Theorem rpnnen1lem2 11814

Description: Lemma for rpnnen1 11820. (Contributed by Mario Carneiro, 12-May-2013.)

Hypotheses

Ref	Expression
rpnnen1lem.1	|- T = { n e. ZZ | ( n / k ) < x }
rpnnen1lem.2	|- F = ( x e. RR |-> ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) )

Assertion

Ref	Expression
rpnnen1lem2	|- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. ZZ )

Distinct variable groups:   k, F, n, x   T, n

Allowed substitution hints:   T( x, k)

Proof of Theorem rpnnen1lem2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpnnen1lem.1	. . 3 |- T = { n e. ZZ | ( n / k ) < x }
2		ssrab2 3687	. . 3 |- { n e. ZZ | ( n / k ) < x } C_ ZZ
3	1, 2	eqsstri 3635	. 2 |- T C_ ZZ
4	3	a1i 11	. . 3 |- ( ( x e. RR /\ k e. NN ) -> T C_ ZZ )
5		nnre 11027	. . . . . . . 8 |- ( k e. NN -> k e. RR )
6		remulcl 10021	. . . . . . . . 9 |- ( ( k e. RR /\ x e. RR ) -> ( k x. x ) e. RR )
7	6	ancoms 469	. . . . . . . 8 |- ( ( x e. RR /\ k e. RR ) -> ( k x. x ) e. RR )
8	5, 7	sylan2 491	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> ( k x. x ) e. RR )
9		btwnz 11479	. . . . . . . 8 |- ( ( k x. x ) e. RR -> ( E. n e. ZZ n < ( k x. x ) /\ E. n e. ZZ ( k x. x ) < n ) )
10	9	simpld 475	. . . . . . 7 |- ( ( k x. x ) e. RR -> E. n e. ZZ n < ( k x. x ) )
11	8, 10	syl 17	. . . . . 6 |- ( ( x e. RR /\ k e. NN ) -> E. n e. ZZ n < ( k x. x ) )
12		zre 11381	. . . . . . . . 9 |- ( n e. ZZ -> n e. RR )
13	12	adantl 482	. . . . . . . 8 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> n e. RR )
14		simpll 790	. . . . . . . 8 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> x e. RR )
15		nngt0 11049	. . . . . . . . . 10 |- ( k e. NN -> 0 < k )
16	5, 15	jca 554	. . . . . . . . 9 |- ( k e. NN -> ( k e. RR /\ 0 < k ) )
17	16	ad2antlr 763	. . . . . . . 8 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( k e. RR /\ 0 < k ) )
18		ltdivmul 10898	. . . . . . . 8 |- ( ( n e. RR /\ x e. RR /\ ( k e. RR /\ 0 < k ) ) -> ( ( n / k ) < x <-> n < ( k x. x ) ) )
19	13, 14, 17, 18	syl3anc 1326	. . . . . . 7 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x <-> n < ( k x. x ) ) )
20	19	rexbidva 3049	. . . . . 6 |- ( ( x e. RR /\ k e. NN ) -> ( E. n e. ZZ ( n / k ) < x <-> E. n e. ZZ n < ( k x. x ) ) )
21	11, 20	mpbird 247	. . . . 5 |- ( ( x e. RR /\ k e. NN ) -> E. n e. ZZ ( n / k ) < x )
22		rabn0 3958	. . . . 5 |- ( { n e. ZZ | ( n / k ) < x } =/= (/) <-> E. n e. ZZ ( n / k ) < x )
23	21, 22	sylibr 224	. . . 4 |- ( ( x e. RR /\ k e. NN ) -> { n e. ZZ | ( n / k ) < x } =/= (/) )
24	1	neeq1i 2858	. . . 4 |- ( T =/= (/) <-> { n e. ZZ | ( n / k ) < x } =/= (/) )
25	23, 24	sylibr 224	. . 3 |- ( ( x e. RR /\ k e. NN ) -> T =/= (/) )
26	1	rabeq2i 3197	. . . . . 6 |- ( n e. T <-> ( n e. ZZ /\ ( n / k ) < x ) )
27	5	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> k e. RR )
28	27, 14, 6	syl2anc 693	. . . . . . . . 9 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( k x. x ) e. RR )
29		ltle 10126	. . . . . . . . 9 |- ( ( n e. RR /\ ( k x. x ) e. RR ) -> ( n < ( k x. x ) -> n <_ ( k x. x ) ) )
30	13, 28, 29	syl2anc 693	. . . . . . . 8 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( n < ( k x. x ) -> n <_ ( k x. x ) ) )
31	19, 30	sylbid 230	. . . . . . 7 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x -> n <_ ( k x. x ) ) )
32	31	impr 649	. . . . . 6 |- ( ( ( x e. RR /\ k e. NN ) /\ ( n e. ZZ /\ ( n / k ) < x ) ) -> n <_ ( k x. x ) )
33	26, 32	sylan2b 492	. . . . 5 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. T ) -> n <_ ( k x. x ) )
34	33	ralrimiva 2966	. . . 4 |- ( ( x e. RR /\ k e. NN ) -> A. n e. T n <_ ( k x. x ) )
35		breq2 4657	. . . . . 6 |- ( y = ( k x. x ) -> ( n <_ y <-> n <_ ( k x. x ) ) )
36	35	ralbidv 2986	. . . . 5 |- ( y = ( k x. x ) -> ( A. n e. T n <_ y <-> A. n e. T n <_ ( k x. x ) ) )
37	36	rspcev 3309	. . . 4 |- ( ( ( k x. x ) e. RR /\ A. n e. T n <_ ( k x. x ) ) -> E. y e. RR A. n e. T n <_ y )
38	8, 34, 37	syl2anc 693	. . 3 |- ( ( x e. RR /\ k e. NN ) -> E. y e. RR A. n e. T n <_ y )
39		suprzcl 11457	. . 3 |- ( ( T C_ ZZ /\ T =/= (/) /\ E. y e. RR A. n e. T n <_ y ) -> sup ( T , RR , < ) e. T )
40	4, 25, 38, 39	syl3anc 1326	. 2 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. T )
41	3, 40	sseldi 3601	1 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. ZZ )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729  (class class class)co 6650   supcsup 8346   RRcr 9935   0cc0 9936   x. cmul 9941   < clt 10074   <_ cle 10075   / cdiv 10684   NNcn 11020   ZZcz 11377

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  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-n0 11293  df-z 11378

This theorem is referenced by:  rpnnen1lem3  11816  rpnnen1lem5  11818  rpnnen1lem3OLD  11822  rpnnen1lem5OLD  11824