Theorem apbtwnz 10209

Description: There is a unique greatest integer less than or equal to a real number which is apart from all integers. (Contributed by Jim Kingdon, 11-May-2022.)

Assertion

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

Distinct variable group:   A, n, x

Proof of Theorem apbtwnz

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 108	. . 3 |- ( ( A e. RR /\ A. n e. ZZ A # n ) -> A e. RR )
2		simpr 109	. . . . 5 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ A < m ) -> A < m )
3	2	olcd 724	. . . 4 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ A < m ) -> ( m <_ A \/ A < m ) )
4		simpr 109	. . . . . . . 8 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> m e. ZZ )
5	4	zred 9313	. . . . . . 7 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> m e. RR )
6	5	adantr 274	. . . . . 6 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ m < A ) -> m e. RR )
7	1	adantr 274	. . . . . . 7 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> A e. RR )
8	7	adantr 274	. . . . . 6 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ m < A ) -> A e. RR )
9		simpr 109	. . . . . 6 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ m < A ) -> m < A )
10	6, 8, 9	ltled 8017	. . . . 5 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ m < A ) -> m <_ A )
11	10	orcd 723	. . . 4 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ m < A ) -> ( m <_ A \/ A < m ) )
12		breq2 3986	. . . . . 6 |- ( n = m -> ( A # n <-> A # m ) )
13		simplr 520	. . . . . 6 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> A. n e. ZZ A # n )
14	12, 13, 4	rspcdva 2835	. . . . 5 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> A # m )
15		reaplt 8486	. . . . . 6 |- ( ( A e. RR /\ m e. RR ) -> ( A # m <-> ( A < m \/ m < A ) ) )
16	7, 5, 15	syl2anc 409	. . . . 5 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> ( A # m <-> ( A < m \/ m < A ) ) )
17	14, 16	mpbid 146	. . . 4 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> ( A < m \/ m < A ) )
18	3, 11, 17	mpjaodan 788	. . 3 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> ( m <_ A \/ A < m ) )
19	1, 18	exbtwnzlemex 10185	. 2 |- ( ( A e. RR /\ A. n e. ZZ A # n ) -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
20	19, 1	exbtwnz 10186	1 |- ( ( A e. RR /\ A. n e. ZZ A # n ) -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   e. wcel 2136   A.wral 2444   E!wreu 2446   class class class wbr 3982  (class class class)co 5842   RRcr 7752   1c1 7754   + caddc 7756   < clt 7933   <_ cle 7934   # cap 8479   ZZcz 9191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-arch 7872

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-inn 8858  df-n0 9115  df-z 9192

This theorem is referenced by:  flapcl  10210