Theorem rebtwnz 11787

Description: There is a unique greatest integer less than or equal to a real number. Exercise 4 of [Apostol] p. 28. (Contributed by NM, 15-Nov-2004.)

Assertion

Ref	Expression
rebtwnz	|- ( A e. RR -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )

Distinct variable group:   x, A

Proof of Theorem rebtwnz

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		renegcl 10344	. . 3 |- ( A e. RR -> -u A e. RR )
2		zbtwnre 11786	. . 3 |- ( -u A e. RR -> E! y e. ZZ ( -u A <_ y /\ y < ( -u A + 1 ) ) )
3	1, 2	syl 17	. 2 |- ( A e. RR -> E! y e. ZZ ( -u A <_ y /\ y < ( -u A + 1 ) ) )
4		znegcl 11412	. . . 4 |- ( x e. ZZ -> -u x e. ZZ )
5		znegcl 11412	. . . . 5 |- ( y e. ZZ -> -u y e. ZZ )
6		zcn 11382	. . . . . 6 |- ( y e. ZZ -> y e. CC )
7		zcn 11382	. . . . . 6 |- ( x e. ZZ -> x e. CC )
8		negcon2 10334	. . . . . 6 |- ( ( y e. CC /\ x e. CC ) -> ( y = -u x <-> x = -u y ) )
9	6, 7, 8	syl2an 494	. . . . 5 |- ( ( y e. ZZ /\ x e. ZZ ) -> ( y = -u x <-> x = -u y ) )
10	5, 9	reuhyp 4896	. . . 4 |- ( y e. ZZ -> E! x e. ZZ y = -u x )
11		breq2 4657	. . . . 5 |- ( y = -u x -> ( -u A <_ y <-> -u A <_ -u x ) )
12		breq1 4656	. . . . 5 |- ( y = -u x -> ( y < ( -u A + 1 ) <-> -u x < ( -u A + 1 ) ) )
13	11, 12	anbi12d 747	. . . 4 |- ( y = -u x -> ( ( -u A <_ y /\ y < ( -u A + 1 ) ) <-> ( -u A <_ -u x /\ -u x < ( -u A + 1 ) ) ) )
14	4, 10, 13	reuxfr 4894	. . 3 |- ( E! y e. ZZ ( -u A <_ y /\ y < ( -u A + 1 ) ) <-> E! x e. ZZ ( -u A <_ -u x /\ -u x < ( -u A + 1 ) ) )
15		zre 11381	. . . . . 6 |- ( x e. ZZ -> x e. RR )
16		leneg 10531	. . . . . . . 8 |- ( ( x e. RR /\ A e. RR ) -> ( x <_ A <-> -u A <_ -u x ) )
17	16	ancoms 469	. . . . . . 7 |- ( ( A e. RR /\ x e. RR ) -> ( x <_ A <-> -u A <_ -u x ) )
18		peano2rem 10348	. . . . . . . . 9 |- ( A e. RR -> ( A - 1 ) e. RR )
19		ltneg 10528	. . . . . . . . 9 |- ( ( ( A - 1 ) e. RR /\ x e. RR ) -> ( ( A - 1 ) < x <-> -u x < -u ( A - 1 ) ) )
20	18, 19	sylan 488	. . . . . . . 8 |- ( ( A e. RR /\ x e. RR ) -> ( ( A - 1 ) < x <-> -u x < -u ( A - 1 ) ) )
21		1re 10039	. . . . . . . . 9 |- 1 e. RR
22		ltsubadd 10498	. . . . . . . . 9 |- ( ( A e. RR /\ 1 e. RR /\ x e. RR ) -> ( ( A - 1 ) < x <-> A < ( x + 1 ) ) )
23	21, 22	mp3an2 1412	. . . . . . . 8 |- ( ( A e. RR /\ x e. RR ) -> ( ( A - 1 ) < x <-> A < ( x + 1 ) ) )
24		recn 10026	. . . . . . . . . . 11 |- ( A e. RR -> A e. CC )
25		ax-1cn 9994	. . . . . . . . . . 11 |- 1 e. CC
26		negsubdi 10337	. . . . . . . . . . 11 |- ( ( A e. CC /\ 1 e. CC ) -> -u ( A - 1 ) = ( -u A + 1 ) )
27	24, 25, 26	sylancl 694	. . . . . . . . . 10 |- ( A e. RR -> -u ( A - 1 ) = ( -u A + 1 ) )
28	27	adantr 481	. . . . . . . . 9 |- ( ( A e. RR /\ x e. RR ) -> -u ( A - 1 ) = ( -u A + 1 ) )
29	28	breq2d 4665	. . . . . . . 8 |- ( ( A e. RR /\ x e. RR ) -> ( -u x < -u ( A - 1 ) <-> -u x < ( -u A + 1 ) ) )
30	20, 23, 29	3bitr3d 298	. . . . . . 7 |- ( ( A e. RR /\ x e. RR ) -> ( A < ( x + 1 ) <-> -u x < ( -u A + 1 ) ) )
31	17, 30	anbi12d 747	. . . . . 6 |- ( ( A e. RR /\ x e. RR ) -> ( ( x <_ A /\ A < ( x + 1 ) ) <-> ( -u A <_ -u x /\ -u x < ( -u A + 1 ) ) ) )
32	15, 31	sylan2 491	. . . . 5 |- ( ( A e. RR /\ x e. ZZ ) -> ( ( x <_ A /\ A < ( x + 1 ) ) <-> ( -u A <_ -u x /\ -u x < ( -u A + 1 ) ) ) )
33	32	bicomd 213	. . . 4 |- ( ( A e. RR /\ x e. ZZ ) -> ( ( -u A <_ -u x /\ -u x < ( -u A + 1 ) ) <-> ( x <_ A /\ A < ( x + 1 ) ) ) )
34	33	reubidva 3125	. . 3 |- ( A e. RR -> ( E! x e. ZZ ( -u A <_ -u x /\ -u x < ( -u A + 1 ) ) <-> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )
35	14, 34	syl5bb 272	. 2 |- ( A e. RR -> ( E! y e. ZZ ( -u A <_ y /\ y < ( -u A + 1 ) ) <-> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )
36	3, 35	mpbid 222	1 |- ( A e. RR -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E!wreu 2914   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   ZZcz 11377

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  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-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688

This theorem is referenced by:  flcl  12596  fllelt  12598  flflp1  12608  flbi  12617  ltflcei  33397