Theorem zltaddlt1le 10410

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 9648	. . . . . 6 |- ( M e. ZZ -> M e. RR )
2	1	adantr 276	. . . . 5 |- ( ( M e. ZZ /\ A e. ( 0 (,) 1 ) ) -> M e. RR )
3		elioore 10314	. . . . . 6 |- ( A e. ( 0 (,) 1 ) -> A e. RR )
4	3	adantl 277	. . . . 5 |- ( ( M e. ZZ /\ A e. ( 0 (,) 1 ) ) -> A e. RR )
5	2, 4	readdcld 8355	. . . 4 |- ( ( M e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + A ) e. RR )
6	5	3adant2 1047	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + A ) e. RR )
7		zre 9648	. . . 4 |- ( N e. ZZ -> N e. RR )
8	7	3ad2ant2 1050	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> N e. RR )
9		ltle 8413	. . 3 |- ( ( ( M + A ) e. RR /\ N e. RR ) -> ( ( M + A ) < N -> ( M + A ) <_ N ) )
10	6, 8, 9	syl2anc 415	. 2 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( ( M + A ) < N -> ( M + A ) <_ N ) )
11		elioo3g 10312	. . . . . 6 |- ( A e. ( 0 (,) 1 ) <-> ( ( 0 e. RR* /\ 1 e. RR* /\ A e. RR* ) /\ ( 0 < A /\ A < 1 ) ) )
12		simpl 109	. . . . . 6 |- ( ( 0 < A /\ A < 1 ) -> 0 < A )
13	11, 12	simplbiim 391	. . . . 5 |- ( A e. ( 0 (,) 1 ) -> 0 < A )
14	3, 13	elrpd 10094	. . . 4 |- ( A e. ( 0 (,) 1 ) -> A e. RR+ )
15		addlelt 10169	. . . 4 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( ( M + A ) <_ N -> M < N ) )
16	1, 7, 14, 15	syl3an 1320	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( ( M + A ) <_ N -> M < N ) )
17		zltp1le 9699	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) )
18	17	3adant3 1048	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M < N <-> ( M + 1 ) <_ N ) )
19	3	3ad2ant3 1051	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> A e. RR )
20		1red 8341	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> 1 e. RR )
21	1	3ad2ant1 1049	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> M e. RR )
22		simpr 110	. . . . . . . 8 |- ( ( 0 < A /\ A < 1 ) -> A < 1 )
23	11, 22	simplbiim 391	. . . . . . 7 |- ( A e. ( 0 (,) 1 ) -> A < 1 )
24	23	3ad2ant3 1051	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> A < 1 )
25	19, 20, 21, 24	ltadd2dd 8750	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + A ) < ( M + 1 ) )
26		peano2z 9680	. . . . . . . 8 |- ( M e. ZZ -> ( M + 1 ) e. ZZ )
27	26	zred 9768	. . . . . . 7 |- ( M e. ZZ -> ( M + 1 ) e. RR )
28	27	3ad2ant1 1049	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + 1 ) e. RR )
29		ltletr 8415	. . . . . 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 1278	. . . . 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 433	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( ( M + 1 ) <_ N -> ( M + A ) < N ) )
32	18, 31	sylbid 150	. . 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 129	1 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( ( M + A ) < N <-> ( M + A ) <_ N ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   RR*cxr 8359   < clt 8360   <_ cle 8361   ZZcz 9644   RR+crp 10054   (,)cioo 10290

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-rp 10055  df-ioo 10294

This theorem is used by:  halfleoddlt  12661