Theorem zltaddlt1le 10341

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 9581	. . . . . 6 |- ( M e. ZZ -> M e. RR )
2	1	adantr 276	. . . . 5 |- ( ( M e. ZZ /\ A e. ( 0 (,) 1 ) ) -> M e. RR )
3		elioore 10245	. . . . . 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 8303	. . . 4 |- ( ( M e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + A ) e. RR )
6	5	3adant2 1043	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + A ) e. RR )
7		zre 9581	. . . 4 |- ( N e. ZZ -> N e. RR )
8	7	3ad2ant2 1046	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> N e. RR )
9		ltle 8361	. . 3 |- ( ( ( M + A ) e. RR /\ N e. RR ) -> ( ( M + A ) < N -> ( M + A ) <_ N ) )
10	6, 8, 9	syl2anc 411	. 2 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( ( M + A ) < N -> ( M + A ) <_ N ) )
11		elioo3g 10243	. . . . . 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 387	. . . . 5 |- ( A e. ( 0 (,) 1 ) -> 0 < A )
14	3, 13	elrpd 10026	. . . 4 |- ( A e. ( 0 (,) 1 ) -> A e. RR+ )
15		addlelt 10101	. . . 4 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( ( M + A ) <_ N -> M < N ) )
16	1, 7, 14, 15	syl3an 1316	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( ( M + A ) <_ N -> M < N ) )
17		zltp1le 9632	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) )
18	17	3adant3 1044	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M < N <-> ( M + 1 ) <_ N ) )
19	3	3ad2ant3 1047	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> A e. RR )
20		1red 8289	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> 1 e. RR )
21	1	3ad2ant1 1045	. . . . . 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 387	. . . . . . 7 |- ( A e. ( 0 (,) 1 ) -> A < 1 )
24	23	3ad2ant3 1047	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> A < 1 )
25	19, 20, 21, 24	ltadd2dd 8696	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + A ) < ( M + 1 ) )
26		peano2z 9613	. . . . . . . 8 |- ( M e. ZZ -> ( M + 1 ) e. ZZ )
27	26	zred 9700	. . . . . . 7 |- ( M e. ZZ -> ( M + 1 ) e. RR )
28	27	3ad2ant1 1045	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + 1 ) e. RR )
29		ltletr 8363	. . . . . 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 1274	. . . . 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 429	. . . 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   RRcr 8126   0cc0 8127   1c1 8128   + caddc 8130   RR*cxr 8307   < clt 8308   <_ cle 8309   ZZcz 9577   RR+crp 9986   (,)cioo 10221

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-rp 9987  df-ioo 10225

This theorem is referenced by:  halfleoddlt  12580