Theorem rpnnen1lem2 11094

Description: Lemma for rpnnen1 11098. (Contributed by Mario Carneiro, 12-May-2013.)

Hypotheses

Ref	Expression
rpnnen1.1	|- T = { n e. ZZ | ( n / k ) < x }
rpnnen1.2	|- 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		rpnnen1.1	. . 3 |- T = { n e. ZZ | ( n / k ) < x }
2		ssrab2 3548	. . 3 |- { n e. ZZ | ( n / k ) < x } C_ ZZ
3	1, 2	eqsstri 3497	. 2 |- T C_ ZZ
4	3	a1i 11	. . 3 |- ( ( x e. RR /\ k e. NN ) -> T C_ ZZ )
5		nnre 10443	. . . . . . . 8 |- ( k e. NN -> k e. RR )
6		remulcl 9481	. . . . . . . . 9 |- ( ( k e. RR /\ x e. RR ) -> ( k x. x ) e. RR )
7	6	ancoms 453	. . . . . . . 8 |- ( ( x e. RR /\ k e. RR ) -> ( k x. x ) e. RR )
8	5, 7	sylan2 474	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> ( k x. x ) e. RR )
9		btwnz 10858	. . . . . . . 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 459	. . . . . . 7 |- ( ( k x. x ) e. RR -> E. n e. ZZ n < ( k x. x ) )
11	8, 10	syl 16	. . . . . 6 |- ( ( x e. RR /\ k e. NN ) -> E. n e. ZZ n < ( k x. x ) )
12		zre 10764	. . . . . . . . 9 |- ( n e. ZZ -> n e. RR )
13	12	adantl 466	. . . . . . . 8 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> n e. RR )
14		simpll 753	. . . . . . . 8 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> x e. RR )
15		nngt0 10465	. . . . . . . . . 10 |- ( k e. NN -> 0 < k )
16	5, 15	jca 532	. . . . . . . . 9 |- ( k e. NN -> ( k e. RR /\ 0 < k ) )
17	16	ad2antlr 726	. . . . . . . 8 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( k e. RR /\ 0 < k ) )
18		ltdivmul 10318	. . . . . . . 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 1219	. . . . . . 7 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x <-> n < ( k x. x ) ) )
20	19	rexbidva 2865	. . . . . 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 232	. . . . 5 |- ( ( x e. RR /\ k e. NN ) -> E. n e. ZZ ( n / k ) < x )
22		rabn0 3768	. . . . 5 |- ( { n e. ZZ | ( n / k ) < x } =/= (/) <-> E. n e. ZZ ( n / k ) < x )
23	21, 22	sylibr 212	. . . 4 |- ( ( x e. RR /\ k e. NN ) -> { n e. ZZ | ( n / k ) < x } =/= (/) )
24	1	neeq1i 2737	. . . 4 |- ( T =/= (/) <-> { n e. ZZ | ( n / k ) < x } =/= (/) )
25	23, 24	sylibr 212	. . 3 |- ( ( x e. RR /\ k e. NN ) -> T =/= (/) )
26	1	rabeq2i 3075	. . . . . 6 |- ( n e. T <-> ( n e. ZZ /\ ( n / k ) < x ) )
27	5	ad2antlr 726	. . . . . . . . . 10 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> k e. RR )
28	27, 14, 6	syl2anc 661	. . . . . . . . 9 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( k x. x ) e. RR )
29		ltle 9577	. . . . . . . . 9 |- ( ( n e. RR /\ ( k x. x ) e. RR ) -> ( n < ( k x. x ) -> n <_ ( k x. x ) ) )
30	13, 28, 29	syl2anc 661	. . . . . . . 8 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( n < ( k x. x ) -> n <_ ( k x. x ) ) )
31	19, 30	sylbid 215	. . . . . . 7 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x -> n <_ ( k x. x ) ) )
32	31	impr 619	. . . . . 6 |- ( ( ( x e. RR /\ k e. NN ) /\ ( n e. ZZ /\ ( n / k ) < x ) ) -> n <_ ( k x. x ) )
33	26, 32	sylan2b 475	. . . . 5 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. T ) -> n <_ ( k x. x ) )
34	33	ralrimiva 2830	. . . 4 |- ( ( x e. RR /\ k e. NN ) -> A. n e. T n <_ ( k x. x ) )
35		breq2 4407	. . . . . 6 |- ( y = ( k x. x ) -> ( n <_ y <-> n <_ ( k x. x ) ) )
36	35	ralbidv 2846	. . . . 5 |- ( y = ( k x. x ) -> ( A. n e. T n <_ y <-> A. n e. T n <_ ( k x. x ) ) )
37	36	rspcev 3179	. . . 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 661	. . 3 |- ( ( x e. RR /\ k e. NN ) -> E. y e. RR A. n e. T n <_ y )
39		suprzcl 10835	. . 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 1219	. 2 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. T )
41	3, 40	sseldi 3465	1 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. ZZ )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2648   A.wral 2799   E.wrex 2800   {crab 2803   C_ wss 3439   (/)c0 3748   class class class wbr 4403   |-> cmpt 4461  (class class class)co 6203   supcsup 7804   RRcr 9395   0cc0 9396   x. cmul 9401   < clt 9532   <_ cle 9533   / cdiv 10107   NNcn 10436   ZZcz 10760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-n0 10694  df-z 10761

This theorem is referenced by:  rpnnen1lem3  11095  rpnnen1lem5  11097