Theorem rpnnen1lem2 11281

Description: Lemma for rpnnen1 11285. (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 3482	. . 3 |- { n e. ZZ | ( n / k ) < x } C_ ZZ
3	1, 2	eqsstri 3430	. 2 |- T C_ ZZ
4	3	a1i 11	. . 3 |- ( ( x e. RR /\ k e. NN ) -> T C_ ZZ )
5		nnre 10605	. . . . . . . 8 |- ( k e. NN -> k e. RR )
6		remulcl 9611	. . . . . . . . 9 |- ( ( k e. RR /\ x e. RR ) -> ( k x. x ) e. RR )
7	6	ancoms 459	. . . . . . . 8 |- ( ( x e. RR /\ k e. RR ) -> ( k x. x ) e. RR )
8	5, 7	sylan2 481	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> ( k x. x ) e. RR )
9		btwnz 11027	. . . . . . . 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 465	. . . . . . 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 10931	. . . . . . . . 9 |- ( n e. ZZ -> n e. RR )
13	12	adantl 472	. . . . . . . 8 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> n e. RR )
14		simpll 765	. . . . . . . 8 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> x e. RR )
15		nngt0 10627	. . . . . . . . . 10 |- ( k e. NN -> 0 < k )
16	5, 15	jca 539	. . . . . . . . 9 |- ( k e. NN -> ( k e. RR /\ 0 < k ) )
17	16	ad2antlr 738	. . . . . . . 8 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( k e. RR /\ 0 < k ) )
18		ltdivmul 10469	. . . . . . . 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 1271	. . . . . . 7 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x <-> n < ( k x. x ) ) )
20	19	rexbidva 2870	. . . . . 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 240	. . . . 5 |- ( ( x e. RR /\ k e. NN ) -> E. n e. ZZ ( n / k ) < x )
22		rabn0 3720	. . . . 5 |- ( { n e. ZZ | ( n / k ) < x } =/= (/) <-> E. n e. ZZ ( n / k ) < x )
23	21, 22	sylibr 217	. . . 4 |- ( ( x e. RR /\ k e. NN ) -> { n e. ZZ | ( n / k ) < x } =/= (/) )
24	1	neeq1i 2688	. . . 4 |- ( T =/= (/) <-> { n e. ZZ | ( n / k ) < x } =/= (/) )
25	23, 24	sylibr 217	. . 3 |- ( ( x e. RR /\ k e. NN ) -> T =/= (/) )
26	1	rabeq2i 3010	. . . . . 6 |- ( n e. T <-> ( n e. ZZ /\ ( n / k ) < x ) )
27	5	ad2antlr 738	. . . . . . . . . 10 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> k e. RR )
28	27, 14, 6	syl2anc 671	. . . . . . . . 9 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( k x. x ) e. RR )
29		ltle 9709	. . . . . . . . 9 |- ( ( n e. RR /\ ( k x. x ) e. RR ) -> ( n < ( k x. x ) -> n <_ ( k x. x ) ) )
30	13, 28, 29	syl2anc 671	. . . . . . . 8 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( n < ( k x. x ) -> n <_ ( k x. x ) ) )
31	19, 30	sylbid 223	. . . . . . 7 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x -> n <_ ( k x. x ) ) )
32	31	impr 629	. . . . . 6 |- ( ( ( x e. RR /\ k e. NN ) /\ ( n e. ZZ /\ ( n / k ) < x ) ) -> n <_ ( k x. x ) )
33	26, 32	sylan2b 482	. . . . 5 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. T ) -> n <_ ( k x. x ) )
34	33	ralrimiva 2790	. . . 4 |- ( ( x e. RR /\ k e. NN ) -> A. n e. T n <_ ( k x. x ) )
35		breq2 4378	. . . . . 6 |- ( y = ( k x. x ) -> ( n <_ y <-> n <_ ( k x. x ) ) )
36	35	ralbidv 2810	. . . . 5 |- ( y = ( k x. x ) -> ( A. n e. T n <_ y <-> A. n e. T n <_ ( k x. x ) ) )
37	36	rspcev 3118	. . . 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 671	. . 3 |- ( ( x e. RR /\ k e. NN ) -> E. y e. RR A. n e. T n <_ y )
39		suprzcl 11005	. . 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 1271	. 2 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. T )
41	3, 40	sseldi 3398	1 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. ZZ )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1448   e. wcel 1891   =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741   C_ wss 3372   (/)c0 3699   class class class wbr 4374   |-> cmpt 4433  (class class class)co 6276   supcsup 7941   RRcr 9525   0cc0 9526   x. cmul 9531   < clt 9662   <_ cle 9663   / cdiv 10258   NNcn 10598   ZZcz 10927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571  ax-resscn 9583  ax-1cn 9584  ax-icn 9585  ax-addcl 9586  ax-addrcl 9587  ax-mulcl 9588  ax-mulrcl 9589  ax-mulcom 9590  ax-addass 9591  ax-mulass 9592  ax-distr 9593  ax-i2m1 9594  ax-1ne0 9595  ax-1rid 9596  ax-rnegex 9597  ax-rrecex 9598  ax-cnre 9599  ax-pre-lttri 9600  ax-pre-lttrn 9601  ax-pre-ltadd 9602  ax-pre-mulgt0 9603  ax-pre-sup 9604

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4169  df-iun 4250  df-br 4375  df-opab 4434  df-mpt 4435  df-tr 4470  df-eprel 4723  df-id 4727  df-po 4733  df-so 4734  df-fr 4771  df-we 4773  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-pred 5359  df-ord 5405  df-on 5406  df-lim 5407  df-suc 5408  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-riota 6238  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-om 6681  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-sup 7943  df-pnf 9664  df-mnf 9665  df-xr 9666  df-ltxr 9667  df-le 9668  df-sub 9849  df-neg 9850  df-div 10259  df-nn 10599  df-n0 10860  df-z 10928

This theorem is referenced by:  rpnnen1lem3  11282  rpnnen1lem5  11284