Theorem apbtwnz 10533

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 109	. . 3 |- ( ( A e. RR /\ A. n e. ZZ A # n ) -> A e. RR )
2		simpr 110	. . . . 5 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ A < m ) -> A < m )
3	2	olcd 741	. . . 4 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ A < m ) -> ( m <_ A \/ A < m ) )
4		simpr 110	. . . . . . . 8 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> m e. ZZ )
5	4	zred 9601	. . . . . . 7 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> m e. RR )
6	5	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ m < A ) -> m e. RR )
7	1	adantr 276	. . . . . . 7 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> A e. RR )
8	7	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ m < A ) -> A e. RR )
9		simpr 110	. . . . . 6 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ m < A ) -> m < A )
10	6, 8, 9	ltled 8297	. . . . 5 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ m < A ) -> m <_ A )
11	10	orcd 740	. . . 4 |- ( ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) /\ m < A ) -> ( m <_ A \/ A < m ) )
12		breq2 4092	. . . . . 6 |- ( n = m -> ( A # n <-> A # m ) )
13		simplr 529	. . . . . 6 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> A. n e. ZZ A # n )
14	12, 13, 4	rspcdva 2915	. . . . 5 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> A # m )
15		reaplt 8767	. . . . . 6 |- ( ( A e. RR /\ m e. RR ) -> ( A # m <-> ( A < m \/ m < A ) ) )
16	7, 5, 15	syl2anc 411	. . . . 5 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> ( A # m <-> ( A < m \/ m < A ) ) )
17	14, 16	mpbid 147	. . . 4 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> ( A < m \/ m < A ) )
18	3, 11, 17	mpjaodan 805	. . 3 |- ( ( ( A e. RR /\ A. n e. ZZ A # n ) /\ m e. ZZ ) -> ( m <_ A \/ A < m ) )
19	1, 18	exbtwnzlemex 10508	. 2 |- ( ( A e. RR /\ A. n e. ZZ A # n ) -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
20	19, 1	exbtwnz 10509	1 |- ( ( A e. RR /\ A. n e. ZZ A # n ) -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   e. wcel 2202   A.wral 2510   E!wreu 2512   class class class wbr 4088  (class class class)co 6017   RRcr 8030   1c1 8032   + caddc 8034   < clt 8213   <_ cle 8214   # cap 8760   ZZcz 9478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-arch 8150

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-inn 9143  df-n0 9402  df-z 9479

This theorem is referenced by:  flapcl  10534