Theorem rpnnen1lem1 11297

Description: Lemma for rpnnen1 11302. (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 10622	. . . 4 |- NN e. _V
2	1	mptex 6151	. . 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 5973	. . 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 675	. 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 3546	. . . . . . 7 |- { n e. ZZ | ( n / k ) < x } C_ ZZ
8	6, 7	eqsstri 3494	. . . . . 6 |- T C_ ZZ
9	8	a1i 11	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> T C_ ZZ )
10		nnre 10623	. . . . . . . . . . . 12 |- ( k e. NN -> k e. RR )
11		remulcl 9631	. . . . . . . . . . . . 13 |- ( ( k e. RR /\ x e. RR ) -> ( k x. x ) e. RR )
12	11	ancoms 454	. . . . . . . . . . . 12 |- ( ( x e. RR /\ k e. RR ) -> ( k x. x ) e. RR )
13	10, 12	sylan2 476	. . . . . . . . . . 11 |- ( ( x e. RR /\ k e. NN ) -> ( k x. x ) e. RR )
14		btwnz 11044	. . . . . . . . . . . 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 460	. . . . . . . . . . 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 10948	. . . . . . . . . . . . 13 |- ( n e. ZZ -> n e. RR )
18	17	adantl 467	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> n e. RR )
19		simpll 758	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> x e. RR )
20		nngt0 10645	. . . . . . . . . . . . . 14 |- ( k e. NN -> 0 < k )
21	10, 20	jca 534	. . . . . . . . . . . . 13 |- ( k e. NN -> ( k e. RR /\ 0 < k ) )
22	21	ad2antlr 731	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( k e. RR /\ 0 < k ) )
23		ltdivmul 10487	. . . . . . . . . . . 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 1264	. . . . . . . . . . 11 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x <-> n < ( k x. x ) ) )
25	24	rexbidva 2933	. . . . . . . . . 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 235	. . . . . . . . 9 |- ( ( x e. RR /\ k e. NN ) -> E. n e. ZZ ( n / k ) < x )
27		rabn0 3782	. . . . . . . . 9 |- ( { n e. ZZ | ( n / k ) < x } =/= (/) <-> E. n e. ZZ ( n / k ) < x )
28	26, 27	sylibr 215	. . . . . . . 8 |- ( ( x e. RR /\ k e. NN ) -> { n e. ZZ | ( n / k ) < x } =/= (/) )
29	6	neeq1i 2705	. . . . . . . 8 |- ( T =/= (/) <-> { n e. ZZ | ( n / k ) < x } =/= (/) )
30	28, 29	sylibr 215	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> T =/= (/) )
31	6	rabeq2i 3077	. . . . . . . . . 10 |- ( n e. T <-> ( n e. ZZ /\ ( n / k ) < x ) )
32	10	ad2antlr 731	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> k e. RR )
33	32, 19, 11	syl2anc 665	. . . . . . . . . . . . 13 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( k x. x ) e. RR )
34		ltle 9729	. . . . . . . . . . . . 13 |- ( ( n e. RR /\ ( k x. x ) e. RR ) -> ( n < ( k x. x ) -> n <_ ( k x. x ) ) )
35	18, 33, 34	syl2anc 665	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( n < ( k x. x ) -> n <_ ( k x. x ) ) )
36	24, 35	sylbid 218	. . . . . . . . . . 11 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x -> n <_ ( k x. x ) ) )
37	36	impr 623	. . . . . . . . . 10 |- ( ( ( x e. RR /\ k e. NN ) /\ ( n e. ZZ /\ ( n / k ) < x ) ) -> n <_ ( k x. x ) )
38	31, 37	sylan2b 477	. . . . . . . . 9 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. T ) -> n <_ ( k x. x ) )
39	38	ralrimiva 2836	. . . . . . . 8 |- ( ( x e. RR /\ k e. NN ) -> A. n e. T n <_ ( k x. x ) )
40		breq2 4427	. . . . . . . . . 10 |- ( y = ( k x. x ) -> ( n <_ y <-> n <_ ( k x. x ) ) )
41	40	ralbidv 2861	. . . . . . . . 9 |- ( y = ( k x. x ) -> ( A. n e. T n <_ y <-> A. n e. T n <_ ( k x. x ) ) )
42	41	rspcev 3182	. . . . . . . 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 665	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> E. y e. RR A. n e. T n <_ y )
44		suprzcl 11022	. . . . . . 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 1264	. . . . . 6 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. T )
46	8, 45	sseldi 3462	. . . . 5 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. ZZ )
47		znq 11275	. . . . 5 |- ( ( sup ( T , RR , < ) e. ZZ /\ k e. NN ) -> ( sup ( T , RR , < ) / k ) e. QQ )
48	46, 47	sylancom 671	. . . 4 |- ( ( x e. RR /\ k e. NN ) -> ( sup ( T , RR , < ) / k ) e. QQ )
49		eqid 2422	. . . 4 |- ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) = ( k e. NN |-> ( sup ( T , RR , < ) / k ) )
50	48, 49	fmptd 6061	. . 3 |- ( x e. RR -> ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) : NN --> QQ )
51		qex 11283	. . . 4 |- QQ e. _V
52	51, 1	elmap 7511	. . 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 215	. 2 |- ( x e. RR -> ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) e. ( QQ ^m NN ) )
54	5, 53	eqeltrd 2507	1 |- ( x e. RR -> ( F ` x ) e. ( QQ ^m NN ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1872   =/= wne 2614   A.wral 2771   E.wrex 2772   {crab 2775   _Vcvv 3080   C_ wss 3436   (/)c0 3761   class class class wbr 4423   |-> cmpt 4482   -->wf 5597   ` cfv 5601  (class class class)co 6305   ^m cmap 7483   supcsup 7963   RRcr 9545   0cc0 9546   x. cmul 9551   < clt 9682   <_ cle 9683   / cdiv 10276   NNcn 10616   ZZcz 10944   QQcq 11271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-map 7485  df-en 7581  df-dom 7582  df-sdom 7583  df-sup 7965  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-n0 10877  df-z 10945  df-q 11272

This theorem is referenced by:  rpnnen1lem3  11299  rpnnen1lem4  11300  rpnnen1lem5  11301  rpnnen1  11302