Theorem rpnnen1lem2 11218

Description: Lemma for rpnnen1 11222. (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 3570	. . 3 |- { n e. ZZ | ( n / k ) < x } C_ ZZ
3	1, 2	eqsstri 3519	. 2 |- T C_ ZZ
4	3	a1i 11	. . 3 |- ( ( x e. RR /\ k e. NN ) -> T C_ ZZ )
5		nnre 10549	. . . . . . . 8 |- ( k e. NN -> k e. RR )
6		remulcl 9580	. . . . . . . . 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 10971	. . . . . . . 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 10874	. . . . . . . . 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 10571	. . . . . . . . . 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 10423	. . . . . . . 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 1229	. . . . . . 7 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x <-> n < ( k x. x ) ) )
20	19	rexbidva 2951	. . . . . 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 3791	. . . . 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 2728	. . . 4 |- ( T =/= (/) <-> { n e. ZZ | ( n / k ) < x } =/= (/) )
25	23, 24	sylibr 212	. . 3 |- ( ( x e. RR /\ k e. NN ) -> T =/= (/) )
26	1	rabeq2i 3092	. . . . . 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 9676	. . . . . . . . 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 2857	. . . 4 |- ( ( x e. RR /\ k e. NN ) -> A. n e. T n <_ ( k x. x ) )
35		breq2 4441	. . . . . 6 |- ( y = ( k x. x ) -> ( n <_ y <-> n <_ ( k x. x ) ) )
36	35	ralbidv 2882	. . . . 5 |- ( y = ( k x. x ) -> ( A. n e. T n <_ y <-> A. n e. T n <_ ( k x. x ) ) )
37	36	rspcev 3196	. . . 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 10948	. . 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 1229	. 2 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. T )
41	3, 40	sseldi 3487	1 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. ZZ )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1383   e. wcel 1804   =/= wne 2638   A.wral 2793   E.wrex 2794   {crab 2797   C_ wss 3461   (/)c0 3770   class class class wbr 4437   |-> cmpt 4495  (class class class)co 6281   supcsup 7902   RRcr 9494   0cc0 9495   x. cmul 9500   < clt 9631   <_ cle 9632   / cdiv 10212   NNcn 10542   ZZcz 10870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-n0 10802  df-z 10871

This theorem is referenced by:  rpnnen1lem3  11219  rpnnen1lem5  11221