Theorem ppiltx 24903

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

Assertion

Ref	Expression
ppiltx	|- ( A e. RR+ -> (π ` A ) < A )

Proof of Theorem ppiltx

Step	Hyp	Ref	Expression
1		rpre 11839	. . . . . 6 |- ( A e. RR+ -> A e. RR )
2		ppicl 24857	. . . . . 6 |- ( A e. RR -> (π ` A ) e. NN0 )
3	1, 2	syl 17	. . . . 5 |- ( A e. RR+ -> (π ` A ) e. NN0 )
4	3	nn0red 11352	. . . 4 |- ( A e. RR+ -> (π ` A ) e. RR )
5	4	adantr 481	. . 3 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> (π ` A ) e. RR )
6		reflcl 12597	. . . . 5 |- ( A e. RR -> ( |_ ` A ) e. RR )
7	1, 6	syl 17	. . . 4 |- ( A e. RR+ -> ( |_ ` A ) e. RR )
8	7	adantr 481	. . 3 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> ( |_ ` A ) e. RR )
9	1	adantr 481	. . 3 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> A e. RR )
10		fzfi 12771	. . . . . 6 |- ( 1 ... ( |_ ` A ) ) e. Fin
11		inss1 3833	. . . . . . 7 |- ( ( 2 ... ( |_ ` A ) ) i^i Prime ) C_ ( 2 ... ( |_ ` A ) )
12		2eluzge1 11734	. . . . . . . . 9 |- 2 e. ( ZZ>= ` 1 )
13		fzss1 12380	. . . . . . . . 9 |- ( 2 e. ( ZZ>= ` 1 ) -> ( 2 ... ( |_ ` A ) ) C_ ( 1 ... ( |_ ` A ) ) )
14	12, 13	mp1i 13	. . . . . . . 8 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> ( 2 ... ( |_ ` A ) ) C_ ( 1 ... ( |_ ` A ) ) )
15		simpr 477	. . . . . . . . . . . 12 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> ( |_ ` A ) e. NN )
16		nnuz 11723	. . . . . . . . . . . 12 |- NN = ( ZZ>= ` 1 )
17	15, 16	syl6eleq 2711	. . . . . . . . . . 11 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> ( |_ ` A ) e. ( ZZ>= ` 1 ) )
18		eluzfz1 12348	. . . . . . . . . . 11 |- ( ( |_ ` A ) e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... ( |_ ` A ) ) )
19	17, 18	syl 17	. . . . . . . . . 10 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> 1 e. ( 1 ... ( |_ ` A ) ) )
20		1lt2 11194	. . . . . . . . . . . 12 |- 1 < 2
21		1re 10039	. . . . . . . . . . . . 13 |- 1 e. RR
22		2re 11090	. . . . . . . . . . . . 13 |- 2 e. RR
23	21, 22	ltnlei 10158	. . . . . . . . . . . 12 |- ( 1 < 2 <-> -. 2 <_ 1 )
24	20, 23	mpbi 220	. . . . . . . . . . 11 |- -. 2 <_ 1
25		elfzle1 12344	. . . . . . . . . . 11 |- ( 1 e. ( 2 ... ( |_ ` A ) ) -> 2 <_ 1 )
26	24, 25	mto 188	. . . . . . . . . 10 |- -. 1 e. ( 2 ... ( |_ ` A ) )
27		nelne1 2890	. . . . . . . . . 10 |- ( ( 1 e. ( 1 ... ( |_ ` A ) ) /\ -. 1 e. ( 2 ... ( |_ ` A ) ) ) -> ( 1 ... ( |_ ` A ) ) =/= ( 2 ... ( |_ ` A ) ) )
28	19, 26, 27	sylancl 694	. . . . . . . . 9 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> ( 1 ... ( |_ ` A ) ) =/= ( 2 ... ( |_ ` A ) ) )
29	28	necomd 2849	. . . . . . . 8 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> ( 2 ... ( |_ ` A ) ) =/= ( 1 ... ( |_ ` A ) ) )
30		df-pss 3590	. . . . . . . 8 |- ( ( 2 ... ( |_ ` A ) ) C. ( 1 ... ( |_ ` A ) ) <-> ( ( 2 ... ( |_ ` A ) ) C_ ( 1 ... ( |_ ` A ) ) /\ ( 2 ... ( |_ ` A ) ) =/= ( 1 ... ( |_ ` A ) ) ) )
31	14, 29, 30	sylanbrc 698	. . . . . . 7 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> ( 2 ... ( |_ ` A ) ) C. ( 1 ... ( |_ ` A ) ) )
32		sspsstr 3712	. . . . . . 7 |- ( ( ( ( 2 ... ( |_ ` A ) ) i^i Prime ) C_ ( 2 ... ( |_ ` A ) ) /\ ( 2 ... ( |_ ` A ) ) C. ( 1 ... ( |_ ` A ) ) ) -> ( ( 2 ... ( |_ ` A ) ) i^i Prime ) C. ( 1 ... ( |_ ` A ) ) )
33	11, 31, 32	sylancr 695	. . . . . 6 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> ( ( 2 ... ( |_ ` A ) ) i^i Prime ) C. ( 1 ... ( |_ ` A ) ) )
34		php3 8146	. . . . . 6 |- ( ( ( 1 ... ( |_ ` A ) ) e. Fin /\ ( ( 2 ... ( |_ ` A ) ) i^i Prime ) C. ( 1 ... ( |_ ` A ) ) ) -> ( ( 2 ... ( |_ ` A ) ) i^i Prime ) ~< ( 1 ... ( |_ ` A ) ) )
35	10, 33, 34	sylancr 695	. . . . 5 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> ( ( 2 ... ( |_ ` A ) ) i^i Prime ) ~< ( 1 ... ( |_ ` A ) ) )
36		fzfi 12771	. . . . . . 7 |- ( 2 ... ( |_ ` A ) ) e. Fin
37		ssfi 8180	. . . . . . 7 |- ( ( ( 2 ... ( |_ ` A ) ) e. Fin /\ ( ( 2 ... ( |_ ` A ) ) i^i Prime ) C_ ( 2 ... ( |_ ` A ) ) ) -> ( ( 2 ... ( |_ ` A ) ) i^i Prime ) e. Fin )
38	36, 11, 37	mp2an 708	. . . . . 6 |- ( ( 2 ... ( |_ ` A ) ) i^i Prime ) e. Fin
39		hashsdom 13170	. . . . . 6 |- ( ( ( ( 2 ... ( |_ ` A ) ) i^i Prime ) e. Fin /\ ( 1 ... ( |_ ` A ) ) e. Fin ) -> ( ( # ` ( ( 2 ... ( |_ ` A ) ) i^i Prime ) ) < ( # ` ( 1 ... ( |_ ` A ) ) ) <-> ( ( 2 ... ( |_ ` A ) ) i^i Prime ) ~< ( 1 ... ( |_ ` A ) ) ) )
40	38, 10, 39	mp2an 708	. . . . 5 |- ( ( # ` ( ( 2 ... ( |_ ` A ) ) i^i Prime ) ) < ( # ` ( 1 ... ( |_ ` A ) ) ) <-> ( ( 2 ... ( |_ ` A ) ) i^i Prime ) ~< ( 1 ... ( |_ ` A ) ) )
41	35, 40	sylibr 224	. . . 4 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> ( # ` ( ( 2 ... ( |_ ` A ) ) i^i Prime ) ) < ( # ` ( 1 ... ( |_ ` A ) ) ) )
42	1	flcld 12599	. . . . . . 7 |- ( A e. RR+ -> ( |_ ` A ) e. ZZ )
43		ppival2 24854	. . . . . . 7 |- ( ( |_ ` A ) e. ZZ -> (π ` ( |_ ` A ) ) = ( # ` ( ( 2 ... ( |_ ` A ) ) i^i Prime ) ) )
44	42, 43	syl 17	. . . . . 6 |- ( A e. RR+ -> (π ` ( |_ ` A ) ) = ( # ` ( ( 2 ... ( |_ ` A ) ) i^i Prime ) ) )
45		ppifl 24886	. . . . . . 7 |- ( A e. RR -> (π ` ( |_ ` A ) ) = (π ` A ) )
46	1, 45	syl 17	. . . . . 6 |- ( A e. RR+ -> (π ` ( |_ ` A ) ) = (π ` A ) )
47	44, 46	eqtr3d 2658	. . . . 5 |- ( A e. RR+ -> ( # ` ( ( 2 ... ( |_ ` A ) ) i^i Prime ) ) = (π ` A ) )
48	47	adantr 481	. . . 4 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> ( # ` ( ( 2 ... ( |_ ` A ) ) i^i Prime ) ) = (π ` A ) )
49		rpge0 11845	. . . . . . 7 |- ( A e. RR+ -> 0 <_ A )
50		flge0nn0 12621	. . . . . . 7 |- ( ( A e. RR /\ 0 <_ A ) -> ( |_ ` A ) e. NN0 )
51	1, 49, 50	syl2anc 693	. . . . . 6 |- ( A e. RR+ -> ( |_ ` A ) e. NN0 )
52		hashfz1 13134	. . . . . 6 |- ( ( |_ ` A ) e. NN0 -> ( # ` ( 1 ... ( |_ ` A ) ) ) = ( |_ ` A ) )
53	51, 52	syl 17	. . . . 5 |- ( A e. RR+ -> ( # ` ( 1 ... ( |_ ` A ) ) ) = ( |_ ` A ) )
54	53	adantr 481	. . . 4 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> ( # ` ( 1 ... ( |_ ` A ) ) ) = ( |_ ` A ) )
55	41, 48, 54	3brtr3d 4684	. . 3 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> (π ` A ) < ( |_ ` A ) )
56		flle 12600	. . . 4 |- ( A e. RR -> ( |_ ` A ) <_ A )
57	9, 56	syl 17	. . 3 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> ( |_ ` A ) <_ A )
58	5, 8, 9, 55, 57	ltletrd 10197	. 2 |- ( ( A e. RR+ /\ ( |_ ` A ) e. NN ) -> (π ` A ) < A )
59	46	adantr 481	. . . 4 |- ( ( A e. RR+ /\ ( |_ ` A ) = 0 ) -> (π ` ( |_ ` A ) ) = (π ` A ) )
60		simpr 477	. . . . . 6 |- ( ( A e. RR+ /\ ( |_ ` A ) = 0 ) -> ( |_ ` A ) = 0 )
61	60	fveq2d 6195	. . . . 5 |- ( ( A e. RR+ /\ ( |_ ` A ) = 0 ) -> (π ` ( |_ ` A ) ) = (π ` 0 ) )
62		2pos 11112	. . . . . 6 |- 0 < 2
63		0re 10040	. . . . . . 7 |- 0 e. RR
64		ppieq0 24902	. . . . . . 7 |- ( 0 e. RR -> ( (π ` 0 ) = 0 <-> 0 < 2 ) )
65	63, 64	ax-mp 5	. . . . . 6 |- ( (π ` 0 ) = 0 <-> 0 < 2 )
66	62, 65	mpbir 221	. . . . 5 |- (π ` 0 ) = 0
67	61, 66	syl6eq 2672	. . . 4 |- ( ( A e. RR+ /\ ( |_ ` A ) = 0 ) -> (π ` ( |_ ` A ) ) = 0 )
68	59, 67	eqtr3d 2658	. . 3 |- ( ( A e. RR+ /\ ( |_ ` A ) = 0 ) -> (π ` A ) = 0 )
69		rpgt0 11844	. . . 4 |- ( A e. RR+ -> 0 < A )
70	69	adantr 481	. . 3 |- ( ( A e. RR+ /\ ( |_ ` A ) = 0 ) -> 0 < A )
71	68, 70	eqbrtrd 4675	. 2 |- ( ( A e. RR+ /\ ( |_ ` A ) = 0 ) -> (π ` A ) < A )
72		elnn0 11294	. . 3 |- ( ( |_ ` A ) e. NN0 <-> ( ( |_ ` A ) e. NN \/ ( |_ ` A ) = 0 ) )
73	51, 72	sylib 208	. 2 |- ( A e. RR+ -> ( ( |_ ` A ) e. NN \/ ( |_ ` A ) = 0 ) )
74	58, 71, 73	mpjaodan 827	1 |- ( A e. RR+ -> (π ` A ) < A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   C_ wss 3574   C. wpss 3575   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   ~< csdm 7954   Fincfn 7955   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   <_ cle 10075   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   ...cfz 12326   |_cfl 12591   #chash 13117   Primecprime 15385  πcppi 24820

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-rep 4771  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-int 4476  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-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  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-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-icc 12182  df-fz 12327  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-prm 15386  df-ppi 24826

This theorem is referenced by:  chtppilimlem1  25162