Theorem rpnnen1lem1 11324

Description: Lemma for rpnnen1 11329. (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		nnex 10648	. . . 4 |- NN e. _V
2	1	mptex 6166	. . 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 5985	. . 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 682	. 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 3526	. . . . . . 7 |- { n e. ZZ | ( n / k ) < x } C_ ZZ
8	6, 7	eqsstri 3474	. . . . . 6 |- T C_ ZZ
9	8	a1i 11	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> T C_ ZZ )
10		nnre 10649	. . . . . . . . . . . 12 |- ( k e. NN -> k e. RR )
11		remulcl 9655	. . . . . . . . . . . . 13 |- ( ( k e. RR /\ x e. RR ) -> ( k x. x ) e. RR )
12	11	ancoms 459	. . . . . . . . . . . 12 |- ( ( x e. RR /\ k e. RR ) -> ( k x. x ) e. RR )
13	10, 12	sylan2 481	. . . . . . . . . . 11 |- ( ( x e. RR /\ k e. NN ) -> ( k x. x ) e. RR )
14		btwnz 11071	. . . . . . . . . . . 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 465	. . . . . . . . . . 11 |- ( ( k x. x ) e. RR -> E. n e. ZZ n < ( k x. x ) )
16	13, 15	syl 17	. . . . . . . . . 10 |- ( ( x e. RR /\ k e. NN ) -> E. n e. ZZ n < ( k x. x ) )
17		zre 10975	. . . . . . . . . . . . 13 |- ( n e. ZZ -> n e. RR )
18	17	adantl 472	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> n e. RR )
19		simpll 765	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> x e. RR )
20		nngt0 10671	. . . . . . . . . . . . . 14 |- ( k e. NN -> 0 < k )
21	10, 20	jca 539	. . . . . . . . . . . . 13 |- ( k e. NN -> ( k e. RR /\ 0 < k ) )
22	21	ad2antlr 738	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( k e. RR /\ 0 < k ) )
23		ltdivmul 10513	. . . . . . . . . . . 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 1276	. . . . . . . . . . 11 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x <-> n < ( k x. x ) ) )
25	24	rexbidva 2910	. . . . . . . . . 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 240	. . . . . . . . 9 |- ( ( x e. RR /\ k e. NN ) -> E. n e. ZZ ( n / k ) < x )
27		rabn0 3764	. . . . . . . . 9 |- ( { n e. ZZ | ( n / k ) < x } =/= (/) <-> E. n e. ZZ ( n / k ) < x )
28	26, 27	sylibr 217	. . . . . . . 8 |- ( ( x e. RR /\ k e. NN ) -> { n e. ZZ | ( n / k ) < x } =/= (/) )
29	6	neeq1i 2700	. . . . . . . 8 |- ( T =/= (/) <-> { n e. ZZ | ( n / k ) < x } =/= (/) )
30	28, 29	sylibr 217	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> T =/= (/) )
31	6	rabeq2i 3054	. . . . . . . . . 10 |- ( n e. T <-> ( n e. ZZ /\ ( n / k ) < x ) )
32	10	ad2antlr 738	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> k e. RR )
33	32, 19, 11	syl2anc 671	. . . . . . . . . . . . 13 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( k x. x ) e. RR )
34		ltle 9753	. . . . . . . . . . . . 13 |- ( ( n e. RR /\ ( k x. x ) e. RR ) -> ( n < ( k x. x ) -> n <_ ( k x. x ) ) )
35	18, 33, 34	syl2anc 671	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( n < ( k x. x ) -> n <_ ( k x. x ) ) )
36	24, 35	sylbid 223	. . . . . . . . . . 11 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x -> n <_ ( k x. x ) ) )
37	36	impr 629	. . . . . . . . . 10 |- ( ( ( x e. RR /\ k e. NN ) /\ ( n e. ZZ /\ ( n / k ) < x ) ) -> n <_ ( k x. x ) )
38	31, 37	sylan2b 482	. . . . . . . . 9 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. T ) -> n <_ ( k x. x ) )
39	38	ralrimiva 2814	. . . . . . . 8 |- ( ( x e. RR /\ k e. NN ) -> A. n e. T n <_ ( k x. x ) )
40		breq2 4422	. . . . . . . . . 10 |- ( y = ( k x. x ) -> ( n <_ y <-> n <_ ( k x. x ) ) )
41	40	ralbidv 2839	. . . . . . . . 9 |- ( y = ( k x. x ) -> ( A. n e. T n <_ y <-> A. n e. T n <_ ( k x. x ) ) )
42	41	rspcev 3162	. . . . . . . 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 671	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> E. y e. RR A. n e. T n <_ y )
44		suprzcl 11049	. . . . . . 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 1276	. . . . . 6 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. T )
46	8, 45	sseldi 3442	. . . . 5 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. ZZ )
47		znq 11302	. . . . 5 |- ( ( sup ( T , RR , < ) e. ZZ /\ k e. NN ) -> ( sup ( T , RR , < ) / k ) e. QQ )
48	46, 47	sylancom 678	. . . 4 |- ( ( x e. RR /\ k e. NN ) -> ( sup ( T , RR , < ) / k ) e. QQ )
49		eqid 2462	. . . 4 |- ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) = ( k e. NN |-> ( sup ( T , RR , < ) / k ) )
50	48, 49	fmptd 6074	. . 3 |- ( x e. RR -> ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) : NN --> QQ )
51		qex 11310	. . . 4 |- QQ e. _V
52	51, 1	elmap 7531	. . 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 217	. 2 |- ( x e. RR -> ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) e. ( QQ ^m NN ) )
54	5, 53	eqeltrd 2540	1 |- ( x e. RR -> ( F ` x ) e. ( QQ ^m NN ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   =/= wne 2633   A.wral 2749   E.wrex 2750   {crab 2753   _Vcvv 3057   C_ wss 3416   (/)c0 3743   class class class wbr 4418   |-> cmpt 4477   -->wf 5601   ` cfv 5605  (class class class)co 6320   ^m cmap 7503   supcsup 7985   RRcr 9569   0cc0 9570   x. cmul 9575   < clt 9706   <_ cle 9707   / cdiv 10302   NNcn 10642   ZZcz 10971   QQcq 11298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647  ax-pre-sup 9648

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-er 7394  df-map 7505  df-en 7601  df-dom 7602  df-sdom 7603  df-sup 7987  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-nn 10643  df-n0 10904  df-z 10972  df-q 11299

This theorem is referenced by:  rpnnen1lem3  11326  rpnnen1lem4  11327  rpnnen1lem5  11328  rpnnen1  11329