Theorem znege1 12346

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 9380	. . . . . 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 9448	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> A e. RR )
6		1red 8041	. . . . 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 9448	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> B e. RR )
9	5, 6, 8	leaddsub2d 8574	. . . 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 8145	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> A <_ B )
13	5, 8, 12	abssuble0d 11342	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> ( abs ` ( A - B ) ) = ( B - A ) )
14	10, 13	breqtrrd 4061	. 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 9380	. . . . . 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 9448	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> B e. RR )
25		1red 8041	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> 1 e. RR )
26	20	zred 9448	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> A e. RR )
27	24, 25, 26	leaddsub2d 8574	. . . 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 8145	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> B <_ A )
30	24, 26, 29	abssubge0d 11341	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> ( abs ` ( A - B ) ) = ( A - B ) )
31	28, 30	breqtrrd 4061	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> 1 <_ ( abs ` ( A - B ) ) )
32		ztri3or 9369	. . 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 4033   ` cfv 5258  (class class class)co 5922   1c1 7880   + caddc 7882   < clt 8061   <_ cle 8062   - cmin 8197   ZZcz 9326   abscabs 11162

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 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997

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 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-n0 9250  df-z 9327  df-uz 9602  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164

This theorem is referenced by: (None)