Theorem congabseq 37541

Description: If two integers are congruent, they are either equal or separated by at least the congruence base. (Contributed by Stefan O'Rear, 4-Oct-2014.)

Assertion

Ref	Expression
congabseq	|- ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) -> ( ( abs ` ( B - C ) ) < A <-> B = C ) )

Proof of Theorem congabseq

Step	Hyp	Ref	Expression
1		zcn 11382	. . . . 5 |- ( B e. ZZ -> B e. CC )
2	1	3ad2ant2 1083	. . . 4 |- ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) -> B e. CC )
3	2	ad2antrr 762	. . 3 |- ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ ( abs ` ( B - C ) ) < A ) -> B e. CC )
4		zcn 11382	. . . . 5 |- ( C e. ZZ -> C e. CC )
5	4	3ad2ant3 1084	. . . 4 |- ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) -> C e. CC )
6	5	ad2antrr 762	. . 3 |- ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ ( abs ` ( B - C ) ) < A ) -> C e. CC )
7		zsubcl 11419	. . . . . . . . . 10 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( B - C ) e. ZZ )
8	7	3adant1 1079	. . . . . . . . 9 |- ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) -> ( B - C ) e. ZZ )
9	8	zcnd 11483	. . . . . . . 8 |- ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) -> ( B - C ) e. CC )
10	9	abscld 14175	. . . . . . 7 |- ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) -> ( abs ` ( B - C ) ) e. RR )
11	10	adantr 481	. . . . . 6 |- ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) -> ( abs ` ( B - C ) ) e. RR )
12		nnre 11027	. . . . . . . 8 |- ( A e. NN -> A e. RR )
13	12	3ad2ant1 1082	. . . . . . 7 |- ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) -> A e. RR )
14	13	adantr 481	. . . . . 6 |- ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) -> A e. RR )
15	11, 14	ltnled 10184	. . . . 5 |- ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) -> ( ( abs ` ( B - C ) ) < A <-> -. A <_ ( abs ` ( B - C ) ) ) )
16	15	biimpa 501	. . . 4 |- ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ ( abs ` ( B - C ) ) < A ) -> -. A <_ ( abs ` ( B - C ) ) )
17		nnz 11399	. . . . . . . . . 10 |- ( A e. NN -> A e. ZZ )
18	17	3ad2ant1 1082	. . . . . . . . 9 |- ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) -> A e. ZZ )
19	18	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ ( abs ` ( B - C ) ) < A ) /\ ( B - C ) =/= 0 ) -> A e. ZZ )
20	8	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ ( abs ` ( B - C ) ) < A ) /\ ( B - C ) =/= 0 ) -> ( B - C ) e. ZZ )
21		simpr 477	. . . . . . . 8 |- ( ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ ( abs ` ( B - C ) ) < A ) /\ ( B - C ) =/= 0 ) -> ( B - C ) =/= 0 )
22	19, 20, 21	3jca 1242	. . . . . . 7 |- ( ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ ( abs ` ( B - C ) ) < A ) /\ ( B - C ) =/= 0 ) -> ( A e. ZZ /\ ( B - C ) e. ZZ /\ ( B - C ) =/= 0 ) )
23		simpllr 799	. . . . . . 7 |- ( ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ ( abs ` ( B - C ) ) < A ) /\ ( B - C ) =/= 0 ) -> A || ( B - C ) )
24		dvdsleabs 15033	. . . . . . 7 |- ( ( A e. ZZ /\ ( B - C ) e. ZZ /\ ( B - C ) =/= 0 ) -> ( A || ( B - C ) -> A <_ ( abs ` ( B - C ) ) ) )
25	22, 23, 24	sylc 65	. . . . . 6 |- ( ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ ( abs ` ( B - C ) ) < A ) /\ ( B - C ) =/= 0 ) -> A <_ ( abs ` ( B - C ) ) )
26	25	ex 450	. . . . 5 |- ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ ( abs ` ( B - C ) ) < A ) -> ( ( B - C ) =/= 0 -> A <_ ( abs ` ( B - C ) ) ) )
27	26	necon1bd 2812	. . . 4 |- ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ ( abs ` ( B - C ) ) < A ) -> ( -. A <_ ( abs ` ( B - C ) ) -> ( B - C ) = 0 ) )
28	16, 27	mpd 15	. . 3 |- ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ ( abs ` ( B - C ) ) < A ) -> ( B - C ) = 0 )
29	3, 6, 28	subeq0d 10400	. 2 |- ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ ( abs ` ( B - C ) ) < A ) -> B = C )
30		oveq1 6657	. . . . . 6 |- ( B = C -> ( B - C ) = ( C - C ) )
31	30	adantl 482	. . . . 5 |- ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ B = C ) -> ( B - C ) = ( C - C ) )
32	5	ad2antrr 762	. . . . . 6 |- ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ B = C ) -> C e. CC )
33	32	subidd 10380	. . . . 5 |- ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ B = C ) -> ( C - C ) = 0 )
34	31, 33	eqtrd 2656	. . . 4 |- ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ B = C ) -> ( B - C ) = 0 )
35	34	abs00bd 14031	. . 3 |- ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ B = C ) -> ( abs ` ( B - C ) ) = 0 )
36		nngt0 11049	. . . . 5 |- ( A e. NN -> 0 < A )
37	36	3ad2ant1 1082	. . . 4 |- ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) -> 0 < A )
38	37	ad2antrr 762	. . 3 |- ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ B = C ) -> 0 < A )
39	35, 38	eqbrtrd 4675	. 2 |- ( ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) /\ B = C ) -> ( abs ` ( B - C ) ) < A )
40	29, 39	impbida 877	1 |- ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) -> ( ( abs ` ( B - C ) ) < A <-> B = C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   ZZcz 11377   abscabs 13974   || cdvds 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984

This theorem is referenced by:  acongeq  37550