . . 5 |- ( ph -> ( A + ( B + -u B ) ) = ( A + 0 ) )
68	66, 67, 49	3eqtrd 2266	. . . 4 |- ( ph -> ( ( A + B ) + -u B ) = A )
69	68	oveq2d 6029	. . 3 |- ( ph -> ( P pCnt ( ( A + B ) + -u B ) ) = ( P pCnt A ) )
70	65, 69	breqtrd 4112	. 2 |- ( ph -> ( P pCnt ( A + B ) ) <_ ( P pCnt A ) )
71	4, 9, 14, 70	xrletrid 10030	1 |- ( ph -> ( P pCnt A ) = ( P pCnt ( A + B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   class class class wbr 4086  (class class class)co 6013   CCcc 8020   0cc0 8022   + caddc 8025   +oocpnf 8201   RR*cxr 8203   < clt 8204   <_ cle 8205   -ucneg 8341   ZZcz 9469   QQcq 9843   Primecprime 12669   pCnt cpc 12847

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-stab 836  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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-2o 6578  df-er 6697  df-en 6905  df-sup 7174  df-inf 7175  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fzo 10368  df-fl 10520  df-mod 10575  df-seqfrec 10700  df-exp 10791  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-dvds 12339  df-gcd 12515  df-prm 12670  df-pc 12848

This theorem is referenced by: (None)