Theorem zltaddlt1le 9789

Description: The sum of an integer and a real number between 0 and 1 is less than or equal to a second integer iff the sum is less than the second integer. (Contributed by AV, 1-Jul-2021.)

Assertion

Ref	Expression
zltaddlt1le	|- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( ( M + A ) < N <-> ( M + A ) <_ N ) )

Proof of Theorem zltaddlt1le

Step	Hyp	Ref	Expression
1		zre 9058	. . . . . 6 |- ( M e. ZZ -> M e. RR )
2	1	adantr 274	. . . . 5 |- ( ( M e. ZZ /\ A e. ( 0 (,) 1 ) ) -> M e. RR )
3		elioore 9695	. . . . . 6 |- ( A e. ( 0 (,) 1 ) -> A e. RR )
4	3	adantl 275	. . . . 5 |- ( ( M e. ZZ /\ A e. ( 0 (,) 1 ) ) -> A e. RR )
5	2, 4	readdcld 7795	. . . 4 |- ( ( M e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + A ) e. RR )
6	5	3adant2 1000	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + A ) e. RR )
7		zre 9058	. . . 4 |- ( N e. ZZ -> N e. RR )
8	7	3ad2ant2 1003	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> N e. RR )
9		ltle 7851	. . 3 |- ( ( ( M + A ) e. RR /\ N e. RR ) -> ( ( M + A ) < N -> ( M + A ) <_ N ) )
10	6, 8, 9	syl2anc 408	. 2 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( ( M + A ) < N -> ( M + A ) <_ N ) )
11		elioo3g 9693	. . . . . 6 |- ( A e. ( 0 (,) 1 ) <-> ( ( 0 e. RR* /\ 1 e. RR* /\ A e. RR* ) /\ ( 0 < A /\ A < 1 ) ) )
12		simpl 108	. . . . . 6 |- ( ( 0 < A /\ A < 1 ) -> 0 < A )
13	11, 12	simplbiim 384	. . . . 5 |- ( A e. ( 0 (,) 1 ) -> 0 < A )
14	3, 13	elrpd 9481	. . . 4 |- ( A e. ( 0 (,) 1 ) -> A e. RR+ )
15		addlelt 9555	. . . 4 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( ( M + A ) <_ N -> M < N ) )
16	1, 7, 14, 15	syl3an 1258	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( ( M + A ) <_ N -> M < N ) )
17		zltp1le 9108	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) )
18	17	3adant3 1001	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M < N <-> ( M + 1 ) <_ N ) )
19	3	3ad2ant3 1004	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> A e. RR )
20		1red 7781	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> 1 e. RR )
21	1	3ad2ant1 1002	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> M e. RR )
22		simpr 109	. . . . . . . 8 |- ( ( 0 < A /\ A < 1 ) -> A < 1 )
23	11, 22	simplbiim 384	. . . . . . 7 |- ( A e. ( 0 (,) 1 ) -> A < 1 )
24	23	3ad2ant3 1004	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> A < 1 )
25	19, 20, 21, 24	ltadd2dd 8184	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + A ) < ( M + 1 ) )
26		peano2z 9090	. . . . . . . 8 |- ( M e. ZZ -> ( M + 1 ) e. ZZ )
27	26	zred 9173	. . . . . . 7 |- ( M e. ZZ -> ( M + 1 ) e. RR )
28	27	3ad2ant1 1002	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + 1 ) e. RR )
29		ltletr 7853	. . . . . 6 |- ( ( ( M + A ) e. RR /\ ( M + 1 ) e. RR /\ N e. RR ) -> ( ( ( M + A ) < ( M + 1 ) /\ ( M + 1 ) <_ N ) -> ( M + A ) < N ) )
30	6, 28, 8, 29	syl3anc 1216	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( ( ( M + A ) < ( M + 1 ) /\ ( M + 1 ) <_ N ) -> ( M + A ) < N ) )
31	25, 30	mpand 425	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( ( M + 1 ) <_ N -> ( M + A ) < N ) )
32	18, 31	sylbid 149	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M < N -> ( M + A ) < N ) )
33	16, 32	syld 45	. 2 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( ( M + A ) <_ N -> ( M + A ) < N ) )
34	10, 33	impbid 128	1 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( ( M + A ) < N <-> ( M + A ) <_ N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   e. wcel 1480   class class class wbr 3929  (class class class)co 5774   RRcr 7619   0cc0 7620   1c1 7621   + caddc 7623   RR*cxr 7799   < clt 7800   <_ cle 7801   ZZcz 9054   RR+crp 9441   (,)cioo 9671

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-rp 9442  df-ioo 9675

This theorem is referenced by:  halfleoddlt  11591