Theorem ppiqltx 16183

Description: The prime-counting function π is strictly less than the identity. (Contributed by Mario Carneiro, 22-Sep-2014.)

Assertion

Ref	Expression
ppiqltx	|- ( ( A e. QQ /\ 0 < A ) -> (π ` A ) < A )

Proof of Theorem ppiqltx

Dummy variables w k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ppiqcl 16163	. . . . 5 |- ( A e. QQ -> (π ` A ) e. NN0 )
2	1	ad2antrr 492	. . . 4 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) e. NN ) -> (π ` A ) e. NN0 )
3	2	nn0red 9625	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) e. NN ) -> (π ` A ) e. RR )
4		simpr 110	. . . 4 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) e. NN ) -> ( |_ ` A ) e. NN )
5	4	nnred 9319	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) e. NN ) -> ( |_ ` A ) e. RR )
6		qre 10034	. . . 4 |- ( A e. QQ -> A e. RR )
7	6	ad2antrr 492	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) e. NN ) -> A e. RR )
8		ppiqfl 16172	. . . . 5 |- ( A e. QQ -> (π ` ( |_ ` A ) ) = (π ` A ) )
9	8	ad2antrr 492	. . . 4 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) e. NN ) -> (π ` ( |_ ` A ) ) = (π ` A ) )
10		fveq2 5695	. . . . . . 7 |- ( w = 1 -> (π ` w ) = (π ` 1 ) )
11		id 19	. . . . . . 7 |- ( w = 1 -> w = 1 )
12	10, 11	breq12d 4143	. . . . . 6 |- ( w = 1 -> ( (π ` w ) < w <-> (π ` 1 ) < 1 ) )
13		fveq2 5695	. . . . . . 7 |- ( w = k -> (π ` w ) = (π ` k ) )
14		id 19	. . . . . . 7 |- ( w = k -> w = k )
15	13, 14	breq12d 4143	. . . . . 6 |- ( w = k -> ( (π ` w ) < w <-> (π ` k ) < k ) )
16		fveq2 5695	. . . . . . 7 |- ( w = ( k + 1 ) -> (π ` w ) = (π ` ( k + 1 ) ) )
17		id 19	. . . . . . 7 |- ( w = ( k + 1 ) -> w = ( k + 1 ) )
18	16, 17	breq12d 4143	. . . . . 6 |- ( w = ( k + 1 ) -> ( (π ` w ) < w <-> (π ` ( k + 1 ) ) < ( k + 1 ) ) )
19		fveq2 5695	. . . . . . 7 |- ( w = ( |_ ` A ) -> (π ` w ) = (π ` ( |_ ` A ) ) )
20		id 19	. . . . . . 7 |- ( w = ( |_ ` A ) -> w = ( |_ ` A ) )
21	19, 20	breq12d 4143	. . . . . 6 |- ( w = ( |_ ` A ) -> ( (π ` w ) < w <-> (π ` ( |_ ` A ) ) < ( |_ ` A ) ) )
22		ppi1 16176	. . . . . . 7 |- (π ` 1 ) = 0
23		0lt1 8454	. . . . . . 7 |- 0 < 1
24	22, 23	eqbrtri 4151	. . . . . 6 |- (π ` 1 ) < 1
25		nnq 10042	. . . . . . . . . . . 12 |- ( k e. NN -> k e. QQ )
26		1z 9674	. . . . . . . . . . . . 13 |- 1 e. ZZ
27		zq 10035	. . . . . . . . . . . . 13 |- ( 1 e. ZZ -> 1 e. QQ )
28	26, 27	ax-mp 5	. . . . . . . . . . . 12 |- 1 e. QQ
29		qaddcl 10044	. . . . . . . . . . . 12 |- ( ( k e. QQ /\ 1 e. QQ ) -> ( k + 1 ) e. QQ )
30	25, 28, 29	sylancl 417	. . . . . . . . . . 11 |- ( k e. NN -> ( k + 1 ) e. QQ )
31		ppiqcl 16163	. . . . . . . . . . 11 |- ( ( k + 1 ) e. QQ -> (π ` ( k + 1 ) ) e. NN0 )
32	30, 31	syl 14	. . . . . . . . . 10 |- ( k e. NN -> (π ` ( k + 1 ) ) e. NN0 )
33	32	adantr 276	. . . . . . . . 9 |- ( ( k e. NN /\ (π ` k ) < k ) -> (π ` ( k + 1 ) ) e. NN0 )
34	33	nn0red 9625	. . . . . . . 8 |- ( ( k e. NN /\ (π ` k ) < k ) -> (π ` ( k + 1 ) ) e. RR )
35		ppiqcl 16163	. . . . . . . . . . . 12 |- ( k e. QQ -> (π ` k ) e. NN0 )
36	25, 35	syl 14	. . . . . . . . . . 11 |- ( k e. NN -> (π ` k ) e. NN0 )
37	36	adantr 276	. . . . . . . . . 10 |- ( ( k e. NN /\ (π ` k ) < k ) -> (π ` k ) e. NN0 )
38	37	nn0red 9625	. . . . . . . . 9 |- ( ( k e. NN /\ (π ` k ) < k ) -> (π ` k ) e. RR )
39		peano2re 8463	. . . . . . . . 9 |- ( (π ` k ) e. RR -> ( (π ` k ) + 1 ) e. RR )
40	38, 39	syl 14	. . . . . . . 8 |- ( ( k e. NN /\ (π ` k ) < k ) -> ( (π ` k ) + 1 ) e. RR )
41		nnre 9313	. . . . . . . . . 10 |- ( k e. NN -> k e. RR )
42	41	adantr 276	. . . . . . . . 9 |- ( ( k e. NN /\ (π ` k ) < k ) -> k e. RR )
43		peano2re 8463	. . . . . . . . 9 |- ( k e. RR -> ( k + 1 ) e. RR )
44	42, 43	syl 14	. . . . . . . 8 |- ( ( k e. NN /\ (π ` k ) < k ) -> ( k + 1 ) e. RR )
45		ppiqp1le 16173	. . . . . . . . . 10 |- ( k e. QQ -> (π ` ( k + 1 ) ) <_ ( (π ` k ) + 1 ) )
46	25, 45	syl 14	. . . . . . . . 9 |- ( k e. NN -> (π ` ( k + 1 ) ) <_ ( (π ` k ) + 1 ) )
47	46	adantr 276	. . . . . . . 8 |- ( ( k e. NN /\ (π ` k ) < k ) -> (π ` ( k + 1 ) ) <_ ( (π ` k ) + 1 ) )
48		1red 8341	. . . . . . . . 9 |- ( ( k e. NN /\ (π ` k ) < k ) -> 1 e. RR )
49		simpr 110	. . . . . . . . 9 |- ( ( k e. NN /\ (π ` k ) < k ) -> (π ` k ) < k )
50	38, 42, 48, 49	ltadd1dd 8885	. . . . . . . 8 |- ( ( k e. NN /\ (π ` k ) < k ) -> ( (π ` k ) + 1 ) < ( k + 1 ) )
51	34, 40, 44, 47, 50	lelttrd 8452	. . . . . . 7 |- ( ( k e. NN /\ (π ` k ) < k ) -> (π ` ( k + 1 ) ) < ( k + 1 ) )
52	51	ex 115	. . . . . 6 |- ( k e. NN -> ( (π ` k ) < k -> (π ` ( k + 1 ) ) < ( k + 1 ) ) )
53	12, 15, 18, 21, 24, 52	nnind 9322	. . . . 5 |- ( ( |_ ` A ) e. NN -> (π ` ( |_ ` A ) ) < ( |_ ` A ) )
54	4, 53	syl 14	. . . 4 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) e. NN ) -> (π ` ( |_ ` A ) ) < ( |_ ` A ) )
55	9, 54	eqbrtrrd 4154	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) e. NN ) -> (π ` A ) < ( |_ ` A ) )
56		flqle 10725	. . . 4 |- ( A e. QQ -> ( |_ ` A ) <_ A )
57	56	ad2antrr 492	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) e. NN ) -> ( |_ ` A ) <_ A )
58	3, 5, 7, 55, 57	ltletrd 8752	. 2 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) e. NN ) -> (π ` A ) < A )
59	8	ad2antrr 492	. . . 4 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) = 0 ) -> (π ` ( |_ ` A ) ) = (π ` A ) )
60		simpr 110	. . . . . 6 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) = 0 ) -> ( |_ ` A ) = 0 )
61	60	fveq2d 5699	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) = 0 ) -> (π ` ( |_ ` A ) ) = (π ` 0 ) )
62		2pos 9397	. . . . . 6 |- 0 < 2
63		0z 9659	. . . . . . 7 |- 0 e. ZZ
64		zq 10035	. . . . . . 7 |- ( 0 e. ZZ -> 0 e. QQ )
65		ppiqeq0 16182	. . . . . . 7 |- ( 0 e. QQ -> ( (π ` 0 ) = 0 <-> 0 < 2 ) )
66	63, 64, 65	mp2b 8	. . . . . 6 |- ( (π ` 0 ) = 0 <-> 0 < 2 )
67	62, 66	mpbir 146	. . . . 5 |- (π ` 0 ) = 0
68	61, 67	eqtrdi 2287	. . . 4 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) = 0 ) -> (π ` ( |_ ` A ) ) = 0 )
69	59, 68	eqtr3d 2273	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) = 0 ) -> (π ` A ) = 0 )
70		simplr 533	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) = 0 ) -> 0 < A )
71	69, 70	eqbrtrd 4152	. 2 |- ( ( ( A e. QQ /\ 0 < A ) /\ ( |_ ` A ) = 0 ) -> (π ` A ) < A )
72		0red 8327	. . . . 5 |- ( ( A e. QQ /\ 0 < A ) -> 0 e. RR )
73	6	adantr 276	. . . . 5 |- ( ( A e. QQ /\ 0 < A ) -> A e. RR )
74		simpr 110	. . . . 5 |- ( ( A e. QQ /\ 0 < A ) -> 0 < A )
75	72, 73, 74	ltled 8446	. . . 4 |- ( ( A e. QQ /\ 0 < A ) -> 0 <_ A )
76		flqge0nn0 10741	. . . 4 |- ( ( A e. QQ /\ 0 <_ A ) -> ( |_ ` A ) e. NN0 )
77	75, 76	syldan 282	. . 3 |- ( ( A e. QQ /\ 0 < A ) -> ( |_ ` A ) e. NN0 )
78		elnn0 9569	. . 3 |- ( ( |_ ` A ) e. NN0 <-> ( ( |_ ` A ) e. NN \/ ( |_ ` A ) = 0 ) )
79	77, 78	sylib 122	. 2 |- ( ( A e. QQ /\ 0 < A ) -> ( ( |_ ` A ) e. NN \/ ( |_ ` A ) = 0 ) )
80	58, 71, 79	mpjaodan 810	1 |- ( ( A e. QQ /\ 0 < A ) -> (π ` A ) < A )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   < clt 8360   <_ cle 8361   NNcn 9306   2c2 9357   NN0cn0 9567   ZZcz 9648   QQcq 10028   |_cfl 10713  πcppi 16152

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-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  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-icc 10307  df-fz 10422  df-fl 10715  df-mod 10773  df-seqfrec 10898  df-exp 10989  df-ihash 11229  df-cj 11621  df-re 11622  df-im 11623  df-rsqrt 11778  df-abs 11779  df-dvds 12571  df-prm 12902  df-ppi 16154

This theorem is used by: (None)