Theorem znege1 11495

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 8865	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B <-> ( A + 1 ) <_ B ) )
2	1	3adant3 964	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) -> ( A < B <-> ( A + 1 ) <_ B ) )
3	2	biimpa 291	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> ( A + 1 ) <_ B )
4		simpl1 947	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> A e. ZZ )
5	4	zred 8929	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> A e. RR )
6		1red 7564	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> 1 e. RR )
7		simpl2 948	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> B e. ZZ )
8	7	zred 8929	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> B e. RR )
9	5, 6, 8	leaddsub2d 8085	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> ( ( A + 1 ) <_ B <-> 1 <_ ( B - A ) ) )
10	3, 9	mpbid 146	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> 1 <_ ( B - A ) )
11		simpr 109	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> A < B )
12	5, 8, 11	ltled 7663	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> A <_ B )
13	5, 8, 12	abssuble0d 10671	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> ( abs ` ( A - B ) ) = ( B - A ) )
14	10, 13	breqtrrd 3877	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> 1 <_ ( abs ` ( A - B ) ) )
15		simpr 109	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A = B ) -> A = B )
16		simpl3 949	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A = B ) -> A =/= B )
17	15, 16	pm2.21ddne 2339	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A = B ) -> 1 <_ ( abs ` ( A - B ) ) )
18		simpr 109	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> B < A )
19		simpl2 948	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> B e. ZZ )
20		simpl1 947	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> A e. ZZ )
21		zltp1le 8865	. . . . . 6 |- ( ( B e. ZZ /\ A e. ZZ ) -> ( B < A <-> ( B + 1 ) <_ A ) )
22	19, 20, 21	syl2anc 404	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> ( B < A <-> ( B + 1 ) <_ A ) )
23	18, 22	mpbid 146	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> ( B + 1 ) <_ A )
24	19	zred 8929	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> B e. RR )
25		1red 7564	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> 1 e. RR )
26	20	zred 8929	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> A e. RR )
27	24, 25, 26	leaddsub2d 8085	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> ( ( B + 1 ) <_ A <-> 1 <_ ( A - B ) ) )
28	23, 27	mpbid 146	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> 1 <_ ( A - B ) )
29	24, 26, 18	ltled 7663	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> B <_ A )
30	24, 26, 29	abssubge0d 10670	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> ( abs ` ( A - B ) ) = ( A - B ) )
31	28, 30	breqtrrd 3877	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> 1 <_ ( abs ` ( A - B ) ) )
32		ztri3or 8854	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B \/ A = B \/ B < A ) )
33	32	3adant3 964	. 2 |- ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) -> ( A < B \/ A = B \/ B < A ) )
34	14, 17, 31, 33	mpjao3dan 1244	1 |- ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) -> 1 <_ ( abs ` ( A - B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ w3o 924   /\ w3a 925   = wceq 1290   e. wcel 1439   =/= wne 2256   class class class wbr 3851   ` cfv 5028  (class class class)co 5666   1c1 7412   + caddc 7414   < clt 7583   <_ cle 7584   - cmin 7714   ZZcz 8811   abscabs 10491

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-frec 6170  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-2 8542  df-n0 8735  df-z 8812  df-uz 9081  df-iseq 9914  df-seq3 9915  df-exp 10016  df-cj 10337  df-re 10338  df-im 10339  df-rsqrt 10492  df-abs 10493

This theorem is referenced by: (None)