Theorem znege1 12934

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 9678	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B <-> ( A + 1 ) <_ B ) )
2	1	3adant3 1048	. . . . 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 1031	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> A e. ZZ )
5	4	zred 9747	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> A e. RR )
6		1red 8331	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> 1 e. RR )
7		simpl2 1032	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> B e. ZZ )
8	7	zred 9747	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> B e. RR )
9	5, 6, 8	leaddsub2d 8865	. . . 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 8435	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> A <_ B )
13	5, 8, 12	abssuble0d 11921	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A < B ) -> ( abs ` ( A - B ) ) = ( B - A ) )
14	10, 13	breqtrrd 4153	. 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 1033	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ A = B ) -> A =/= B )
17	15, 16	pm2.21ddne 2503	. 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 1032	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> B e. ZZ )
20		simpl1 1031	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> A e. ZZ )
21		zltp1le 9678	. . . . . 6 |- ( ( B e. ZZ /\ A e. ZZ ) -> ( B < A <-> ( B + 1 ) <_ A ) )
22	19, 20, 21	syl2anc 415	. . . . 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 9747	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> B e. RR )
25		1red 8331	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> 1 e. RR )
26	20	zred 9747	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> A e. RR )
27	24, 25, 26	leaddsub2d 8865	. . . 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 8435	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> B <_ A )
30	24, 26, 29	abssubge0d 11920	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> ( abs ` ( A - B ) ) = ( A - B ) )
31	28, 30	breqtrrd 4153	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) /\ B < A ) -> 1 <_ ( abs ` ( A - B ) ) )
32		ztri3or 9666	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B \/ A = B \/ B < A ) )
33	32	3adant3 1048	. 2 |- ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) -> ( A < B \/ A = B \/ B < A ) )
34	14, 17, 31, 33	mpjao3dan 1348	1 |- ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) -> 1 <_ ( abs ` ( A - B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 1008   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   1c1 8170   + caddc 8172   < clt 8350   <_ cle 8351   - cmin 8487   ZZcz 9623   abscabs 11741

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743

This theorem is referenced by: (None)