Theorem zltaddlt1le 10389

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 9627	. . . . . 6 |- ( M e. ZZ -> M e. RR )
2	1	adantr 276	. . . . 5 |- ( ( M e. ZZ /\ A e. ( 0 (,) 1 ) ) -> M e. RR )
3		elioore 10293	. . . . . 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 8345	. . . 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 9627	. . . 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 8403	. . 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 10291	. . . . . 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 10073	. . . 4 |- ( A e. ( 0 (,) 1 ) -> A e. RR+ )
15		addlelt 10148	. . . 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 9678	. . . . 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 8331	. . . . . 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 8740	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + A ) < ( M + 1 ) )
26		peano2z 9659	. . . . . . . 8 |- ( M e. ZZ -> ( M + 1 ) e. ZZ )
27	26	zred 9747	. . . . . . 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 8405	. . . . . 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   RRcr 8168   0cc0 8169   1c1 8170   + caddc 8172   RR*cxr 8349   < clt 8350   <_ cle 8351   ZZcz 9623   RR+crp 10033   (,)cioo 10269

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 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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-rp 10034  df-ioo 10273

This theorem is referenced by:  halfleoddlt  12639