Theorem zltaddlt1le 10164

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 9411	. . . . . 6 |- ( M e. ZZ -> M e. RR )
2	1	adantr 276	. . . . 5 |- ( ( M e. ZZ /\ A e. ( 0 (,) 1 ) ) -> M e. RR )
3		elioore 10069	. . . . . 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 8137	. . . 4 |- ( ( M e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + A ) e. RR )
6	5	3adant2 1019	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + A ) e. RR )
7		zre 9411	. . . 4 |- ( N e. ZZ -> N e. RR )
8	7	3ad2ant2 1022	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> N e. RR )
9		ltle 8195	. . 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 10067	. . . . . 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 9850	. . . 4 |- ( A e. ( 0 (,) 1 ) -> A e. RR+ )
15		addlelt 9925	. . . 4 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( ( M + A ) <_ N -> M < N ) )
16	1, 7, 14, 15	syl3an 1292	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( ( M + A ) <_ N -> M < N ) )
17		zltp1le 9462	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) )
18	17	3adant3 1020	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M < N <-> ( M + 1 ) <_ N ) )
19	3	3ad2ant3 1023	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> A e. RR )
20		1red 8122	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> 1 e. RR )
21	1	3ad2ant1 1021	. . . . . 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 1023	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> A < 1 )
25	19, 20, 21, 24	ltadd2dd 8530	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + A ) < ( M + 1 ) )
26		peano2z 9443	. . . . . . . 8 |- ( M e. ZZ -> ( M + 1 ) e. ZZ )
27	26	zred 9530	. . . . . . 7 |- ( M e. ZZ -> ( M + 1 ) e. RR )
28	27	3ad2ant1 1021	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( M + 1 ) e. RR )
29		ltletr 8197	. . . . . 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 1250	. . . . 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 981   e. wcel 2178   class class class wbr 4059  (class class class)co 5967   RRcr 7959   0cc0 7960   1c1 7961   + caddc 7963   RR*cxr 8141   < clt 8142   <_ cle 8143   ZZcz 9407   RR+crp 9810   (,)cioo 10045

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-id 4358  df-po 4361  df-iso 4362  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-rp 9811  df-ioo 10049

This theorem is referenced by:  halfleoddlt  12320