Theorem znege1 12371

Description: The absolute value of the difference between two unequal integers is at least one. (Contributed by Jim Kingdon, 31-Jan-2022.)

Assertion

Ref	Expression
znege1	|- ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) -> 1 <_ ( abs ` ( A - B ) ) )

Proof of Theorem znege1

Step	Hyp	Ref	Expression
1		zltp1le 9397	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B <-> ( A + 1 ) <_ B ) )
2	1	3adant3 1019	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) -> ( A < B <-> ( A + 1 ) <_ B ) )
3	2	biimpa 296	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> ( A + 1 ) <_ B )
4		simpl1 1002	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> A e. ZZ )
5	4	zred 9465	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> A e. RR )
6		1red 8058	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> 1 e. RR )
7		simpl2 1003	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> B e. ZZ )
8	7	zred 9465	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> B e. RR )
9	5, 6, 8	leaddsub2d 8591	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> ( ( A + 1 ) <_ B <-> 1 <_ ( B - A ) ) )
10	3, 9	mpbid 147	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> 1 <_ ( B - A ) )
11		simpr 110	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> A < B )
12	5, 8, 11	ltled 8162	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> A <_ B )
13	5, 8, 12	abssuble0d 11359	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> ( abs ` ( A - B ) ) = ( B - A ) )
14	10, 13	breqtrrd 4062	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> 1 <_ ( abs ` ( A - B ) ) )
15		simpr 110	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A = B ) -> A = B )
16		simpl3 1004	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A = B ) -> A =/= B )
17	15, 16	pm2.21ddne 2450	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A = B ) -> 1 <_ ( abs ` ( A - B ) ) )
18		simpr 110	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> B < A )
19		simpl2 1003	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> B e. ZZ )
20		simpl1 1002	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> A e. ZZ )
21		zltp1le 9397	. . . . . 6 |- ( ( B e. ZZ /\ A e. ZZ ) -> ( B < A <-> ( B + 1 ) <_ A ) )
22	19, 20, 21	syl2anc 411	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> ( B < A <-> ( B + 1 ) <_ A ) )
23	18, 22	mpbid 147	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> ( B + 1 ) <_ A )
24	19	zred 9465	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> B e. RR )
25		1red 8058	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> 1 e. RR )
26	20	zred 9465	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> A e. RR )
27	24, 25, 26	leaddsub2d 8591	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> ( ( B + 1 ) <_ A <-> 1 <_ ( A - B ) ) )
28	23, 27	mpbid 147	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> 1 <_ ( A - B ) )
29	24, 26, 18	ltled 8162	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> B <_ A )
30	24, 26, 29	abssubge0d 11358	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> ( abs ` ( A - B ) ) = ( A - B ) )
31	28, 30	breqtrrd 4062	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> 1 <_ ( abs ` ( A - B ) ) )
32		ztri3or 9386	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B \/ A = B \/ B < A ) )
33	32	3adant3 1019	. 2 |- ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) -> ( A < B \/ A = B \/ B < A ) )
34	14, 17, 31, 33	mpjao3dan 1318	1 |- ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) -> 1 <_ ( abs ` ( A - B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 979   /\ w3a 980   = wceq 1364   e. wcel 2167   =/= wne 2367   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   1c1 7897   + caddc 7899   < clt 8078   <_ cle 8079   - cmin 8214   ZZcz 9343   abscabs 11179

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-n0 9267  df-z 9344  df-uz 9619  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181

This theorem is referenced by: (None)