Theorem rpnnen1lem2 11210

Description: Lemma for rpnnen1 11214. (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 3571	. . 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 10538	. . . . . . . 8 |- ( k e. NN -> k e. RR )
6		remulcl 9566	. . . . . . . . 9 |- ( ( k e. RR /\ x e. RR ) -> ( k x. x ) e. RR )
7	6	ancoms 451	. . . . . . . 8 |- ( ( x e. RR /\ k e. RR ) -> ( k x. x ) e. RR )
8	5, 7	sylan2 472	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> ( k x. x ) e. RR )
9		btwnz 10962	. . . . . . . 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 457	. . . . . . 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 10864	. . . . . . . . 9 |- ( n e. ZZ -> n e. RR )
13	12	adantl 464	. . . . . . . 8 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> n e. RR )
14		simpll 751	. . . . . . . 8 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> x e. RR )
15		nngt0 10560	. . . . . . . . . 10 |- ( k e. NN -> 0 < k )
16	5, 15	jca 530	. . . . . . . . 9 |- ( k e. NN -> ( k e. RR /\ 0 < k ) )
17	16	ad2antlr 724	. . . . . . . 8 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( k e. RR /\ 0 < k ) )
18		ltdivmul 10413	. . . . . . . 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 1226	. . . . . . 7 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x <-> n < ( k x. x ) ) )
20	19	rexbidva 2962	. . . . . 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 3804	. . . . 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 2739	. . . 4 |- ( T =/= (/) <-> { n e. ZZ | ( n / k ) < x } =/= (/) )
25	23, 24	sylibr 212	. . 3 |- ( ( x e. RR /\ k e. NN ) -> T =/= (/) )
26	1	rabeq2i 3103	. . . . . 6 |- ( n e. T <-> ( n e. ZZ /\ ( n / k ) < x ) )
27	5	ad2antlr 724	. . . . . . . . . 10 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> k e. RR )
28	27, 14, 6	syl2anc 659	. . . . . . . . 9 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( k x. x ) e. RR )
29		ltle 9662	. . . . . . . . 9 |- ( ( n e. RR /\ ( k x. x ) e. RR ) -> ( n < ( k x. x ) -> n <_ ( k x. x ) ) )
30	13, 28, 29	syl2anc 659	. . . . . . . 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 617	. . . . . 6 |- ( ( ( x e. RR /\ k e. NN ) /\ ( n e. ZZ /\ ( n / k ) < x ) ) -> n <_ ( k x. x ) )
33	26, 32	sylan2b 473	. . . . 5 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. T ) -> n <_ ( k x. x ) )
34	33	ralrimiva 2868	. . . 4 |- ( ( x e. RR /\ k e. NN ) -> A. n e. T n <_ ( k x. x ) )
35		breq2 4443	. . . . . 6 |- ( y = ( k x. x ) -> ( n <_ y <-> n <_ ( k x. x ) ) )
36	35	ralbidv 2893	. . . . 5 |- ( y = ( k x. x ) -> ( A. n e. T n <_ y <-> A. n e. T n <_ ( k x. x ) ) )
37	36	rspcev 3207	. . . 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 659	. . 3 |- ( ( x e. RR /\ k e. NN ) -> E. y e. RR A. n e. T n <_ y )
39		suprzcl 10938	. . 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 1226	. 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 367   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808   C_ wss 3461   (/)c0 3783   class class class wbr 4439   |-> cmpt 4497  (class class class)co 6270   supcsup 7892   RRcr 9480   0cc0 9481   x. cmul 9486   < clt 9617   <_ cle 9618   / cdiv 10202   NNcn 10531   ZZcz 10860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-n0 10792  df-z 10861

This theorem is referenced by:  rpnnen1lem3  11211  rpnnen1lem5  11213