Theorem rpnnen1lem1 11815

Description: Lemma for rpnnen1 11820. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
rpnnen1lem.1	|- T = { n e. ZZ | ( n / k ) < x }
rpnnen1lem.2	|- F = ( x e. RR |-> ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) )
rpnnen1lem.n	|- NN e. _V
rpnnen1lem.q	|- QQ e. _V

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		rpnnen1lem.n	. . . 4 |- NN e. _V
2	1	mptex 6486	. . 3 |- ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) e. _V
3		rpnnen1lem.2	. . . 4 |- F = ( x e. RR |-> ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) )
4	3	fvmpt2 6291	. . 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 707	. 2 |- ( x e. RR -> ( F ` x ) = ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) )
6		rpnnen1lem.1	. . . . . . 7 |- T = { n e. ZZ | ( n / k ) < x }
7		ssrab2 3687	. . . . . . 7 |- { n e. ZZ | ( n / k ) < x } C_ ZZ
8	6, 7	eqsstri 3635	. . . . . 6 |- T C_ ZZ
9	8	a1i 11	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> T C_ ZZ )
10		nnre 11027	. . . . . . . . . . . 12 |- ( k e. NN -> k e. RR )
11		remulcl 10021	. . . . . . . . . . . . 13 |- ( ( k e. RR /\ x e. RR ) -> ( k x. x ) e. RR )
12	11	ancoms 469	. . . . . . . . . . . 12 |- ( ( x e. RR /\ k e. RR ) -> ( k x. x ) e. RR )
13	10, 12	sylan2 491	. . . . . . . . . . 11 |- ( ( x e. RR /\ k e. NN ) -> ( k x. x ) e. RR )
14		btwnz 11479	. . . . . . . . . . . 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 475	. . . . . . . . . . 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 11381	. . . . . . . . . . . . 13 |- ( n e. ZZ -> n e. RR )
18	17	adantl 482	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> n e. RR )
19		simpll 790	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> x e. RR )
20		nngt0 11049	. . . . . . . . . . . . . 14 |- ( k e. NN -> 0 < k )
21	10, 20	jca 554	. . . . . . . . . . . . 13 |- ( k e. NN -> ( k e. RR /\ 0 < k ) )
22	21	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( k e. RR /\ 0 < k ) )
23		ltdivmul 10898	. . . . . . . . . . . 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 1326	. . . . . . . . . . 11 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x <-> n < ( k x. x ) ) )
25	24	rexbidva 3049	. . . . . . . . . 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 247	. . . . . . . . 9 |- ( ( x e. RR /\ k e. NN ) -> E. n e. ZZ ( n / k ) < x )
27		rabn0 3958	. . . . . . . . 9 |- ( { n e. ZZ | ( n / k ) < x } =/= (/) <-> E. n e. ZZ ( n / k ) < x )
28	26, 27	sylibr 224	. . . . . . . 8 |- ( ( x e. RR /\ k e. NN ) -> { n e. ZZ | ( n / k ) < x } =/= (/) )
29	6	neeq1i 2858	. . . . . . . 8 |- ( T =/= (/) <-> { n e. ZZ | ( n / k ) < x } =/= (/) )
30	28, 29	sylibr 224	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> T =/= (/) )
31	6	rabeq2i 3197	. . . . . . . . . 10 |- ( n e. T <-> ( n e. ZZ /\ ( n / k ) < x ) )
32	10	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> k e. RR )
33	32, 19, 11	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( k x. x ) e. RR )
34		ltle 10126	. . . . . . . . . . . . 13 |- ( ( n e. RR /\ ( k x. x ) e. RR ) -> ( n < ( k x. x ) -> n <_ ( k x. x ) ) )
35	18, 33, 34	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( n < ( k x. x ) -> n <_ ( k x. x ) ) )
36	24, 35	sylbid 230	. . . . . . . . . . 11 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. ZZ ) -> ( ( n / k ) < x -> n <_ ( k x. x ) ) )
37	36	impr 649	. . . . . . . . . 10 |- ( ( ( x e. RR /\ k e. NN ) /\ ( n e. ZZ /\ ( n / k ) < x ) ) -> n <_ ( k x. x ) )
38	31, 37	sylan2b 492	. . . . . . . . 9 |- ( ( ( x e. RR /\ k e. NN ) /\ n e. T ) -> n <_ ( k x. x ) )
39	38	ralrimiva 2966	. . . . . . . 8 |- ( ( x e. RR /\ k e. NN ) -> A. n e. T n <_ ( k x. x ) )
40		breq2 4657	. . . . . . . . . 10 |- ( y = ( k x. x ) -> ( n <_ y <-> n <_ ( k x. x ) ) )
41	40	ralbidv 2986	. . . . . . . . 9 |- ( y = ( k x. x ) -> ( A. n e. T n <_ y <-> A. n e. T n <_ ( k x. x ) ) )
42	41	rspcev 3309	. . . . . . . 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 693	. . . . . . 7 |- ( ( x e. RR /\ k e. NN ) -> E. y e. RR A. n e. T n <_ y )
44		suprzcl 11457	. . . . . . 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 1326	. . . . . 6 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. T )
46	8, 45	sseldi 3601	. . . . 5 |- ( ( x e. RR /\ k e. NN ) -> sup ( T , RR , < ) e. ZZ )
47		znq 11792	. . . . 5 |- ( ( sup ( T , RR , < ) e. ZZ /\ k e. NN ) -> ( sup ( T , RR , < ) / k ) e. QQ )
48	46, 47	sylancom 701	. . . 4 |- ( ( x e. RR /\ k e. NN ) -> ( sup ( T , RR , < ) / k ) e. QQ )
49		eqid 2622	. . . 4 |- ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) = ( k e. NN |-> ( sup ( T , RR , < ) / k ) )
50	48, 49	fmptd 6385	. . 3 |- ( x e. RR -> ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) : NN --> QQ )
51		rpnnen1lem.q	. . . 4 |- QQ e. _V
52	51, 1	elmap 7886	. . 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 224	. 2 |- ( x e. RR -> ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) e. ( QQ ^m NN ) )
54	5, 53	eqeltrd 2701	1 |- ( x e. RR -> ( F ` x ) e. ( QQ ^m NN ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   supcsup 8346   RRcr 9935   0cc0 9936   x. cmul 9941   < clt 10074   <_ cle 10075   / cdiv 10684   NNcn 11020   ZZcz 11377   QQcq 11788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-q 11789

This theorem is referenced by:  rpnnen1lem3  11816  rpnnen1lem4  11817  rpnnen1lem5  11818  rpnnen1lem6  11819