Theorem rpnnen1lem1 11220

Description: Lemma for rpnnen1 11225. (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 10550	. . . 4 |- NN e. _V
2	1	mptex 6142	. . 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 5964	. . 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 3590	. . . . . . 7 |- { n e. ZZ | ( n / k ) < x } C_ ZZ
8	6, 7	eqsstri 3539	. . . . . 6 |- T C_ ZZ
9	8	a1i 11	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> T C_ ZZ )
10		nnre 10555	. . . . . . . . . . . 12 |- ( k e. NN -> k e. RR )
11		remulcl 9589	. . . . . . . . . . . . 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 10975	. . . . . . . . . . . 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 10880	. . . . . . . . . . . . 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 10577	. . . . . . . . . . . . . 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 10429	. . . . . . . . . . . 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 1228	. . . . . . . . . . 11 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x <-> n < ( k x. x ) ) )
25	24	rexbidva 2975	. . . . . . . . . 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 3810	. . . . . . . . 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 2752	. . . . . . . 8 |- ( T =/= (/) <-> { n e. ZZ | ( n / k ) < x } =/= (/) )
30	28, 29	sylibr 212	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> T =/= (/) )
31	6	rabeq2i 3115	. . . . . . . . . 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 9685	. . . . . . . . . . . . 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 2881	. . . . . . . 8 |- ( ( x e. RR /\ k e. NN ) -> A. n e. T n <_ ( k x. x ) )
40		breq2 4457	. . . . . . . . . 10 |- ( y = ( k x. x ) -> ( n <_ y <-> n <_ ( k x. x ) ) )
41	40	ralbidv 2906	. . . . . . . . 9 |- ( y = ( k x. x ) -> ( A. n e. T n <_ y <-> A. n e. T n <_ ( k x. x ) ) )
42	41	rspcev 3219	. . . . . . . 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 10952	. . . . . . 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 1228	. . . . . 6 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. T )
46	8, 45	sseldi 3507	. . . . 5 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. ZZ )
47		znq 11198	. . . . 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 2467	. . . 4 |- ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) = ( k e. NN |-> ( sup ( T , RR , < ) / k ) )
50	48, 49	fmptd 6056	. . 3 |- ( x e. RR -> ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) : NN --> QQ )
51		qexALT 11209	. . . 4 |- QQ e. _V
52	51, 1	elmap 7459	. . 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 2555	1 |- ( x e. RR -> ( F ` x ) e. ( QQ ^m NN ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2817   E.wrex 2818   {crab 2821   _Vcvv 3118   C_ wss 3481   (/)c0 3790   class class class wbr 4453   |-> cmpt 4511   -->wf 5590   ` cfv 5594  (class class class)co 6295   ^m cmap 7432   supcsup 7912   RRcr 9503   0cc0 9504   x. cmul 9509   < clt 9640   <_ cle 9641   / cdiv 10218   NNcn 10548   ZZcz 10876   QQcq 11194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-n0 10808  df-z 10877  df-q 11195

This theorem is referenced by:  rpnnen1lem3  11222  rpnnen1lem4  11223  rpnnen1lem5  11224  rpnnen1  11225