Theorem znege1 11701

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 9012	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B <-> ( A + 1 ) <_ B ) )
2	1	3adant3 984	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) -> ( A < B <-> ( A + 1 ) <_ B ) )
3	2	biimpa 292	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> ( A + 1 ) <_ B )
4		simpl1 967	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> A e. ZZ )
5	4	zred 9077	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> A e. RR )
6		1red 7705	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> 1 e. RR )
7		simpl2 968	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> B e. ZZ )
8	7	zred 9077	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> B e. RR )
9	5, 6, 8	leaddsub2d 8227	. . . 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 7804	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> A <_ B )
13	5, 8, 12	abssuble0d 10841	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> ( abs ` ( A - B ) ) = ( B - A ) )
14	10, 13	breqtrrd 3921	. 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 969	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A = B ) -> A =/= B )
17	15, 16	pm2.21ddne 2365	. 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 968	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> B e. ZZ )
20		simpl1 967	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> A e. ZZ )
21		zltp1le 9012	. . . . . 6 |- ( ( B e. ZZ /\ A e. ZZ ) -> ( B < A <-> ( B + 1 ) <_ A ) )
22	19, 20, 21	syl2anc 406	. . . . 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 9077	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> B e. RR )
25		1red 7705	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> 1 e. RR )
26	20	zred 9077	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> A e. RR )
27	24, 25, 26	leaddsub2d 8227	. . . 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 7804	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> B <_ A )
30	24, 26, 29	abssubge0d 10840	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> ( abs ` ( A - B ) ) = ( A - B ) )
31	28, 30	breqtrrd 3921	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> 1 <_ ( abs ` ( A - B ) ) )
32		ztri3or 9001	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B \/ A = B \/ B < A ) )
33	32	3adant3 984	. 2 |- ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) -> ( A < B \/ A = B \/ B < A ) )
34	14, 17, 31, 33	mpjao3dan 1268	1 |- ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) -> 1 <_ ( abs ` ( A - B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ w3o 944   /\ w3a 945   = wceq 1314   e. wcel 1463   =/= wne 2282   class class class wbr 3895   ` cfv 5081  (class class class)co 5728   1c1 7548   + caddc 7550   < clt 7724   <_ cle 7725   - cmin 7856   ZZcz 8958   abscabs 10661

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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-2 8689  df-n0 8882  df-z 8959  df-uz 9229  df-seqfrec 10112  df-exp 10186  df-cj 10507  df-re 10508  df-im 10509  df-rsqrt 10662  df-abs 10663

This theorem is referenced by: (None)