Theorem gcdneg 12274

Description: Negating one operand of the gcd operator does not alter the result. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
gcdneg	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd -u N ) = ( M gcd N ) )

Proof of Theorem gcdneg

Step	Hyp	Ref	Expression
1		oveq12 5952	. . . . 5 |- ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = ( 0 gcd 0 ) )
2	1	adantl 277	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( 0 gcd 0 ) )
3		zcn 9376	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
4	3	negeq0d 8374	. . . . . . . 8 |- ( N e. ZZ -> ( N = 0 <-> -u N = 0 ) )
5	4	anbi2d 464	. . . . . . 7 |- ( N e. ZZ -> ( ( M = 0 /\ N = 0 ) <-> ( M = 0 /\ -u N = 0 ) ) )
6	5	adantl 277	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 /\ N = 0 ) <-> ( M = 0 /\ -u N = 0 ) ) )
7		oveq12 5952	. . . . . 6 |- ( ( M = 0 /\ -u N = 0 ) -> ( M gcd -u N ) = ( 0 gcd 0 ) )
8	6, 7	biimtrdi 163	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 /\ N = 0 ) -> ( M gcd -u N ) = ( 0 gcd 0 ) ) )
9	8	imp 124	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd -u N ) = ( 0 gcd 0 ) )
10	2, 9	eqtr4d 2240	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( M gcd -u N ) )
11		gcddvds 12255	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
12		gcdcl 12258	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
13	12	nn0zd 9492	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. ZZ )
14		dvdsnegb 12090	. . . . . . . . 9 |- ( ( ( M gcd N ) e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || N <-> ( M gcd N ) || -u N ) )
15	13, 14	sylancom 420	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || N <-> ( M gcd N ) || -u N ) )
16	15	anbi2d 464	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || N ) <-> ( ( M gcd N ) || M /\ ( M gcd N ) || -u N ) ) )
17	11, 16	mpbid 147	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || -u N ) )
18	6	notbid 668	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ N = 0 ) <-> -. ( M = 0 /\ -u N = 0 ) ) )
19		simpl 109	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
20		znegcl 9402	. . . . . . . . . 10 |- ( N e. ZZ -> -u N e. ZZ )
21	20	adantl 277	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> -u N e. ZZ )
22		dvdslegcd 12256	. . . . . . . . . 10 |- ( ( ( ( M gcd N ) e. ZZ /\ M e. ZZ /\ -u N e. ZZ ) /\ -. ( M = 0 /\ -u N = 0 ) ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || -u N ) -> ( M gcd N ) <_ ( M gcd -u N ) ) )
23	22	ex 115	. . . . . . . . 9 |- ( ( ( M gcd N ) e. ZZ /\ M e. ZZ /\ -u N e. ZZ ) -> ( -. ( M = 0 /\ -u N = 0 ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || -u N ) -> ( M gcd N ) <_ ( M gcd -u N ) ) ) )
24	13, 19, 21, 23	syl3anc 1249	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ -u N = 0 ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || -u N ) -> ( M gcd N ) <_ ( M gcd -u N ) ) ) )
25	18, 24	sylbid 150	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ N = 0 ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || -u N ) -> ( M gcd N ) <_ ( M gcd -u N ) ) ) )
26	25	com12 30	. . . . . 6 |- ( -. ( M = 0 /\ N = 0 ) -> ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || -u N ) -> ( M gcd N ) <_ ( M gcd -u N ) ) ) )
27	17, 26	mpdi 43	. . . . 5 |- ( -. ( M = 0 /\ N = 0 ) -> ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) <_ ( M gcd -u N ) ) )
28	27	impcom 125	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) <_ ( M gcd -u N ) )
29		gcddvds 12255	. . . . . . . 8 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || -u N ) )
30	20, 29	sylan2 286	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || -u N ) )
31		gcdcl 12258	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M gcd -u N ) e. NN0 )
32	31	nn0zd 9492	. . . . . . . . . 10 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M gcd -u N ) e. ZZ )
33	20, 32	sylan2 286	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd -u N ) e. ZZ )
34		dvdsnegb 12090	. . . . . . . . 9 |- ( ( ( M gcd -u N ) e. ZZ /\ N e. ZZ ) -> ( ( M gcd -u N ) || N <-> ( M gcd -u N ) || -u N ) )
35	33, 34	sylancom 420	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd -u N ) || N <-> ( M gcd -u N ) || -u N ) )
36	35	anbi2d 464	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || N ) <-> ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || -u N ) ) )
37	30, 36	mpbird 167	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || N ) )
38		simpr 110	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
39		dvdslegcd 12256	. . . . . . . . 9 |- ( ( ( ( M gcd -u N ) e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || N ) -> ( M gcd -u N ) <_ ( M gcd N ) ) )
40	39	ex 115	. . . . . . . 8 |- ( ( ( M gcd -u N ) e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ N = 0 ) -> ( ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || N ) -> ( M gcd -u N ) <_ ( M gcd N ) ) ) )
41	33, 19, 38, 40	syl3anc 1249	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ N = 0 ) -> ( ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || N ) -> ( M gcd -u N ) <_ ( M gcd N ) ) ) )
42	41	com12 30	. . . . . 6 |- ( -. ( M = 0 /\ N = 0 ) -> ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || N ) -> ( M gcd -u N ) <_ ( M gcd N ) ) ) )
43	37, 42	mpdi 43	. . . . 5 |- ( -. ( M = 0 /\ N = 0 ) -> ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd -u N ) <_ ( M gcd N ) ) )
44	43	impcom 125	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd -u N ) <_ ( M gcd N ) )
45	13	zred 9494	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. RR )
46	33	zred 9494	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd -u N ) e. RR )
47	45, 46	letri3d 8187	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) = ( M gcd -u N ) <-> ( ( M gcd N ) <_ ( M gcd -u N ) /\ ( M gcd -u N ) <_ ( M gcd N ) ) ) )
48	47	adantr 276	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( M gcd N ) = ( M gcd -u N ) <-> ( ( M gcd N ) <_ ( M gcd -u N ) /\ ( M gcd -u N ) <_ ( M gcd N ) ) ) )
49	28, 44, 48	mpbir2and 946	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( M gcd -u N ) )
50		gcdmndc 12247	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
51		exmiddc 837	. . . 4 |- (DECID ( M = 0 /\ N = 0 ) -> ( ( M = 0 /\ N = 0 ) \/ -. ( M = 0 /\ N = 0 ) ) )
52	50, 51	syl 14	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 /\ N = 0 ) \/ -. ( M = 0 /\ N = 0 ) ) )
53	10, 49, 52	mpjaodan 799	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( M gcd -u N ) )
54	53	eqcomd 2210	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd -u N ) = ( M gcd N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1372   e. wcel 2175   class class class wbr 4043  (class class class)co 5943   0cc0 7924   <_ cle 8107   -ucneg 8243   ZZcz 9371   || cdvds 12069   gcd cgcd 12245

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-sup 7085  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-z 9372  df-uz 9648  df-q 9740  df-rp 9775  df-fz 10130  df-fzo 10264  df-fl 10411  df-mod 10466  df-seqfrec 10591  df-exp 10682  df-cj 11124  df-re 11125  df-im 11126  df-rsqrt 11280  df-abs 11281  df-dvds 12070  df-gcd 12246

This theorem is referenced by:  neggcd  12275  gcdabs  12280