Theorem rpnnen1lem1 10984

Description: Lemma for rpnnen1 10989. (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
rpnnen1lem1	|- ( x e. RR -> ( F ` x ) e. ( QQ ^m NN ) )

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

Allowed substitution hints:   T( x, k)

Proof of Theorem rpnnen1lem1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnexALT 10329	. . . 4 |- NN e. _V
2	1	mptex 5953	. . 3 |- ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) e. _V
3		rpnnen1.2	. . . 4 |- F = ( x e. RR |-> ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) )
4	3	fvmpt2 5786	. . 3 |- ( ( x e. RR /\ ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) e. _V ) -> ( F ` x ) = ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) )
5	2, 4	mpan2 671	. 2 |- ( x e. RR -> ( F ` x ) = ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) )
6		rpnnen1.1	. . . . . . 7 |- T = { n e. ZZ | ( n / k ) < x }
7		ssrab2 3442	. . . . . . 7 |- { n e. ZZ | ( n / k ) < x } C_ ZZ
8	6, 7	eqsstri 3391	. . . . . 6 |- T C_ ZZ
9	8	a1i 11	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> T C_ ZZ )
10		nnre 10334	. . . . . . . . . . . 12 |- ( k e. NN -> k e. RR )
11		remulcl 9372	. . . . . . . . . . . . 13 |- ( ( k e. RR /\ x e. RR ) -> ( k x. x ) e. RR )
12	11	ancoms 453	. . . . . . . . . . . 12 |- ( ( x e. RR /\ k e. RR ) -> ( k x. x ) e. RR )
13	10, 12	sylan2 474	. . . . . . . . . . 11 |- ( ( x e. RR /\ k e. NN ) -> ( k x. x ) e. RR )
14		btwnz 10749	. . . . . . . . . . . 12 |- ( ( k x. x ) e. RR -> ( E. n e. ZZ n < ( k x. x ) /\ E. n e. ZZ ( k x. x ) < n ) )
15	14	simpld 459	. . . . . . . . . . 11 |- ( ( k x. x ) e. RR -> E. n e. ZZ n < ( k x. x ) )
16	13, 15	syl 16	. . . . . . . . . 10 |- ( ( x e. RR /\ k e. NN ) -> E. n e. ZZ n < ( k x. x ) )
17		zre 10655	. . . . . . . . . . . . 13 |- ( n e. ZZ -> n e. RR )
18	17	adantl 466	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> n e. RR )
19		simpll 753	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> x e. RR )
20		nngt0 10356	. . . . . . . . . . . . . 14 |- ( k e. NN -> 0 < k )
21	10, 20	jca 532	. . . . . . . . . . . . 13 |- ( k e. NN -> ( k e. RR /\ 0 < k ) )
22	21	ad2antlr 726	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( k e. RR /\ 0 < k ) )
23		ltdivmul 10209	. . . . . . . . . . . 12 |- ( ( n e. RR /\ x e. RR /\ ( k e. RR /\ 0 < k ) ) -> ( ( n / k ) < x <-> n < ( k x. x ) ) )
24	18, 19, 22, 23	syl3anc 1218	. . . . . . . . . . 11 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x <-> n < ( k x. x ) ) )
25	24	rexbidva 2737	. . . . . . . . . 10 |- ( ( x e. RR /\ k e. NN ) -> ( E. n e. ZZ ( n / k ) < x <-> E. n e. ZZ n < ( k x. x ) ) )
26	16, 25	mpbird 232	. . . . . . . . 9 |- ( ( x e. RR /\ k e. NN ) -> E. n e. ZZ ( n / k ) < x )
27		rabn0 3662	. . . . . . . . 9 |- ( { n e. ZZ | ( n / k ) < x } =/= (/) <-> E. n e. ZZ ( n / k ) < x )
28	26, 27	sylibr 212	. . . . . . . 8 |- ( ( x e. RR /\ k e. NN ) -> { n e. ZZ | ( n / k ) < x } =/= (/) )
29	6	neeq1i 2623	. . . . . . . 8 |- ( T =/= (/) <-> { n e. ZZ | ( n / k ) < x } =/= (/) )
30	28, 29	sylibr 212	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> T =/= (/) )
31	6	rabeq2i 2974	. . . . . . . . . 10 |- ( n e. T <-> ( n e. ZZ /\ ( n / k ) < x ) )
32	10	ad2antlr 726	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> k e. RR )
33	32, 19, 11	syl2anc 661	. . . . . . . . . . . . 13 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( k x. x ) e. RR )
34		ltle 9468	. . . . . . . . . . . . 13 |- ( ( n e. RR /\ ( k x. x ) e. RR ) -> ( n < ( k x. x ) -> n <_ ( k x. x ) ) )
35	18, 33, 34	syl2anc 661	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( n < ( k x. x ) -> n <_ ( k x. x ) ) )
36	24, 35	sylbid 215	. . . . . . . . . . 11 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x -> n <_ ( k x. x ) ) )
37	36	impr 619	. . . . . . . . . 10 |- ( ( ( x e. RR /\ k e. NN ) /\ ( n e. ZZ /\ ( n / k ) < x ) ) -> n <_ ( k x. x ) )
38	31, 37	sylan2b 475	. . . . . . . . 9 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. T ) -> n <_ ( k x. x ) )
39	38	ralrimiva 2804	. . . . . . . 8 |- ( ( x e. RR /\ k e. NN ) -> A. n e. T n <_ ( k x. x ) )
40		breq2 4301	. . . . . . . . . 10 |- ( y = ( k x. x ) -> ( n <_ y <-> n <_ ( k x. x ) ) )
41	40	ralbidv 2740	. . . . . . . . 9 |- ( y = ( k x. x ) -> ( A. n e. T n <_ y <-> A. n e. T n <_ ( k x. x ) ) )
42	41	rspcev 3078	. . . . . . . 8 |- ( ( ( k x. x ) e. RR /\ A. n e. T n <_ ( k x. x ) ) -> E. y e. RR A. n e. T n <_ y )
43	13, 39, 42	syl2anc 661	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> E. y e. RR A. n e. T n <_ y )
44		suprzcl 10726	. . . . . . 7 |- ( ( T C_ ZZ /\ T =/= (/) /\ E. y e. RR A. n e. T n <_ y ) -> sup ( T , RR , < ) e. T )
45	9, 30, 43, 44	syl3anc 1218	. . . . . 6 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. T )
46	8, 45	sseldi 3359	. . . . 5 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. ZZ )
47		znq 10962	. . . . 5 |- ( ( sup ( T , RR , < ) e. ZZ /\ k e. NN ) -> ( sup ( T , RR , < ) / k ) e. QQ )
48	46, 47	sylancom 667	. . . 4 |- ( ( x e. RR /\ k e. NN ) -> ( sup ( T , RR , < ) / k ) e. QQ )
49		eqid 2443	. . . 4 |- ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) = ( k e. NN |-> ( sup ( T , RR , < ) / k ) )
50	48, 49	fmptd 5872	. . 3 |- ( x e. RR -> ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) : NN --> QQ )
51		qexALT 10973	. . . 4 |- QQ e. _V
52	51, 1	elmap 7246	. . 3 |- ( ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) e. ( QQ ^m NN ) <-> ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) : NN --> QQ )
53	50, 52	sylibr 212	. 2 |- ( x e. RR -> ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) e. ( QQ ^m NN ) )
54	5, 53	eqeltrd 2517	1 |- ( x e. RR -> ( F ` x ) e. ( QQ ^m NN ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2611   A.wral 2720   E.wrex 2721   {crab 2724   _Vcvv 2977   C_ wss 3333   (/)c0 3642   class class class wbr 4297   e. cmpt 4355   -->wf 5419   ` cfv 5423  (class class class)co 6096   ^m cmap 7219   supcsup 7695   RRcr 9286   0cc0 9287   x. cmul 9292   < clt 9423   <_ cle 9424   / cdiv 9998   NNcn 10327   ZZcz 10651   QQcq 10958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-n0 10585  df-z 10652  df-q 10959

This theorem is referenced by:  rpnnen1lem3  10986  rpnnen1lem4  10987  rpnnen1lem5  10988  rpnnen1  10989