Theorem pcadd2 12535

Description: The inequality of pcadd 12534 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 12505	. . 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 9726	. . . 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 12505	. . 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 12505	. . . . 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 9891	. . 3 |- ( ph -> ( P pCnt A ) <_ ( P pCnt B ) )
14	1, 2, 5, 13	pcadd 12534	. 2 |- ( ph -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) )
15		qnegcl 9727	. . . . 5 |- ( B e. QQ -> -u B e. QQ )
16	5, 15	syl 14	. . . 4 |- ( ph -> -u B e. QQ )
17		pcxqcl 12506	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. QQ ) -> ( ( P pCnt A ) e. ZZ \/ ( P pCnt A ) = +oo ) )
18		zq 9717	. . . . . . . . . . . . 13 |- ( ( P pCnt A ) e. ZZ -> ( P pCnt A ) e. QQ )
19	18	orim1i 761	. . . . . . . . . . . 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 12506	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ B e. QQ ) -> ( ( P pCnt B ) e. ZZ \/ ( P pCnt B ) = +oo ) )
23		zq 9717	. . . . . . . . . . . . 13 |- ( ( P pCnt B ) e. ZZ -> ( P pCnt B ) e. QQ )
24	23	orim1i 761	. . . . . . . . . . . 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 10374	. . . . . . . . . 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 12519	. . . . . . . . . . . . . 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 4044	. . . . . . . . . . . 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 12534	. . . . . . . . . 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 9725	. . . . . . . . . . . . . . 15 |- ( B e. QQ -> B e. CC )
40	5, 39	syl 14	. . . . . . . . . . . . . 14 |- ( ph -> B e. CC )
41	40	negcld 8341	. . . . . . . . . . . . 13 |- ( ph -> -u B e. CC )
42		qcn 9725	. . . . . . . . . . . . . 14 |- ( A e. QQ -> A e. CC )
43	2, 42	syl 14	. . . . . . . . . . . . 13 |- ( ph -> A e. CC )
44	41, 43, 40	add12d 8210	. . . . . . . . . . . 12 |- ( ph -> ( -u B + ( A + B ) ) = ( A + ( -u B + B ) ) )
45	41, 40	addcomd 8194	. . . . . . . . . . . . . 14 |- ( ph -> ( -u B + B ) = ( B + -u B ) )
46	40	negidd 8344	. . . . . . . . . . . . . 14 |- ( ph -> ( B + -u B ) = 0 )
47	45, 46	eqtrd 2229	. . . . . . . . . . . . 13 |- ( ph -> ( -u B + B ) = 0 )
48	47	oveq2d 5941	. . . . . . . . . . . 12 |- ( ph -> ( A + ( -u B + B ) ) = ( A + 0 ) )
49	43	addridd 8192	. . . . . . . . . . . 12 |- ( ph -> ( A + 0 ) = A )
50	44, 48, 49	3eqtrd 2233	. . . . . . . . . . 11 |- ( ph -> ( -u B + ( A + B ) ) = A )
51	50	oveq2d 5941	. . . . . . . . . 10 |- ( ph -> ( P pCnt ( -u B + ( A + B ) ) ) = ( P pCnt A ) )
52	34, 51	breq12d 4047	. . . . . . . . 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 664	. . . . . . 7 |- ( ph -> -. ( P pCnt B ) <_ ( P pCnt ( A + B ) ) )
55		pcxqcl 12506	. . . . . . . . . 10 |- ( ( P e. Prime /\ ( A + B ) e. QQ ) -> ( ( P pCnt ( A + B ) ) e. ZZ \/ ( P pCnt ( A + B ) ) = +oo ) )
56		zq 9717	. . . . . . . . . . 11 |- ( ( P pCnt ( A + B ) ) e. ZZ -> ( P pCnt ( A + B ) ) e. QQ )
57	56	orim1i 761	. . . . . . . . . 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 10374	. . . . . . . 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 9891	. . . . 5 |- ( ph -> ( P pCnt ( A + B ) ) <_ ( P pCnt B ) )
64	63, 34	breqtrrd 4062	. . . 4 |- ( ph -> ( P pCnt ( A + B ) ) <_ ( P pCnt -u B ) )
65	1, 7, 16, 64	pcadd 12534	. . 3 |- ( ph -> ( P pCnt ( A + B ) ) <_ ( P pCnt ( ( A + B ) + -u B ) ) )
66	43, 40, 41	addassd 8066	. . . . 5 |- ( ph -> ( ( A + B ) + -u B ) = ( A + ( B + -u B ) ) )
67	46	oveq2d 5941	. . . . 5 |- ( ph -> ( A + ( B + -u B ) ) = ( A + 0 ) )
68	66, 67, 49	3eqtrd 2233	. . . 4 |- ( ph -> ( ( A + B ) + -u B ) = A )
69	68	oveq2d 5941	. . 3 |- ( ph -> ( P pCnt ( ( A + B ) + -u B ) ) = ( P pCnt A ) )
70	65, 69	breqtrd 4060	. 2 |- ( ph -> ( P pCnt ( A + B ) ) <_ ( P pCnt A ) )
71	4, 9, 14, 70	xrletrid 9897	1 |- ( ph -> ( P pCnt A ) = ( P pCnt ( A + B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2167   class class class wbr 4034  (class class class)co 5925   CCcc 7894   0cc0 7896   + caddc 7899   +oocpnf 8075   RR*cxr 8077   < clt 8078   <_ cle 8079   -ucneg 8215   ZZcz 9343   QQcq 9710   Primecprime 12300   pCnt cpc 12478

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-2o 6484  df-er 6601  df-en 6809  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-dvds 11970  df-gcd 12146  df-prm 12301  df-pc 12479

This theorem is referenced by: (None)